From 7f19efb163f774494ed6592941c44ba817c1309e Mon Sep 17 00:00:00 2001 From: giovsaraceno Date: Fri, 22 Nov 2024 08:22:48 +0100 Subject: [PATCH] update news --- NEWS.md | 25 +++--- docs/articles/Introduction.html | 80 +++++++++--------- .../figure-html/unnamed-chunk-17-1.png | Bin 80664 -> 81947 bytes .../figure-html/unnamed-chunk-17-2.png | Bin 80079 -> 81377 bytes .../figure-html/unnamed-chunk-17-3.png | Bin 86438 -> 84962 bytes .../figure-html/unnamed-chunk-17-4.png | Bin 63946 -> 61342 bytes .../figure-html/unnamed-chunk-4-1.png | Bin 52120 -> 53582 bytes docs/articles/generate_rpkb.html | 4 +- docs/articles/kSample_test.html | 12 +-- .../figure-html/unnamed-chunk-4-1.png | Bin 71058 -> 70242 bytes docs/articles/wireless_clustering.html | 22 +++-- docs/news/index.html | 24 +++--- docs/pkgdown.yml | 2 +- docs/reference/select_h-1.png | Bin 89553 -> 90605 bytes docs/reference/select_h-2.png | Bin 89553 -> 90605 bytes docs/reference/select_h.html | 42 ++++----- docs/search.json | 2 +- src/QuadratiK.dll | Bin 1166848 -> 1166848 bytes vignettes/Introduction.html | 76 ++++++++--------- vignettes/TwoSample_test.html | 2 +- vignettes/generate_rpkb.html | 6 +- vignettes/kSample_test.html | 14 +-- vignettes/wireless_clustering.html | 22 +++-- 23 files changed, 173 insertions(+), 160 deletions(-) diff --git a/NEWS.md b/NEWS.md index 990a3d9..43c1119 100644 --- a/NEWS.md +++ b/NEWS.md @@ -3,30 +3,29 @@ QuadratiK 1.1.3 (Development version) ### NEW FEATURES - * the function rpkb() returns directly the generated data set and not a list + * the function rpkb() returns directly the generated data set and not a list. ### MINOR IMPROVEMENTS * packages ggplot2 and foreach not called as full packages but using - importFrom - - * nClust argument of pkbc function has no default value - - * usage of standardGeneric (see https://adv-r.hadley.nz/s4.html#s4-generics) + importFrom. + * nClust argument of pkbc function has no default value. + * usage of standardGeneric (see https://adv-r.hadley.nz/s4.html#s4-generics). ### BUG FIXES - * goodpractice sugestions + * goodpractice suggestions. + * specify that rho in pk.test() cannot be equal 1. ### DOCUMENTATION FIXES * Uniform notation: d is dimension in Eucldean space with spherical data lying on the (d-1)-dimensional sphere - * Specify dimension of input matrix x in clustering functions. - - * Improve explaination of Average Silhouette Width + * Improve explanation of Average Silhouette Width. + * Better explanation of motivation for L2 normalization in the wireless data + set application. QuadratiK 1.1.2 (2024-10-29) @@ -34,7 +33,7 @@ QuadratiK 1.1.2 (2024-10-29) ### NEW FEATURES - * Added NEWS.md file + * Added NEWS.md file. * More detailed information in the README file and documentation (the methods are more usable and clearer for a broader audience). @@ -42,7 +41,7 @@ QuadratiK 1.1.2 (2024-10-29) ### MINOR IMPROVEMENTS * Added badges for rOpenSci review status, test coverage, R CMD CHECK, - CRAN downloads + CRAN downloads. * Addition of test codes with testthat package. * In particular, testing that random sampling from the PKBD and the wrapped Cauchy distribution is equivalent. @@ -57,7 +56,7 @@ QuadratiK 1.1.2 (2024-10-29) * The help documentation for the methods defined for the classes in the package can be accessed directly (addition of roxygen tag name and - aliases) + aliases). QuadratiK 1.1.1 (2024-06-05) diff --git a/docs/articles/Introduction.html b/docs/articles/Introduction.html index 2ef07e7..8548a2a 100644 --- a/docs/articles/Introduction.html +++ b/docs/articles/Introduction.html @@ -187,9 +187,9 @@

Goodness-of-Fit Tests## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ -## Test Statistic: -1.497559 0.5626296 -## Critical Value: 1.962161 8.901682 -## H0 is rejected: FALSE FALSE +## Test Statistic: 3.298167 1.714125 +## Critical Value: 1.815007 8.901682 +## H0 is rejected: TRUE FALSE ## Selected tuning parameter h: 0.4

If needed, we can specify \(\mu\) and \(\Sigma\), otherwise the standard @@ -202,8 +202,8 @@

Goodness-of-Fit Tests## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ -## Test Statistic: 0.03172812 0.921138 -## Critical Value: 2.12372 8.901682 +## Test Statistic: -0.8648377 0.7204634 +## Critical Value: 1.803737 8.901682 ## H0 is rejected: FALSE FALSE ## Selected tuning parameter h: 0.4

Two-sample test

@@ -220,8 +220,8 @@

Goodness-of-Fit Tests## Kernel-based quadratic distance two-sample test ## U-statistic Dn Trace ## ------------------------------------------------ -## Test Statistic: 5.930108 10.54562 -## Critical Value: 0.4755583 0.8466447 +## Test Statistic: 5.668556 10.60736 +## Critical Value: 0.7071142 1.324683 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4 @@ -240,8 +240,8 @@

Goodness-of-Fit Tests## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ -## Test Statistic: 7.790354 11.66877 -## Critical Value: 0.8164556 1.22384 +## Test Statistic: 7.540686 10.45173 +## Critical Value: 0.7273207 1.008851 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4 @@ -269,13 +269,13 @@

Test for uniformity on the sphere## U-statistic: ## ## H0 is rejected: FALSE -## Statistic Un: -0.3463395 -## Critical value: 1.837585 +## Statistic Un: -0.04448323 +## Critical value: 1.814438 ## ## V-statistic: ## ## H0 is rejected: FALSE -## Statistic Vn: 16.82915 +## Statistic Vn: 17.75278 ## Critical value: 23.22949 @@ -297,23 +297,23 @@

Poisson kernel-based distributio rho <- 0.9 x <- rpkb(n = 100, mu = mu, rho = rho) head(x) -
##           [,1]        [,2]        [,3]
-## [1,] 0.9658979  0.06344496 -0.25102973
-## [2,] 0.9874521 -0.01877415  0.15679906
-## [3,] 0.9895562  0.12852790  0.06526165
-## [4,] 0.9947932  0.09516170  0.03647805
-## [5,] 0.8935936  0.11475933 -0.43395944
-## [6,] 0.9210925  0.38551171 -0.05449194
+
##           [,1]         [,2]        [,3]
+## [1,] 0.9889098  0.144026221  0.03624804
+## [2,] 0.9979113 -0.059248435  0.02574338
+## [3,] 0.9996806  0.010460509 -0.02300656
+## [4,] 0.9963903 -0.001169547  0.08488232
+## [5,] 0.9933974  0.113643034  0.01570940
+## [6,] 0.9756830 -0.079833339  0.20413065
 dens_x <- dpkb(x, mu = mu, rho = rho)
 head(dens_x)
##           [,1]
-## [1,] 0.7927658
-## [2,] 2.5703461
-## [3,] 3.0937250
-## [4,] 5.6075970
-## [5,] 0.1671203
-## [6,] 0.2550555
+## [1,] 2.915268 +## [2,] 9.367626 +## [3,] 13.903580 +## [4,] 7.135393 +## [5,] 4.670202 +## [6,] 1.212625

Clustering Algorithm for Spherical Data @@ -343,38 +343,38 @@

Clustering Algorithm for Spheri ## ## Summary: ## nClust LogLik WCSS -## [1,] 2 -624.9802 391.4818 -## [2,] 3 -323.2378 323.0939 -## [3,] 4 -311.0624 323.1053 +## [1,] 2 -657.1994 398.0559 +## [2,] 3 -392.3710 328.9308 +## [3,] 4 -380.0957 328.7125 ## ## Results for 2 clusters: ## Estimated Mixing Proportions (alpha): -## [1] 0.3038688 0.6961312 +## [1] 0.2724978 0.7275022 ## ## Clustering table: ## ## 1 2 -## 92 208 +## 85 215 ## ## ## Results for 3 clusters: ## Estimated Mixing Proportions (alpha): -## [1] 0.3165424 0.3342900 0.3491676 +## [1] 0.3252225 0.3426152 0.3321622 ## ## Clustering table: ## ## 1 2 3 -## 94 102 104 +## 94 106 100 ## ## ## Results for 4 clusters: ## Estimated Mixing Proportions (alpha): -## [1] 0.01175708 0.31663714 0.33714297 0.33446281 +## [1] 0.332925621 0.008442903 0.343746968 0.314884508 ## ## Clustering table: ## ## 1 2 3 4 -## 3 93 102 102 +## 100 3 106 91

The software includes additional graphical functions, aiding users in validating and representing the cluster results as well as enhancing the interpretability and usability of the analysis.

@@ -383,27 +383,27 @@

Clustering Algorithm for Spheri x_new <- rpkb(n = 10, mu = c(1,0,0), rho = rho) memb_mew <- predict(cluster_res, k = 3, newdata = x_new) memb_mew$Memb

-
##  [1] 2 2 3 2 2 2 2 2 2 2
+
##  [1] 3 3 3 3 3 3 3 3 1 3
 # Compute measures for evaluating the clustering results
 val_res <- pkbc_validation(cluster_res)
 val_res
## $metrics
-##             2         3         4
-## ASW 0.5014643 0.6876098 0.3228634
+##             2        3         4
+## ASW 0.4763106 0.650476 0.4810613
 ## 
 ## $IGP
 ## $IGP[[1]]
 ## NULL
 ## 
 ## $IGP[[2]]
-## [1] 1.000000 0.993865
+## [1] 1.0000000 0.9948454
 ## 
 ## $IGP[[3]]
-## [1] 1.00 1.00 0.99
+## [1] 1.0000000 1.0000000 0.9803922
 ## 
 ## $IGP[[4]]
-## [1] 1.0000000 1.0000000 0.9827586 1.0000000
+## [1] 0.9803922 1.0000000 1.0000000 1.0000000 
 # Plot method for the pkbc object:
 # - scatter plot of data points on the sphere
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diff --git a/docs/articles/kSample_test.html b/docs/articles/kSample_test.html
index 90ac41f..351883d 100644
--- a/docs/articles/kSample_test.html
+++ b/docs/articles/kSample_test.html
@@ -175,7 +175,7 @@
 

 
 h_k$h_sel

-
## [1] 1.2

+
## [1] 1.6

 
The select_h function has also generated a figure
 displaying the obtained power versus the considered \(h\), for each value of alternative \(\delta\) considered.

 
We can now perform the \(k\)-sample
@@ -188,11 +188,11 @@
 ##  Kernel-based quadratic distance k-sample test 
 ## U-statistic   Dn          Trace 
 ## ------------------------------------------------
-## Test Statistic:   3.76127     5.552056 
-## Critical Value:   1.270525    1.876836 
+## Test Statistic:   4.238404    6.834896 
+## Critical Value:   1.113956    1.797719 
 ## H0 is rejected:   TRUE        TRUE 
 ## CV method:  subsampling 
-## Selected tuning parameter h:  1.2
+## Selected tuning parameter h: 1.6

The function kb.test() returns an object of class kb.test. The show method for the kb.test object shows the computed statistics with @@ -208,8 +208,8 @@ 

## 
 ##  Kernel-based quadratic distance k-sample test 
 ##   Statistic    Value Critical_Value Reject_H0
-## 1        Dn 3.761270       1.270525      TRUE
-## 2     Trace 5.552056       1.876836      TRUE
+## 1 Dn 4.238404 1.113956 TRUE +## 2 Trace 6.834896 1.797719 TRUE 
 summary_ktest$summary_tables
## [[1]]
diff --git a/docs/articles/kSample_test_files/figure-html/unnamed-chunk-4-1.png b/docs/articles/kSample_test_files/figure-html/unnamed-chunk-4-1.png
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diff --git a/docs/articles/wireless_clustering.html b/docs/articles/wireless_clustering.html
index 1a25c87..1edf01f 100644
--- a/docs/articles/wireless_clustering.html
+++ b/docs/articles/wireless_clustering.html
@@ -112,14 +112,20 @@ 
The Wireless Indoor Localization
 Notice that, the Wi-Fi signal strength is measured in dBm, decibel
 milliwatts, which is expressed as a negative value ranging from -100 to
 0. In total, we have 500 observations for each room.

-
Given that Wi-Fi signal strength values are inherently bounded within
-a certain range, it is possible to consider the spherically transformed
-data points using \(L_2\)
-normalization. This transformation maps the data onto the surface of a
-7-dimensional sphere, ensuring that each observation has a uniform
-length. By projecting the data onto this high-dimensional sphere, we can
-take advantage of spherical geometry, and consequently perform the
-proposed clustering algorithm.

+
In many wireless applications, the relative signal strengths across
+routers are more relevant to the underlying spatial patterns and device
+positioning than the absolute magnitudes. Additionally, absolute signal
+strength can be affected by noise, device orientation or environmental
+factors. In this case, it is reasonable to consider the spherically
+transformed data points using L2 normalization. This transformation maps
+the data onto the surface of a 6-dimensional sphere, ensuring that each
+observation has a uniform length. Given that absolute signal strength is
+not critical to the research question, the spherical representation
+provides a meaningful and interpretable framework for studying the data
+set. In general, it is appropriate to consider spherically transformed
+data points when: (i) the absolute length of the measurements is
+irrelevant or too noisy; (ii) if we are more interested in the relative
+distributions (or angular relationships) between data points.

 
We perform the clustering algorithm on the wireless data
 set. We consider the \(K= 3, 4, 5\) as
 possible values for the number of clusters.

diff --git a/docs/news/index.html b/docs/news/index.html
index 89ad792..f134901 100644
--- a/docs/news/index.html
+++ b/docs/news/index.html
@@ -59,35 +59,37 @@
 
QuadratiK 1.1.3 (Development version)

 

 
NEW FEATURES

-
  • the function rpkb() returns directly the generated data set and not a list
    • 
+
    • the function rpkb() returns directly the generated data set and not a list.
      • 
 

 

 
MINOR IMPROVEMENTS

-

 

 
BUG FIXES

-
  • goodpractice sugestions
    • 
+
    • goodpractice suggestions.
      • 
+
    • specify that rho in pk.test() cannot be equal 1.
      • 
 

 

 
DOCUMENTATION FIXES

-
  • Uniform notation: d is dimension in Eucldean space with spherical data lying on the (d-1)-dimensional sphere
    • 
-
  • Specify dimension of input matrix x in clustering functions.
    • 
-
  • Improve explaination of Average Silhouette Width
    • 
+
    • Uniform notation: d is dimension in Eucldean space with spherical data lying on the (d-1)-dimensional sphere
      • 
+
    • Specify dimension of input matrix x in clustering functions.
      • 
+
    • Improve explanation of Average Silhouette Width.
      • 
+
    • Better explanation of motivation for L2 normalization in the wireless data set application.
      •

QuadratiK 1.1.2 (2024-10-29)

CRAN release: 2024-10-29

NEW FEATURES

-

  • Added NEWS.md file

    • +

    • Added NEWS.md file.

    • More detailed information in the README file and documentation (the methods are more usable and clearer for a broader audience).

MINOR IMPROVEMENTS

-
  • Added badges for rOpenSci review status, test coverage, R CMD CHECK, CRAN downloads
    • +
    • Added badges for rOpenSci review status, test coverage, R CMD CHECK, CRAN downloads.
    • Addition of test codes with testthat package.
    • In particular, testing that random sampling from the PKBD and the wrapped Cauchy distribution is equivalent.
@@ -98,7 +100,7 @@

BUG FIXES

DOCUMENTATION FIXES

-
  • The help documentation for the methods defined for the classes in the package can be accessed directly (addition of roxygen tag name and aliases)
    • +
    • The help documentation for the methods defined for the classes in the package can be accessed directly (addition of roxygen tag name and aliases).

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0.6 0.40 +#> 21 2.0 0.6 0.38 +#> 22 2.4 0.6 0.34 +#> 24 3.2 0.6 0.34 +#> 13 2.0 0.3 0.32 #> 19 1.2 0.6 0.30 -#> 17 0.4 0.6 0.18 -#> 14 2.4 0.3 0.18 -#> 10 0.8 0.3 0.16 -#> 12 1.6 0.3 0.16 -#> 13 2.0 0.3 0.12 -#> 15 2.8 0.3 0.12 -#> 16 3.2 0.3 0.12 -#> 3 1.2 0.2 0.10 -#> 6 2.4 0.2 0.10 -#> 2 0.8 0.2 0.08 -#> 8 3.2 0.2 0.08 +#> 18 0.8 0.6 0.26 +#> 15 2.8 0.3 0.22 +#> 17 0.4 0.6 0.20 +#> 12 1.6 0.3 0.20 +#> 4 1.6 0.2 0.18 +#> 16 3.2 0.3 0.18 +#> 6 2.4 0.2 0.16 +#> 9 0.4 0.3 0.12 +#> 5 2.0 0.2 0.12 +#> 14 2.4 0.3 0.12 +#> 10 0.8 0.3 0.10 +#> 11 1.2 0.3 0.10 +#> 7 2.8 0.2 0.10 #> 1 0.4 0.2 0.06 -#> 9 0.4 0.3 0.06 -#> 11 1.2 0.3 0.06 -#> 5 2.0 0.2 0.06 -#> 7 2.8 0.2 0.06 -#> 4 1.6 0.2 0.02 +#> 3 1.2 0.2 0.06 +#> 8 3.2 0.2 0.06 +#> 2 0.8 0.2 0.04 #> #> $power.plot diff --git a/docs/search.json b/docs/search.json index 3639f38..21035df 100644 --- a/docs/search.json +++ b/docs/search.json @@ -1 +1 @@ -[{"path":"/articles/generate_rpkb.html","id":"note","dir":"Articles","previous_headings":"","what":"Note","title":"Random sampling from the Poisson kernel-based density","text":"limitation rejvmf method ensure computational feasibility sampler \\(\\rho\\) approaching 1.","code":""},{"path":"/articles/generate_rpkb.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Random sampling from the Poisson kernel-based density","text":"Golzy, M. Markatou, M. (2020). Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29(4), 758-770. DOI: 10.1080/10618600.2020.1740713. Sablica, L., Hornik, K. Leydold, J. (2023). “Efficient sampling PKBD distribution”, Electronic Journal Statistics, 17(2), 2180-2209. DOI: 10.1214/23-EJS2149","code":""},{"path":"/articles/Introduction.html","id":"overview","dir":"Articles","previous_headings":"","what":"Overview","title":"Introduction to the QuadratiK Package","text":"QuadratiK package provides first implementation, R Python, comprehensive set goodness--fit tests clustering technique spherical data using kernel-based quadratic distances. primary goal QuadratiK offer flexible tools testing multivariate high-dimensional data uniformity, normality, comparing two samples. package includes several novel algorithms designed handle spherical data, often encountered fields like directional statistics, geospatial data analysis, signal processing. particular, offers functions clustering spherical data efficiently, computing density value generating random samples Poisson kernel-based density.","code":""},{"path":"/articles/Introduction.html","id":"installation","dir":"Articles","previous_headings":"Overview","what":"Installation","title":"Introduction to the QuadratiK Package","text":"can install version published CRAN QuadratiK: development version GitHub: QuadratiK package also available Python PyPI Dashboard application. Usage instruction Dashboard can found https://quadratik.readthedocs.io/en/latest/user_guide/dashboard_application_usage.html.","code":"install.packages(\"QuadratiK\") library(devtools) devtools::install_github('giovsaraceno/QuadratiK-package')"},{"path":"/articles/Introduction.html","id":"citation","dir":"Articles","previous_headings":"Overview","what":"Citation","title":"Introduction to the QuadratiK Package","text":"use package research work, please cite follows: Saraceno G, Markatou M, Mukhopadhyay R, Golzy M (2024). QuadratiK: Collection Methods Constructed using Kernel-Based Quadratic Distances. https://cran.r-project.org/package=QuadratiK associated paper: Saraceno Giovanni, Markatou Marianthi, Mukhopadhyay Raktim, Golzy Mojgan (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. arXiv preprint arXiv:2402.02290.","code":"@Manual{saraceno2024QuadratiK, title = {QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances}, author = {Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year = {2024}, note = {, , } } @misc{saraceno2024package, title={Goodness-of-Fit and Clustering of Spherical Data: the QuadratiK package in R and Python}, author={Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year={2024}, eprint={2402.02290}, archivePrefix={arXiv}, primaryClass={stat.CO}, url={} }"},{"path":[]},{"path":"/articles/Introduction.html","id":"goodness-of-fit-tests","dir":"Articles","previous_headings":"Key features and basic usage","what":"Goodness-of-Fit Tests","title":"Introduction to the QuadratiK Package","text":"software implements one, two, k-sample tests goodness fit, offering efficient mathematically sound way assess fit probability distributions. tests particularly useful large, high dimensional data sets assessment fit probability models interest. provided goodness--fit tests can performed using kb.test() function. kernel-based quadratic distance tests constructed using normal kernel depends tuning parameter \\(h\\). value \\(h\\) provided, function perform select_h() algorithm searching optimal value. details please visit relative help documentations. proposed tests perform well terms level power contiguous alternatives, heavy tailed distributions higher dimensions. Test normality test null hypothesis normality \\(H_0:F=\\mathcal{N}_d(\\mu, \\Sigma)\\) needed, can specify \\(\\mu\\) \\(\\Sigma\\), otherwise standard normal distribution considered. Two-sample test case want compare two samples \\(X \\sim F\\) \\(Y \\sim G\\) null hypothesis \\(H_0:F=G\\) vs \\(H_1:F\\=G\\). k-sample test case want compare \\(k\\) samples, \\(k>2\\), \\(H_0:F_1=F_2=\\ldots=F_k\\) vs \\(H_1:F_i\\=F_j\\) \\(\\= j\\).","code":"library(QuadratiK) ?kb.test ?select_h x <- matrix(rnorm(100), ncol = 2) # Does x come from a multivariate standard normal distribution? kb.test(x, h=0.4) ## ## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ ## Test Statistic: -1.497559 0.5626296 ## Critical Value: 1.962161 8.901682 ## H0 is rejected: FALSE FALSE ## Selected tuning parameter h: 0.4 x <- matrix(rnorm(100,4), ncol = 2) # Does x come from the specified multivariate normal distribution? kb.test(x, mu_hat = c(4,4), Sigma_hat = diag(2), h = 0.4) ## ## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ ## Test Statistic: 0.03172812 0.921138 ## Critical Value: 2.12372 8.901682 ## H0 is rejected: FALSE FALSE ## Selected tuning parameter h: 0.4 x <- matrix(rnorm(100), ncol = 2) y <- matrix(rnorm(100,mean = 5), ncol = 2) # Do x and y come from the same distribution? kb.test(x, y, h = 0.4) ## ## Kernel-based quadratic distance two-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 5.930108 10.54562 ## Critical Value: 0.4755583 0.8466447 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4 x1 <- matrix(rnorm(100), ncol = 2) x2 <- matrix(rnorm(100), ncol = 2) x3 <- matrix(rnorm(100, mean = 5), ncol = 2) y <- rep(c(1, 2, 3), each = 50) # Do x1, x2 and x3 come from the same distribution? x <- rbind(x1, x2, x3) kb.test(x, y, h = 0.4) ## ## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 7.790354 11.66877 ## Critical Value: 0.8164556 1.22384 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4"},{"path":"/articles/Introduction.html","id":"test-for-uniformity-on-the-sphere","dir":"Articles","previous_headings":"Key features and basic usage > Goodness-of-Fit Tests","what":"Test for uniformity on the sphere","title":"Introduction to the QuadratiK Package","text":"Expanded capabilities include supporting tests uniformity (d-1)-dimensional Sphere based Poisson kernel. Poisson kernel depends concentration parameter \\(\\rho\\) location vector \\(\\mu\\). details please visit help documentation pk.test() function. test null hypothesis uniformity \\((d-1)\\)-dimensional sphere \\(\\mathcal{S}^{d-1} = \\{x \\\\mathbb{R}^d : ||x||=1 \\}\\)","code":"?pk.test # Generate points on the sphere from the uniform ditribution x <- sample_hypersphere(d = 3, n_points = 100) # Does x come from the uniform distribution on the sphere? pk.test(x, rho = 0.7) ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Selected consentration parameter rho: 0.7 ## ## U-statistic: ## ## H0 is rejected: FALSE ## Statistic Un: -0.3463395 ## Critical value: 1.837585 ## ## V-statistic: ## ## H0 is rejected: FALSE ## Statistic Vn: 16.82915 ## Critical value: 23.22949"},{"path":"/articles/Introduction.html","id":"poisson-kernel-based-distribution-pkbd","dir":"Articles","previous_headings":"Key features and basic usage","what":"Poisson kernel-based distribution (PKBD)","title":"Introduction to the QuadratiK Package","text":"package offers functions computing density value generating random samples PKBD. Poisson kernel-based densities based normalized Poisson kernel defined \\((d-1)\\)-dimensional unit sphere. details please visit help documentation dpkb() rpkb() functions. Example","code":"?dpkb ?rpkb mu <- c(1,0,0) rho <- 0.9 x <- rpkb(n = 100, mu = mu, rho = rho) head(x) ## [,1] [,2] [,3] ## [1,] 0.9658979 0.06344496 -0.25102973 ## [2,] 0.9874521 -0.01877415 0.15679906 ## [3,] 0.9895562 0.12852790 0.06526165 ## [4,] 0.9947932 0.09516170 0.03647805 ## [5,] 0.8935936 0.11475933 -0.43395944 ## [6,] 0.9210925 0.38551171 -0.05449194 dens_x <- dpkb(x, mu = mu, rho = rho) head(dens_x) ## [,1] ## [1,] 0.7927658 ## [2,] 2.5703461 ## [3,] 3.0937250 ## [4,] 5.6075970 ## [5,] 0.1671203 ## [6,] 0.2550555"},{"path":"/articles/Introduction.html","id":"clustering-algorithm-for-spherical-data","dir":"Articles","previous_headings":"Key features and basic usage","what":"Clustering Algorithm for Spherical Data","title":"Introduction to the QuadratiK Package","text":"package incorporates unique clustering algorithm specifically tailored spherical data especially useful presence noise data presence non-negligible overlap clusters. algorithm leverages mixture Poisson kernel-based densities Sphere, enabling effective clustering spherical data data spherically transformed. details please visit help documentation pkbc() function. Example software includes additional graphical functions, aiding users validating representing cluster results well enhancing interpretability usability analysis.","code":"?pkbc # Generate 3 samples from the PKBD with different location directions x1 <- rpkb(n = 100, mu = c(1,0,0), rho = rho) x2 <- rpkb(n = 100, mu = c(-1,0,0), rho = rho) x3 <- rpkb(n = 100, mu = c(0,0,1), rho = rho) x <- rbind(x1, x2, x3) # Perform the clustering algorithm # Serch for 2, 3 or 4 clusters cluster_res <- pkbc(dat = x, nClust = c(2, 3, 4)) summary(cluster_res) ## Poisson Kernel-Based Clustering on the Sphere (pkbc) Results ## ------------------------------------------------------------ ## ## Summary: ## nClust LogLik WCSS ## [1,] 2 -624.9802 391.4818 ## [2,] 3 -323.2378 323.0939 ## [3,] 4 -311.0624 323.1053 ## ## Results for 2 clusters: ## Estimated Mixing Proportions (alpha): ## [1] 0.3038688 0.6961312 ## ## Clustering table: ## ## 1 2 ## 92 208 ## ## ## Results for 3 clusters: ## Estimated Mixing Proportions (alpha): ## [1] 0.3165424 0.3342900 0.3491676 ## ## Clustering table: ## ## 1 2 3 ## 94 102 104 ## ## ## Results for 4 clusters: ## Estimated Mixing Proportions (alpha): ## [1] 0.01175708 0.31663714 0.33714297 0.33446281 ## ## Clustering table: ## ## 1 2 3 4 ## 3 93 102 102 # Predict the membership of new data with respect to the clustering results x_new <- rpkb(n = 10, mu = c(1,0,0), rho = rho) memb_mew <- predict(cluster_res, k = 3, newdata = x_new) memb_mew$Memb ## [1] 2 2 3 2 2 2 2 2 2 2 # Compute measures for evaluating the clustering results val_res <- pkbc_validation(cluster_res) val_res ## $metrics ## 2 3 4 ## ASW 0.5014643 0.6876098 0.3228634 ## ## $IGP ## $IGP[[1]] ## NULL ## ## $IGP[[2]] ## [1] 1.000000 0.993865 ## ## $IGP[[3]] ## [1] 1.00 1.00 0.99 ## ## $IGP[[4]] ## [1] 1.0000000 1.0000000 0.9827586 1.0000000 # Plot method for the pkbc object: # - scatter plot of data points on the sphere # - elbow plot for helping the choice of the number of clusters plot(cluster_res)"},{"path":"/articles/Introduction.html","id":"additional-resources","dir":"Articles","previous_headings":"","what":"Additional Resources","title":"Introduction to the QuadratiK Package","text":"detailed information QuadratiK package, can explore following resources: Package Documentation CRAN – Official package documentation CRAN. GitHub Repository – GitHub repository development version, issues, community discussions. QuadratiK Package Website – dedicated website additional tutorials examples. ’re new package, recommend starting available vignettes: Two-sample test k-sample test Test uniformity Clustering sphere Generate PKBD","code":""},{"path":"/articles/Introduction.html","id":"references","dir":"Articles","previous_headings":"Additional Resources","what":"References","title":"Introduction to the QuadratiK Package","text":"information methods implemented package, refer associated research papers: Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” arXiv:2407.16374 Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: 10.5705/ss.202022.0347. Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713.","code":""},{"path":"/articles/kSample_test.html","id":"note","dir":"Articles","previous_headings":"","what":"Note","title":"k-sample test","text":"value \\(h\\) provided kb.test(), function performs function select_h automatic search optimal value \\(h\\) use. . following code shows usage, executed since obtain results. details visit help documentation select_h() function. kb.test() function, critical value can computed subsampling, bootstrap permutation algorithm. default method set subsampling since needs less computational time. details sampling algorithm see documentation kb.test() function following reference. proposed tests exhibit high power asymmetric alternatives close null hypothesis small sample size, well \\(k \\ge 3\\) sample comparison, dimension \\(d>2\\) sample sizes. details, see extensive simulation study reported following reference.","code":"k_test_h <- kb.test(x=x, y=y) help(select_h)"},{"path":"/articles/kSample_test.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"k-sample test","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.”https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":"/articles/TwoSample_test.html","id":"non-parametric-two-sample-test","dir":"Articles","previous_headings":"","what":"Non-parametric two-sample test","title":"Non-parametric Two-sample test","text":"Let \\(x_1, x_2, \\ldots, x_{n_1} \\sim F\\) \\(y_1, y_2, \\ldots, y_{n_2} \\sim G\\) random samples distributions \\(F\\) \\(G\\), respectively. test null hypothesis two samples generated unknown distribution \\(\\bar{F}\\), : \\[ H_0: F = G = \\bar{F} \\] versus alternative hypothesis two distributions different, \\[ H_1: F \\= G. \\] compute kernel-based quadratic distance (KBQD) tests \\[ \\mathrm{trace}_n = \\frac{1}{n_1(n_1-1)}\\sum_{=1}^{n_1} \\sum_{j \\=}^{n_1} K_{\\bar{F}}(\\mathbf{x}_i,\\mathbf{x}_j) + \\frac{1}{n_2(n_2-1)}\\sum_{=1}^{n_2} \\sum_{j \\=}^{n_2} K_{\\bar{F}}(\\mathbf{y}_i,\\mathbf{y}_j), \\] \\[ D_{n} = \\frac{1}{n_1(n_1-1)}\\sum_{=1}^{n_1} \\sum_{j \\=}^{n_2} K_{\\bar{F}}(\\mathbf{x}_i,\\mathbf{x}_j) - \\frac{2}{n_1 n_2}\\sum_{=1}^{n_1} \\sum_{j =1}^{n_2} K_{\\bar{F}}(\\mathbf{x}_i,\\mathbf{y}_j) + \\frac{1}{n_2(n_2-1)}\\sum_{=1}^{n_2} \\sum_{j \\=}^{n_2} K_{\\bar{F}}(\\mathbf{y}_i,\\mathbf{y}_j). \\nonumber \\] \\(K_{\\bar{F}}\\) denotes Normal kernel \\(K\\) defined \\[ K(\\mathbf{s}, \\mathbf{t}) = (2 \\pi)^{-d/2} \\left(\\det{\\mathbf{\\Sigma}_h}\\right)^{-\\frac{1}{2}} \\exp\\left\\{-\\frac{1}{2}(\\mathbf{s} - \\mathbf{t})^\\top \\mathbf{\\Sigma}_h^{-1}(\\mathbf{s} - \\mathbf{t})\\right\\}, \\] every \\(\\mathbf{s}, \\mathbf{t} \\\\mathbb{R}^d \\times \\mathbb{R}^d\\), covariance matrix \\(\\mathbf{\\Sigma}_h = h^2 \\) tuning parameter \\(h\\), centered respect \\(\\bar{F} = \\frac{n_1F + n_2G}{n_1 + n_2}\\). information centering kernel, see documentation kb.test() function. KBQD tests exhibit high power asymmetric alternatives close null hypothesis small sample size. consider example scenario. generate samples \\(x = (x_1, \\ldots,x_n)\\) standard normal distribution \\(N_d(0,I_d)\\) \\(y = (y_1, \\ldots,y_n)\\) skew-normal distribution \\(SN_d(0,I_d, \\lambda)\\), \\(d=4\\), \\(n=100\\) \\(\\lambda= (0.5,\\ldots,0.5)\\). two-sample test can performed providing two samples compared x y kb.test() function. value \\(h\\) provided, function automatically performs function select_h. can display chosen optimal value \\(h\\) together power plot obtained versus considered \\(h\\), alternatives \\(\\delta\\) select_h() function. details visit help documentation select_h() function. two-sample case, summary function provides results test list tables standard descriptive statistics variable, computed per group overall. Additionally, generates qq-plots comparing quantiles two groups variable.","code":"help(kb.test) library(sn) library(mvtnorm) library(QuadratiK) n <- 100 d <- 4 skewness_y <- 0.5 set.seed(2468) x_2 <- rmvnorm(n, mean = rep(0,d)) y_2 <- rmsn(n=n, xi=0, Omega = diag(d), alpha=rep(skewness_y,d)) set.seed(2468) two_test <- kb.test(x=x_2, y=y_2) two_test ## ## Kernel-based quadratic distance two-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 1.679763 2.312427 ## Critical Value: 1.087455 1.497868 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 1.6 two_test@h$h_sel ## [1] 1.6 two_test@h$power.plot help(select_h) summary_two <- summary(two_test) ## ## Kernel-based quadratic distance two-sample test ## Statistic Value Critical_Value Reject_H0 ## 1 Dn 1.679763 1.087455 TRUE ## 2 Trace 2.312427 1.497868 TRUE summary_two$summary_tables ## [[1]] ## Group 1 Group 2 Overall ## mean -0.005393522 0.3197861 0.1571963 ## sd 1.039119207 0.9094193 0.9875137 ## median -0.019317321 0.4448058 0.1601955 ## IQR 1.562613453 1.3612937 1.4292426 ## min -2.675477796 -1.4256211 -2.6754778 ## max 2.151784802 2.9375947 2.9375947 ## ## [[2]] ## Group 1 Group 2 Overall ## mean -0.10005083 0.1936138 0.04678149 ## sd 1.10476260 1.0556439 1.08777010 ## median -0.07955849 0.2235325 0.10130247 ## IQR 1.48816630 1.4716179 1.41498342 ## min -3.22222061 -2.4336333 -3.22222061 ## max 2.50192633 2.6879362 2.68793623 ## ## [[3]] ## Group 1 Group 2 Overall ## mean -0.006524772 0.1701261 0.08180065 ## sd 0.958942739 0.9524916 0.95742170 ## median -0.039301279 0.1887394 0.11877637 ## IQR 1.329868172 1.4657107 1.40312077 ## min -2.860006689 -2.1762183 -2.86000669 ## max 1.923763114 2.4237195 2.42371949 ## ## [[4]] ## Group 1 Group 2 Overall ## mean -0.06757686 0.2236458 0.07803449 ## sd 0.98684958 0.9862135 0.99481815 ## median -0.03258747 0.1097711 0.05517931 ## IQR 1.30933016 1.4088334 1.39890664 ## min -2.29625537 -2.1827156 -2.29625537 ## max 2.40795077 2.9929942 2.99299420"},{"path":"/articles/TwoSample_test.html","id":"select-h","dir":"Articles","previous_headings":"Non-parametric two-sample test","what":"Select h","title":"Non-parametric Two-sample test","text":"search optimal value tuning parameter \\(h\\) can performed independently test computation using select_h function. requires two samples, provided x y, considered family alternatives. code evaluated since obtain results.","code":"set.seed(2468) two_test_h <- select_h(x=x_2, y=y_2, alternative = \"skewness\")"},{"path":"/articles/TwoSample_test.html","id":"note","dir":"Articles","previous_headings":"Non-parametric two-sample test","what":"Note","title":"Non-parametric Two-sample test","text":"Notice test statistics two-sample testing coincide \\(k\\)-sample test statistics \\(k=2\\). Hence, alternatively two sample tests can performed providing two samples together x indicating membership groups argument y. See k-sample test vignette details. kb.test() function, critical value can computed subsampling, bootstrap permutation algorithm. default method set subsampling since needs less computational time. details sampling algorithm see documentation kb.test() function. details level power performance considered two-sample tests, see extensive simulation study reported following reference.","code":"x_pool <- rbind(x_2,y_2) y_memb <- rep(c(1,2),each=n) h <- two_test@h$h_sel set.seed(2468) kb.test(x=x_pool, y=y_memb, h=h) ## ## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 1.679763 2.312427 ## Critical Value: 1.087455 1.497868 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 1.6"},{"path":"/articles/TwoSample_test.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Non-parametric Two-sample test","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.”https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":"/articles/uniformity.html","id":"uniformity-test-on-the-sphere","dir":"Articles","previous_headings":"","what":"Uniformity test on the Sphere","title":"Uniformity test on the Sphere","text":"Let \\(x_1, x_2, \\ldots, x_n \\sim F\\) random sample empirical distribution function \\(\\hat{F}\\). test null hypothesis uniformity \\(d\\)-dimensional sphere, .e., \\(H_0: F = G\\) \\(G\\) uniform distribution \\(d\\)-dimensional sphere \\(\\mathcal{S}^{d-1}\\). compute U-statistic estimate sample KBQD (Kernel-Based Quadratic Distance) \\[ U_{n}=\\frac{2}{n(n-1)}\\sum_{=2}^{n}\\sum_{j=1}^{-1}K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}), \\] first test statistic given \\[T_{n}=\\frac{U_{n}}{\\sqrt{Var(U_{n})}},\\] \\[ Var(U_{n})= \\frac{2}{n(n-1)}\\left[\\frac{1+\\rho^{2}}{(1-\\rho^{2})^{d-1}}-1\\right], \\] V-statistic estimate KBQD\\[ V_{n} = \\frac{1}{n}\\sum_{=1}^{n}\\sum_{j=1}^{n}K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}), \\] \\(K_{cen}\\) denotes Poisson kernel \\(K_\\rho\\) centered respect uniform distribution \\(d\\)-dimensional sphere, \\[ K_{cen}(\\mathbf{u}, \\mathbf{v}) = K_\\rho(\\mathbf{u}, \\mathbf{v}) -1 \\] \\[ K_\\rho(\\mathbf{u}, \\mathbf{v}) = \\frac{1-\\rho^{2}}{\\left(1+\\rho^{2}-2\\rho (\\mathbf{u}\\cdot \\mathbf{v})\\right)^{d/2}}, \\] every \\(\\mathbf{u}, \\mathbf{v} \\\\mathcal{S}^{d-1} \\times \\mathcal{S}^{d-1}\\). generated \\(n=200\\) observations uniform distribution \\(S^{d-1}\\), \\(d=3\\). pk.test used testing uniformity generated sample, providing data matrix x value concentration parameter rho. pk.test() function returns object class pk.test. show function displays computed statistics corresponding critical values. test correctly reject null hypothesis uniformity. summary function pk.test output object provides results performed test, generates figure showing qq-plots uniform distribution variable table standard descriptive statistics. figure automatically generated summary function result test uniformity displays qq-plots given samples uniform distribution table standard descriptive statistics variable.","code":"library(QuadratiK) n <- 200 d <- 3 set.seed(2468) z <- matrix(rnorm(n * d), n, d) dat_sphere <- z/sqrt(rowSums(z^2)) rho <- 0.7 set.seed(2468) res_unif <- pk.test(x=dat_sphere, rho=rho) show(res_unif) ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Selected consentration parameter rho: 0.7 ## ## U-statistic: ## ## H0 is rejected: FALSE ## Statistic Un: -0.9756673 ## Critical value: 1.725683 ## ## V-statistic: ## ## H0 is rejected: FALSE ## Statistic Vn: 14.89598 ## Critical value: 23.22949 summary_unif <- summary(res_unif) ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Test_Statistics Critical_Value Reject_H0 ## 1 -0.9756673 1.725683 FALSE ## 2 14.8959834 23.229487 FALSE"},{"path":"/articles/uniformity.html","id":"multimodal-example","dir":"Articles","previous_headings":"Uniformity test on the Sphere","what":"Multimodal example","title":"Uniformity test on the Sphere","text":"Poisson kernel-based test uniformity exhibits excellent results especially case multimodal distributions. example, generate data points mixture 4 von Mises-Fisher distributions 2 dimensions. direction mean vectors distributions set (1, 0), (0, 1), (-1, 0), (0, -1), concentration parameter \\(\\kappa=5\\). now compare results pk.test function Ajne Bingham tests using sphunif package. example, Poisson kernel-based test statistics reject null hypothesis, Bingham test Ajne test obtain p-values equal 0.0532 0.3156, respectively.","code":"# Load necessary libraries library(movMF) library(sphunif) set.seed(2468) # Define the mean directions of the 4 von Mises-Fisher distributions means <- rbind( c(1, 0), c(0, 1), c(-1,0), c(0, -1) ) # Define the concentration parameter (kappa) kappa <- 5 # Generate 100 samples from a mixture of 4 von Mises-Fisher distributions samples <- matrix(rmovMF(100, theta = kappa * means), ncol=2) # Run the pk.test from the QuadratiK package to test the data pk_test_result <- pk.test(samples, rho=0.8) # Run the Bingham and Ajne tests from the sphunif package other_test_result <- unif_test(samples, type = c(\"Bingham\", \"Ajne\")) pk_test_result ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Selected consentration parameter rho: 0.8 ## ## U-statistic: ## ## H0 is rejected: TRUE ## Statistic Un: 2.692594 ## Critical value: 1.612663 ## ## V-statistic: ## ## H0 is rejected: TRUE ## Statistic Vn: 15.14426 ## Critical value: 12.8308 other_test_result ## $Bingham ## ## Bingham test of circular uniformity ## ## data: samples ## statistic = 5.8674, p-value = 0.0532 ## alternative hypothesis: scatter matrix different from constant ## ## ## $Ajne ## ## Ajne test of circular uniformity ## ## data: samples ## statistic = 0.28268, p-value = 0.3156 ## alternative hypothesis: any non-axial alternative to circular uniformity"},{"path":"/articles/uniformity.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Uniformity test on the Sphere","text":"Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi:10.5705/ss.202022.0347.","code":""},{"path":"/articles/wireless_clustering.html","id":"the-wireless-indoor-localization-data","dir":"Articles","previous_headings":"","what":"The Wireless Indoor Localization Data","title":"Clustering algorithm on the Wireless data","text":"consider Wireless Indoor Localization Data Set, publicly available UCI Machine Learning Repository’s website. data set used study performance different indoor localization algorithms. available within QuadratiK package wireless. Wireless Indoor Localization data set contains measurements Wi-Fi signal strength different indoor rooms. consists data frame 2000 rows 8 columns. first 7 variables report values Wi-Fi signal strength received 7 different Wi-Fi routers office location Pittsburgh (USA). last column indicates class labels, 1 4, indicating different rooms. Notice , Wi-Fi signal strength measured dBm, decibel milliwatts, expressed negative value ranging -100 0. total, 500 observations room. Given Wi-Fi signal strength values inherently bounded within certain range, possible consider spherically transformed data points using \\(L_2\\) normalization. transformation maps data onto surface 7-dimensional sphere, ensuring observation uniform length. projecting data onto high-dimensional sphere, can take advantage spherical geometry, consequently perform proposed clustering algorithm. perform clustering algorithm wireless data set. consider \\(K= 3, 4, 5\\) possible values number clusters. pkbc function creates object class pkbc containing clustering results value number clusters considered. guide choice number clusters, function pkbc_validation provides cluster validation measures graphical tools. Specifically, returns object InGroup Proportion (IGP), metrics, table computed evaluation measures. table includes Average Silhouette Width (ASW) , true labels provided, measures adjusted rand index (ARI), Macro-Precision Macro-Recall. clusters identified \\(k=4\\) achieve high values ARI, Macro Precision Macro Recall. brief description reported evaluation measures, corresponding references, please visit help documentation pkbc_validation function. plot method pkbc class can used display scatter plot data points Elbow plot computed within-cluster sum squares values. scatter plot, \\(d=2\\) \\(d=3\\), observations displayed unit circle unit sphere, respectively. \\(d>3\\), spherical PCA applied data set, first 3 principal components used visualizing data points sphere. generated scatter plot specified number clusters, data points colored assigned membership. Elbow plots reported metrics suggest \\(K=4\\) number clusters. accordance known ground truth. Additionally, true labels available provided plot method, scatter plot display data points colored respect true labels assigned memberships two adjacent plots. plot points using principal components also shows identified cluster follows initial labels. number clusters selected, method summary_stat can used obtain additional summary information respect clustering results. particular, function provides mean, standard deviation, median, inter-quantile range, minimum maximum computed variable, overall assigned membership.","code":"library(QuadratiK) head(wireless) ## V1 V2 V3 V4 V5 V6 V7 V8 ## 1 -64 -56 -61 -66 -71 -82 -81 1 ## 2 -68 -57 -61 -65 -71 -85 -85 1 ## 3 -63 -60 -60 -67 -76 -85 -84 1 ## 4 -61 -60 -68 -62 -77 -90 -80 1 ## 5 -63 -65 -60 -63 -77 -81 -87 1 ## 6 -64 -55 -63 -66 -76 -88 -83 1 wire <- wireless[,-8] labels <- wireless[,8] wire_norm <- wire/sqrt(rowSums(wire^2)) set.seed(2468) res_pk <- pkbc(as.matrix(wire_norm),3:5) set.seed(2468) res_validation <- pkbc_validation(res_pk, true_label = labels) res_validation$IGP ## [[1]] ## NULL ## ## [[2]] ## NULL ## ## [[3]] ## [1] 0.9860558 0.9706215 0.9429038 ## ## [[4]] ## [1] 0.9662698 0.9733607 0.9526627 0.9880240 ## ## [[5]] ## [1] 0.9713701 0.7727273 0.9880240 0.9639831 0.9433498 round(res_validation$metrics, 5) ## 3 4 5 ## ASW 0.35326 0.38031 0.30240 ## ARI 0.69526 0.94031 0.91409 ## Macro_Precision 0.18389 0.97719 0.00120 ## Macro_Recall 0.26000 0.97700 0.00150 help(pkbc_validation) plot(res_pk, k = 4) plot(res_pk, k=4, true_label = labels) summary_clust <- stats_clusters(res_pk,4) summary_clust ## [[1]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.33976246 -0.23423419 -0.29470203 -0.34377323 -0.30363137 ## sd 0.01283014 0.05324085 0.01581506 0.01298448 0.05251967 ## median -0.33952107 -0.24604104 -0.29625176 -0.34242063 -0.31847692 ## IQR 0.01674475 0.03014941 0.02182651 0.01835030 0.06412536 ## min -0.37455424 -0.30643954 -0.34994496 -0.39357081 -0.39357081 ## max -0.29339739 -0.06308050 -0.23847076 -0.31226867 -0.06308050 ## ## [[2]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.30636324 -0.35918738 -0.32636022 -0.31540450 -0.32655905 ## sd 0.01392354 0.02052666 0.01842337 0.01608549 0.02635941 ## median -0.30600005 -0.35809184 -0.32487137 -0.31538238 -0.32221067 ## IQR 0.01958541 0.02460137 0.02426854 0.02263003 0.03624468 ## min -0.35600342 -0.45578394 -0.40026324 -0.36631517 -0.45578394 ## max -0.26903743 -0.30605235 -0.27775661 -0.27586207 -0.26903743 ## ## [[3]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.32905221 -0.35786158 -0.31417147 -0.28929074 -0.32231793 ## sd 0.01563476 0.02459888 0.01710971 0.02094437 0.03163402 ## median -0.32915518 -0.35575396 -0.31237754 -0.29072716 -0.32209343 ## IQR 0.01957377 0.03437142 0.02422092 0.02851547 0.03973413 ## min -0.38248214 -0.43623816 -0.38676345 -0.33639128 -0.43623816 ## max -0.28256746 -0.28237524 -0.26460827 -0.23485570 -0.23485570 ## ## [[4]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.34918727 -0.24111635 -0.30079691 -0.35013402 -0.31082318 ## sd 0.01475930 0.04839750 0.01973759 0.01679759 0.05251032 ## median -0.34858664 -0.24898566 -0.30107248 -0.35004522 -0.32519736 ## IQR 0.01898212 0.03356955 0.02672633 0.02286452 0.06907237 ## min -0.40129910 -0.31690470 -0.35542814 -0.41273961 -0.41273961 ## max -0.30779911 -0.07399736 -0.23119131 -0.30906423 -0.07399736 ## ## [[5]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.38226045 -0.43319403 -0.37583631 -0.28257085 -0.36807675 ## sd 0.01770626 0.02787579 0.01814674 0.02110273 0.05820777 ## median -0.37991233 -0.43009295 -0.37549563 -0.28337199 -0.37738801 ## IQR 0.02531565 0.04108696 0.02541214 0.02710382 0.06966093 ## min -0.45385132 -0.52580164 -0.42808192 -0.34353824 -0.52580164 ## max -0.34027255 -0.36916034 -0.33333333 -0.21464345 -0.21464345 ## ## [[6]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.45125600 -0.46327257 -0.48366352 -0.49713921 -0.47391075 ## sd 0.01446455 0.02195263 0.01814601 0.01412842 0.02488735 ## median -0.45081811 -0.46519225 -0.48319020 -0.49649227 -0.47411000 ## IQR 0.01833254 0.02770315 0.02503768 0.01895624 0.03825696 ## min -0.51475369 -0.53938031 -0.53130759 -0.53840040 -0.53938031 ## max -0.41652096 -0.40313011 -0.44107352 -0.45551482 -0.40313011 ## ## [[7]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.45728346 -0.46863418 -0.48984740 -0.49701329 -0.47827795 ## sd 0.01710219 0.02396734 0.01993870 0.01587742 0.02515570 ## median -0.45703017 -0.46920446 -0.49013407 -0.49711582 -0.47914650 ## IQR 0.02311965 0.03050579 0.02784124 0.02030659 0.03643335 ## min -0.50728615 -0.56835661 -0.54571977 -0.53648350 -0.56835661 ## max -0.41326597 -0.38995633 -0.43815927 -0.44423198 -0.38995633"},{"path":"/articles/wireless_clustering.html","id":"note","dir":"Articles","previous_headings":"","what":"Note","title":"Clustering algorithm on the Wireless data","text":"number cluster k provided plot function, one scatter plot displayed possible number clusters available object class pkbc.","code":""},{"path":"/articles/wireless_clustering.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Clustering algorithm on the Wireless data","text":"Golzy, M. Markatou, M. (2020). “Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling,” Journal Computational Graphical Statistics, 29(4), 758-770. DOI: 10.1080/10618600.2020.1740713.","code":""},{"path":"/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Giovanni Saraceno. Author, maintainer. ORCID 000-0002-1753-2367 Marianthi Markatou. Author. Raktim Mukhopadhyay. Author. Mojgan Golzy. Author. Hingee Kassel. Reviewer.","code":""},{"path":"/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Saraceno G, Markatou M, Mukhopadhyay R, Golzy M (2024). QuadratiK: Collection Methods Constructed using Kernel-Based Quadratic Distances. Version 1.1.2, https://github.com/giovsaraceno/QuadratiK-package. Markatou M, Saraceno G (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” arXiv preprint. https://doi.org/10.48550/arXiv.2407.16374.","code":"@Manual{, title = {QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances}, author = {Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year = {2024}, note = {Version 1.1.2}, url = {https://github.com/giovsaraceno/QuadratiK-package}, } @Article{, title = {A Unified Framework for Multivariate Two- and k-Sample Kernel-based Quadratic Distance Goodness-of-Fit Tests}, author = {Marianthi Markatou and Giovanni Saraceno}, journal = {arXiv preprint}, year = {2024}, note = {https://doi.org/10.48550/arXiv.2407.16374}, }"},{"path":[]},{"path":"/CODE_OF_CONDUCT.html","id":"our-pledge","dir":"","previous_headings":"","what":"Our Pledge","title":"Contributor Covenant Code of Conduct","text":"interest fostering open welcoming environment, contributors maintainers pledge make participation project community harassment-free experience everyone, regardless age, body size, disability, ethnicity, sex characteristics, gender identity expression, level experience, education, socio-economic status, nationality, personal appearance, race, religion, sexual identity orientation.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"our-standards","dir":"","previous_headings":"","what":"Our Standards","title":"Contributor Covenant Code of Conduct","text":"Examples behavior contributes creating positive environment include: Using welcoming inclusive language respectful differing viewpoints experiences Gracefully accepting constructive criticism Focusing best community Showing empathy towards community members Examples unacceptable behavior participants include: use sexualized language imagery unwelcome sexual attention advances Trolling, insulting/derogatory comments, personal political attacks Public private harassment Publishing others’ private information, physical electronic address, without explicit permission conduct reasonably considered inappropriate professional setting","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"our-responsibilities","dir":"","previous_headings":"","what":"Our Responsibilities","title":"Contributor Covenant Code of Conduct","text":"Project maintainers responsible clarifying standards acceptable behavior expected take appropriate fair corrective action response instances unacceptable behavior. Project maintainers right responsibility remove, edit, reject comments, commits, code, wiki edits, issues, contributions aligned Code Conduct, ban temporarily permanently contributor behaviors deem inappropriate, threatening, offensive, harmful.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"scope","dir":"","previous_headings":"","what":"Scope","title":"Contributor Covenant Code of Conduct","text":"Code Conduct applies within project spaces, also applies individual representing project community public spaces. Examples representing project community include using official project e-mail address, posting via official social media account, acting appointed representative online offline event. Representation project may defined clarified project maintainers.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"enforcement","dir":"","previous_headings":"","what":"Enforcement","title":"Contributor Covenant Code of Conduct","text":"Instances abusive, harassing, otherwise unacceptable behavior may reported contacting project team gsaracen buffalo.edu. complaints reviewed investigated result response deemed necessary appropriate circumstances. project team obligated maintain confidentiality regard reporter incident. details specific enforcement policies may posted separately. Project maintainers follow enforce Code Conduct good faith may face temporary permanent repercussions determined members project’s leadership.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"attribution","dir":"","previous_headings":"","what":"Attribution","title":"Contributor Covenant Code of Conduct","text":"Code Conduct adapted Contributor Covenant, version 1.4, available https://www.contributor-covenant.org/version/1/4/code--conduct.html answers common questions code conduct, see https://www.contributor-covenant.org/faq","code":""},{"path":"/CONTRIBUTING.html","id":null,"dir":"","previous_headings":"","what":"Contributing to QuadratiK package","title":"Contributing to QuadratiK package","text":"Version: 1.1.2 First , thanks considering contributing QuadratiK! 👍 QuadratiK comprehensive statistical analysis package providing comprehensive set goodness--fit tests clustering technique spherical data using kernel-based quadratic distances.","code":""},{"path":"/CONTRIBUTING.html","id":"code-of-conduct","dir":"","previous_headings":"","what":"Code of conduct","title":"Contributing to QuadratiK package","text":"Please note project released Contributor Code Conduct. participating project agree abide terms.","code":""},{"path":"/CONTRIBUTING.html","id":"share-the-love-️","dir":"","previous_headings":"Code of conduct","what":"Share the love ❤️","title":"Contributing to QuadratiK package","text":"Think QuadratiK useful? Let others discover , telling person, via Twitter, ResearchGate blog post. Using QuadratiK paper writing? Cite .","code":""},{"path":"/CONTRIBUTING.html","id":"ask-a-question","dir":"","previous_headings":"Code of conduct","what":"Ask a question","title":"Contributing to QuadratiK package","text":"Using QuadratiK got stuck? Browse documentation see can find solution. Post questions requests issue GitHub. ’ll try best address , questions often lead better documentation discovery bugs. Want ask question private? Contact package maintainer gsaracen@buffalo.edu.","code":""},{"path":"/CONTRIBUTING.html","id":"propose-an-idea-","dir":"","previous_headings":"Code of conduct","what":"Propose an idea 💡","title":"Contributing to QuadratiK package","text":"idea new QuadratiK feature? Take look documentation issue list see isn’t included suggested yet. , suggest idea issue GitHub. Please Explain detail work. Keep scope narrow possible. Functions described comments near code translated documentation using roxygen2. Write documentation using roxygen2 package. Test changes goodpractice submitting. try follow GitHub flow development. Fork repo clone computer. learn process, see guide. forked cloned project since worked , pull changes original repo clone using git pull upstream master. Open RStudio project file (.Rproj). Write code. Test code (bonus points adding unit tests). Document code (see function documentation ). Check code devtools::check() aim 0 errors warnings. Commit push changes. Submit pull request.","code":""},{"path":"/CONTRIBUTING.html","id":"report-a-bug","dir":"","previous_headings":"Code of conduct","what":"Report a bug","title":"Contributing to QuadratiK package","text":"Using QuadratiK discovered bug? ’s annoying! Don’t let others experience report issue GitHub can fix . good bug report makes easier us , please include: operating system name version (e.g. Mac OS 10.13.6). details local setup might helpful troubleshooting. Detailed steps reproduce bug.","code":""},{"path":"/CONTRIBUTING.html","id":"the-website","dir":"","previous_headings":"Code of conduct > Report a bug","what":"The website","title":"Contributing to QuadratiK package","text":"website generated pkgdown. means don’t write html: content pulled together documentation code, vignettes, Markdown files, package DESCRIPTION _pkgdown.yml settings. know way around pkgdown, can propose file change improve documentation. , report issue can point right direction.","code":""},{"path":"/CONTRIBUTING.html","id":"workflow","dir":"","previous_headings":"","what":"Workflow","title":"Contributing to QuadratiK package","text":"QuadratiK package engineered combination R C++ code balance usability high-speed computation, catering complex data analysis needs. package undergone extensive testing various examples. Continuous integration practices employed maintain high standards reliability functionality. QuadratiK first released CRAN February 23, 2024. Regular updates bug fixes planned continually enhance package’s functionality user experience. actively planning include additional methods based kernel-based quadratic-distance. One primary goals make QuadratiK increasingly user-friendly, improvements user experience layout outputs. User feedback highly valued key driver future development. Life Cycle Statement subject periodic review updated reflect evolving nature QuadratiK.","code":""},{"path":"/index.html","id":"collection-of-methods-constructed-using-the-kernel-based-quadratic-distances","dir":"","previous_headings":"","what":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"QuadratiK provides first implementation, R Python, comprehensive set goodness--fit tests clustering technique d-dimensional spherical data d ≥ 2 using kernel-based quadratic distances. includes: Goodness--Fit Tests: software implements one, two, k-sample tests goodness fit, offering efficient mathematically sound way assess fit probability distributions. tests particularly useful large, high dimensional data sets assessment fit probability models interest. Specifically, offer tests multivariate normality, well two- k-sample tests, testing equality two distributions interest, H0 : F1 = F2 H0 : F1 = … = Fk respectively. proposed tests perform well terms level power contiguous alternatives, heavy tailed distributions higher dimensions. Expanded capabilities include supporting tests uniformity d-dimensional Sphere based Poisson kernel, exhibiting excellent results especially case multimodal distributions. Poisson kernel-based distribution (PKBD): package offers functions computing density value generating random samples PKBD. Poisson kernel-based densities based normalized Poisson kernel defined d-dimensional unit sphere. Given vector μ ∈ 𝒮d − 1, parameter ρ 0 < ρ < 1, probability density function d-variate Poisson kernel-based density defined : $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d},$$ μ vector orienting center distribution, ρ parameter control concentration distribution around vector μ related variance distribution. Furthermore, ωd = 2πd/2[Γ(d/2)]−1 surface area unit sphere ℝd (see Golzy Markatou, 2020). Clustering Algorithm Spherical Data: package incorporates unique clustering algorithm specifically tailored d-dimensional spherical data especially useful presence noise data presence non-negligible overlap clusters. algorithm leverages mixture Poisson kernel-based densities d-dimensional Sphere, enabling effective clustering spherical data data spherically transformed. Additional Features: Alongside functionalities, software includes additional graphical functions, aiding users validating representing cluster results well enhancing interpretability usability analysis. introduction usage QuadratiK see vignette Introduction QuadratiK Package.","code":""},{"path":"/index.html","id":"installation","dir":"","previous_headings":"","what":"Installation","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"can install version published CRAN QuadratiK development version GitHub QuadratiK package also available Python PyPI https://pypi.org/project/QuadratiK/ also Dashboard application. Usage instruction Dashboard can found https://quadratik.readthedocs.io/en/latest/user_guide/dashboard_application_usage.html.","code":"install.packages(\"QuadratiK\") library(devtools) install_github('giovsaraceno/QuadratiK-package')"},{"path":"/index.html","id":"authors","dir":"","previous_headings":"","what":"Authors","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"Giovanni Saraceno, Marianthi Markatou, Raktim Mukhopadhyay, Mojgan Golzy Maintainer: Giovanni Saraceno ","code":""},{"path":"/index.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"use package research work, please cite follows: Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). QuadratiK: Collection Methods Constructed using Kernel-Based Quadratic Distances. https://cran.r-project.org/package=QuadratiK, https://github.com/giovsaraceno/QuadratiK-package, https://giovsaraceno.github.io/QuadratiK-package/. associated paper: Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. arXiv preprint arXiv:2402.02290v2.","code":"@Manual{saraceno2024QuadratiK, title = {QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances}, author = {Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year = {2024}, note = {, , }, } @misc{saraceno2024package, title={Goodness-of-Fit and Clustering of Spherical Data: the QuadratiK package in R and Python}, author={Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year={2024}, eprint={2402.02290}, archivePrefix={arXiv}, primaryClass={stat.CO}, url={https://arxiv.org/abs/2402.02290} }"},{"path":"/index.html","id":"references","dir":"","previous_headings":"","what":"References","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: 10.5705/ss.202022.0347. Golzy, M. & Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713. Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” arXiv:2407.16374","code":""},{"path":"/index.html","id":"details","dir":"","previous_headings":"","what":"Details","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"work supported Kaleida Health Foundation National Science Foundation.","code":""},{"path":"/LICENSE.html","id":null,"dir":"","previous_headings":"","what":"GNU General Public License","title":"GNU General Public License","text":"Version 3, 29 June 2007Copyright © 2007 Free Software Foundation, Inc.  Everyone permitted copy distribute verbatim copies license document, changing allowed.","code":""},{"path":"/LICENSE.html","id":"preamble","dir":"","previous_headings":"","what":"Preamble","title":"GNU General Public License","text":"GNU General Public License free, copyleft license software kinds works. licenses software practical works designed take away freedom share change works. contrast, GNU General Public License intended guarantee freedom share change versions program–make sure remains free software users. , Free Software Foundation, use GNU General Public License software; applies also work released way authors. can apply programs, . speak free software, referring freedom, price. 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Nothing License shall construed excluding limiting implied license defenses infringement may otherwise available applicable patent law.","code":""},{"path":"/LICENSE.html","id":"id_12-no-surrender-of-others-freedom","dir":"","previous_headings":"TERMS AND CONDITIONS","what":"12. No Surrender of Others’ Freedom","title":"GNU General Public License","text":"conditions imposed (whether court order, agreement otherwise) contradict conditions License, excuse conditions License. convey covered work satisfy simultaneously obligations License pertinent obligations, consequence may convey . example, agree terms obligate collect royalty conveying convey Program, way satisfy terms License refrain entirely conveying Program.","code":""},{"path":"/LICENSE.html","id":"id_13-use-with-the-gnu-affero-general-public-license","dir":"","previous_headings":"TERMS AND CONDITIONS","what":"13. Use with the GNU Affero General Public License","title":"GNU General Public License","text":"Notwithstanding provision License, permission link combine covered work work licensed version 3 GNU Affero General Public License single combined work, convey resulting work. terms License continue apply part covered work, special requirements GNU Affero General Public License, section 13, concerning interaction network apply combination .","code":""},{"path":"/LICENSE.html","id":"id_14-revised-versions-of-this-license","dir":"","previous_headings":"TERMS AND CONDITIONS","what":"14. Revised Versions of this License","title":"GNU General Public License","text":"Free Software Foundation may publish revised /new versions GNU General Public License time time. new versions similar spirit present version, may differ detail address new problems concerns. version given distinguishing version number. Program specifies certain numbered version GNU General Public License “later version” applies , option following terms conditions either numbered version later version published Free Software Foundation. Program specify version number GNU General Public License, may choose version ever published Free Software Foundation. Program specifies proxy can decide future versions GNU General Public License can used, proxy’s public statement acceptance version permanently authorizes choose version Program. Later license versions may give additional different permissions. However, additional obligations imposed author copyright holder result choosing follow later version.","code":""},{"path":"/LICENSE.html","id":"id_15-disclaimer-of-warranty","dir":"","previous_headings":"TERMS AND CONDITIONS","what":"15. Disclaimer of Warranty","title":"GNU General Public License","text":"WARRANTY PROGRAM, EXTENT PERMITTED APPLICABLE LAW. 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END TERMS CONDITIONS","code":""},{"path":"/LICENSE.html","id":"how-to-apply-these-terms-to-your-new-programs","dir":"","previous_headings":"","what":"How to Apply These Terms to Your New Programs","title":"GNU General Public License","text":"develop new program, want greatest possible use public, best way achieve make free software everyone can redistribute change terms. , attach following notices program. safest attach start source file effectively state exclusion warranty; file least “copyright” line pointer full notice found. Also add information contact electronic paper mail. program terminal interaction, make output short notice like starts interactive mode: hypothetical commands show w show c show appropriate parts General Public License. course, program’s commands might different; GUI interface, use “box”. also get employer (work programmer) school, , sign “copyright disclaimer” program, necessary. information , apply follow GNU GPL, see . GNU General Public License permit incorporating program proprietary programs. program subroutine library, may consider useful permit linking proprietary applications library. want , use GNU Lesser General Public License instead License. first, please read .","code":" Copyright (C) This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . Copyright (C) This program comes with ABSOLUTELY NO WARRANTY; for details type 'show w'. This is free software, and you are welcome to redistribute it under certain conditions; type 'show c' for details."},{"path":"/reference/breast_cancer.html","id":null,"dir":"Reference","previous_headings":"","what":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"breast_cancer Wisconsin data 569 rows 31 columns. first 30 variables report features computed digitized image fine needle aspirate (FNA) breast mass. describe characteristics cell nuclei present image. last column indicates class labels (Benign = 0 Malignant = 1).","code":""},{"path":"/reference/breast_cancer.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"","code":"breast_cancer"},{"path":"/reference/breast_cancer.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"data frame 569 observations 31 variables.","code":""},{"path":"/reference/breast_cancer.html","id":"source","dir":"Reference","previous_headings":"","what":"Source","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"Wolberg, W., Mangasarian, O., Street, N., & Street, W. (1993). Breast Cancer Wisconsin (Diagnostic). UCI Machine Learning Repository. https://doi.org/10.24432/C5DW2B.","code":""},{"path":"/reference/breast_cancer.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"Street, W. N., Wolberg, W. H., & Mangasarian, O. L. (1993, July). Nuclear feature extraction breast tumor diagnosis. Biomedical image processing biomedical visualization (Vol. 1905, pp. 861-870). SPIE.","code":""},{"path":"/reference/breast_cancer.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"","code":"data(breast_cancer) summary(breast_cancer) #> radius1 texture1 perimeter1 area1 #> Min. : 6.981 Min. : 9.71 Min. : 43.79 Min. : 143.5 #> 1st Qu.:11.700 1st Qu.:16.17 1st Qu.: 75.17 1st Qu.: 420.3 #> Median :13.370 Median :18.84 Median : 86.24 Median : 551.1 #> Mean :14.127 Mean :19.29 Mean : 91.97 Mean : 654.9 #> 3rd Qu.:15.780 3rd Qu.:21.80 3rd Qu.:104.10 3rd Qu.: 782.7 #> Max. :28.110 Max. :39.28 Max. :188.50 Max. :2501.0 #> smoothness1 compactness1 concavity1 concave_points1 #> Min. :0.05263 Min. :0.01938 Min. :0.00000 Min. :0.00000 #> 1st Qu.:0.08637 1st Qu.:0.06492 1st Qu.:0.02956 1st Qu.:0.02031 #> Median :0.09587 Median :0.09263 Median :0.06154 Median :0.03350 #> Mean :0.09636 Mean :0.10434 Mean :0.08880 Mean :0.04892 #> 3rd Qu.:0.10530 3rd Qu.:0.13040 3rd Qu.:0.13070 3rd Qu.:0.07400 #> Max. :0.16340 Max. :0.34540 Max. :0.42680 Max. :0.20120 #> symmetry1 fractal_dimension1 radius2 texture2 #> Min. :0.1060 Min. :0.04996 Min. :0.1115 Min. :0.3602 #> 1st Qu.:0.1619 1st Qu.:0.05770 1st Qu.:0.2324 1st Qu.:0.8339 #> Median :0.1792 Median :0.06154 Median :0.3242 Median :1.1080 #> Mean :0.1812 Mean :0.06280 Mean :0.4052 Mean :1.2169 #> 3rd Qu.:0.1957 3rd Qu.:0.06612 3rd Qu.:0.4789 3rd Qu.:1.4740 #> Max. :0.3040 Max. :0.09744 Max. :2.8730 Max. :4.8850 #> perimeter2 area2 smoothness2 compactness2 #> Min. : 0.757 Min. : 6.802 Min. :0.001713 Min. :0.002252 #> 1st Qu.: 1.606 1st Qu.: 17.850 1st Qu.:0.005169 1st Qu.:0.013080 #> Median : 2.287 Median : 24.530 Median :0.006380 Median :0.020450 #> Mean : 2.866 Mean : 40.337 Mean :0.007041 Mean :0.025478 #> 3rd Qu.: 3.357 3rd Qu.: 45.190 3rd Qu.:0.008146 3rd Qu.:0.032450 #> Max. :21.980 Max. :542.200 Max. :0.031130 Max. :0.135400 #> concavity2 concave_points2 symmetry2 fractal_dimension2 #> Min. :0.00000 Min. :0.000000 Min. :0.007882 Min. :0.0008948 #> 1st Qu.:0.01509 1st Qu.:0.007638 1st Qu.:0.015160 1st Qu.:0.0022480 #> Median :0.02589 Median :0.010930 Median :0.018730 Median :0.0031870 #> Mean :0.03189 Mean :0.011796 Mean :0.020542 Mean :0.0037949 #> 3rd Qu.:0.04205 3rd Qu.:0.014710 3rd Qu.:0.023480 3rd Qu.:0.0045580 #> Max. :0.39600 Max. :0.052790 Max. :0.078950 Max. :0.0298400 #> radius3 texture3 perimeter3 area3 #> Min. : 7.93 Min. :12.02 Min. : 50.41 Min. : 185.2 #> 1st Qu.:13.01 1st Qu.:21.08 1st Qu.: 84.11 1st Qu.: 515.3 #> Median :14.97 Median :25.41 Median : 97.66 Median : 686.5 #> Mean :16.27 Mean :25.68 Mean :107.26 Mean : 880.6 #> 3rd Qu.:18.79 3rd Qu.:29.72 3rd Qu.:125.40 3rd Qu.:1084.0 #> Max. :36.04 Max. :49.54 Max. :251.20 Max. :4254.0 #> smoothness3 compactness3 concavity3 concave_points3 #> Min. :0.07117 Min. :0.02729 Min. :0.0000 Min. :0.00000 #> 1st Qu.:0.11660 1st Qu.:0.14720 1st Qu.:0.1145 1st Qu.:0.06493 #> Median :0.13130 Median :0.21190 Median :0.2267 Median :0.09993 #> Mean :0.13237 Mean :0.25427 Mean :0.2722 Mean :0.11461 #> 3rd Qu.:0.14600 3rd Qu.:0.33910 3rd Qu.:0.3829 3rd Qu.:0.16140 #> Max. :0.22260 Max. :1.05800 Max. :1.2520 Max. :0.29100 #> symmetry3 fractal_dimension3 y #> Min. :0.1565 Min. :0.05504 Length:569 #> 1st Qu.:0.2504 1st Qu.:0.07146 Class :character #> Median :0.2822 Median :0.08004 Mode :character #> Mean :0.2901 Mean :0.08395 #> 3rd Qu.:0.3179 3rd Qu.:0.09208 #> Max. :0.6638 Max. :0.20750"},{"path":"/reference/compare_qq.html","id":null,"dir":"Reference","previous_headings":"","what":"QQ-plot of given two samples using ggplot2 — compare_qq","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"QQ-plot given two samples using ggplot2","code":""},{"path":"/reference/compare_qq.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"","code":"compare_qq(sample1, sample2, main_title)"},{"path":"/reference/compare_qq.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"sample1 matrix observations sample 1 sample2 matrix observations sample 2 main_title title generated plot","code":""},{"path":"/reference/compare_qq.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"QQ-plot given samples","code":""},{"path":"/reference/compute_CV.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for two-sample KBQD tests — compute_CV","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"function computes critical value two-sample kernel tests centered Gaussian kernel using one three methods: bootstrap, permutation, subsampling.","code":""},{"path":"/reference/compute_CV.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"","code":"compute_CV( B, Quantile, data_pool, size_x, size_y, h, method, b = 1, compute_variance )"},{"path":"/reference/compute_CV.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"B number bootstrap/permutation/subsampling samples generate. Quantile quantile bootstrap/permutation/subsampling distribution use critical value. data_pool matrix containing data used test. size_x number rows data_pool matrix corresponding group X. size_y number rows data_pool matrix corresponding group Y. h tuning parameter kernel test. method method use computing critical value (one \"bootstrap\", \"permutation\", \"subsampling\"). b subsampling block size (used method \"subsampling\"). compute_variance indicates nonparametric variance computed. Default TRUE.","code":""},{"path":"/reference/compute_CV.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"critical value specified method significance level.","code":""},{"path":"/reference/compute_CV.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"Markatou Marianthi & Saraceno Giovanni (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":"/reference/compute_stats.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute and display some descriptive statistics for the two sample tests — compute_stats","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"Compute display descriptive statistics two sample tests","code":""},{"path":"/reference/compute_stats.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"","code":"compute_stats(var1, var2, var_name, eps = 3)"},{"path":"/reference/compute_stats.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"var1 vector observations given variable sample 1 var2 vector observations given variable sample 2 var_name Name variable displayed eps precision displayed statistics","code":""},{"path":"/reference/compute_stats.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"Computed statistics plot","code":""},{"path":"/reference/cv_ksample.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for the KBQD k-sample tests — cv_ksample","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"function computes empirical critical value k-sample KBQD tests using centered Gaussian kernel, bootstrap, permutation, subsampling.","code":""},{"path":"/reference/cv_ksample.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"","code":"cv_ksample( x, y, h, B = 150, b = 0.9, Quantile = 0.95, method = \"subsampling\", compute_variance = TRUE )"},{"path":"/reference/cv_ksample.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"x matrix containing observations used k-sample test y vector indicating sample observation h tuning parameter test using Gaussian kernel B number bootstrap/permutation/subsampling samples generate b subsampling block size (used method \"subsampling\") Quantile quantile bootstrap/permutation/subsampling distribution use critical value method method use computing critical value (one \"bootstrap\", \"permutation\") compute_variance indicates nonparametric variance computed. Default TRUE.","code":""},{"path":"/reference/cv_ksample.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"vector two critical values corresponding different formulation k-sample test statistics.","code":""},{"path":"/reference/DOF.html","id":null,"dir":"Reference","previous_headings":"","what":"Degrees of freedom (DOF) for the Poisson kernel — DOF","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"Compute Degrees Freedom (DOF) Poisson Kernel given dimension d concentration parameter rho","code":""},{"path":"/reference/DOF.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"","code":"DOF(d, rho)"},{"path":"/reference/DOF.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"d number dimensions rho concentration parameter","code":""},{"path":"/reference/DOF.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"list containing DOF coefficient c asymptotic distribution","code":""},{"path":"/reference/DOF_norm.html","id":null,"dir":"Reference","previous_headings":"","what":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"Compute Degrees Freedom (DOF) normal Kernel centered respect standard normal distribution, given dimension d bandwidth parameter h.","code":""},{"path":"/reference/DOF_norm.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"","code":"DOF_norm(Sigma_h, V)"},{"path":"/reference/DOF_norm.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"Sigma_h covariance matrix gaussian kernel V Covariance matrix tested distribution G","code":""},{"path":"/reference/DOF_norm.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"list containing DOF coefficient c asymptotic distribution","code":""},{"path":"/reference/dpkb.html","id":null,"dir":"Reference","previous_headings":"","what":"The Poisson kernel-based Distribution (PKBD) — dpkb","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"Poisson kernel-based densities based normalized Poisson kernel defined \\((d-1)\\)-dimensional unit sphere. Given vector \\(\\mathbf{\\mu} \\\\mathcal{S}^{d-1}\\), \\(\\mathcal{S}^{d-1}= \\{x \\\\mathbb{R}^d : ||x|| = 1\\}\\), parameter \\(\\rho\\) \\(0 < \\rho < 1\\), probability density function \\(d\\)-variate Poisson kernel-based density defined : $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d},$$ \\(\\mu\\) vector orienting center distribution, \\(\\rho\\) parameter control concentration distribution around vector \\(\\mu\\) related variance distribution. Recall , \\(x = (x_1, \\ldots, x_d) \\\\mathbb{R}^d\\), \\(||x|| = \\sqrt{x_1^2 + \\ldots + x_d^2}\\). Furthermore, \\(\\omega_d = 2\\pi^{d/2} [\\Gamma(d/2)]^{-1}\\) surface area unit sphere \\(\\mathbb{R}^d\\) (see Golzy Markatou, 2020). \\(\\rho \\0\\), Poisson kernel-based density tends uniform density sphere. Connections PKBDs distributions discussed detail Golzy Markatou (2020). note \\(d=2\\), PKBDs reduce wrapped Cauchy distribution. Additionally, precise choice parameters \\(\\rho\\) \\(\\mu\\) two-dimensional PKBD becomes two-dimensional projected normal distribution. However, connection \\(d\\)-dimensional projected normal distributions carry beyond \\(d=2\\). Golzy Markatou (2020) proposed acceptance-rejection method simulating data PKBD using von Mises-Fisher envelops (rejvmf method). Furthermore Sablica, Hornik Leydold (2023) proposed new ways simulating PKBD, using angular central Gaussian envelops (rejacg) using projected Saw distributions (rejpsaw).","code":""},{"path":"/reference/dpkb.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"","code":"dpkb(x, mu, rho, logdens = FALSE) rpkb( n, mu, rho, method = \"rejacg\", tol.eps = .Machine$double.eps^0.25, max.iter = 1000 )"},{"path":"/reference/dpkb.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"x \\(n \\times d\\)-matrix (data.frame) \\(n\\) data point sphere \\(\\mathcal{S}^{d-1}\\), \\(d \\ge 2\\). mu location vector parameter length indicating dimension generated points. rho Concentration parameter, \\(0 \\le\\) rho \\(< 1\\). logdens Logical; 'TRUE', densities returned logarithmic scale. n number observations. method string indicates method used sampling observations. available methods 'rejvmf' acceptance-rejection algorithm using von Mises-Fisher envelops (Algorithm Table 2 Golzy Markatou 2020); 'rejacg' using angular central Gaussian envelops (Algorithm Table 1 Sablica et al. 2023); 'rejpsaw' using projected Saw distributions (Algorithm Table 2 Sablica et al. 2023). tol.eps desired accuracy convergence tolerance ('rejacg' method). max.iter maximum number iterations ('rejacg' method).","code":""},{"path":"/reference/dpkb.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"dpkb gives density value; rpkb generates random observations PKBD.","code":""},{"path":"/reference/dpkb.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"function dpkb() computes density value given point x Poisson kernel-based distribution mean direction vector mu concentration parameter rho. number observations generated determined n rpkb(). function returns \\((n \\times d)\\)-matrix generated \\(n\\) observations \\(\\mathcal{S}^{(d-1)}\\). limitation rejvmf method ensure computational feasibility sampler \\(\\rho\\) approaching 1. chosen method 'rejacg', function uniroot, stat package, used estimate beta parameter. case, complete results provided output.","code":""},{"path":"/reference/dpkb.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"required packages (movMF rejvmf method, Tinflex rejpsaw) installed, function display message asking user install missing package(s).","code":""},{"path":"/reference/dpkb.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713. Sablica L., Hornik K. Leydold J. (2023) \"Efficient sampling PKBD distribution\", Electronic Journal Statistics, 17(2), 2180-2209.","code":""},{"path":"/reference/dpkb.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"","code":"# Generate some data from pkbd density pkbd_dat <- rpkb(10, c(0.5,0), 0.5) # Calculate the PKBD density values dens_val <- dpkb(pkbd_dat, c(0.5,0.5),0.5)"},{"path":"/reference/extract_stats.html","id":null,"dir":"Reference","previous_headings":"","what":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"Method objects class pkbc computes descriptive variable respect detected groups. Method objects class pkbc computes descriptive variable respect detected groups.","code":""},{"path":"/reference/extract_stats.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"","code":"extract_stats(object, ...) # S4 method for pkbc extract_stats(object, k)"},{"path":"/reference/extract_stats.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"object Object class pkbc. ... possible additional inputs k Number clusters used.","code":""},{"path":"/reference/extract_stats.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"List computed descriptive statistics variable.","code":""},{"path":"/reference/extract_stats.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"function computes mean, standard deviation, median, inter-quantile range, minimum maximum variable data set given final membership assigned clustering algorithm.","code":""},{"path":"/reference/extract_stats.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions dat<-matrix(rnorm(300),ncol=3) #Perform the clustering algorithm pkbc_res<- pkbc(dat, 3) extract_stats(pkbc_res, 3) #> [[1]] #> Group 1 Group 2 Group 3 Overall #> mean 0.08920104 -0.91948554 -0.730958308 0.03245039 #> sd 0.56580990 0.02016632 0.005881618 0.59346106 #> median 0.04157567 -0.92107038 -0.730958308 0.01132900 #> IQR 0.96291884 0.01823508 0.004158932 1.02074711 #> min -0.97048825 -0.94215818 -0.735117240 -0.97048825 #> max 0.99916163 -0.89364323 -0.726799376 0.99916163 #> #> [[2]] #> Group 1 Group 2 Group 3 Overall #> mean -0.04517405 0.26661864 0.130846021 -0.02918194 #> sd 0.57552347 0.09128171 0.005606366 0.56187290 #> median -0.09048645 0.30696292 0.130846021 -0.05715153 #> IQR 0.94679985 0.05778760 0.003964299 0.86955159 #> min -0.98329467 0.13035633 0.126881722 -0.98329467 #> max 0.93041624 0.32219239 0.134810320 0.93041624 #> #> [[3]] #> Group 1 Group 2 Group 3 Overall #> mean -0.02838592 -0.25722878 0.669725381 -0.02357741 #> sd 0.59030936 0.11966575 0.005324043 0.58285373 #> median -0.00847352 -0.29000844 0.669725381 -0.02467391 #> IQR 1.05195509 0.13113221 0.003764667 1.03465610 #> min -0.95321117 -0.35654011 0.665960714 -0.95321117 #> max 0.98834993 -0.09235811 0.673490048 0.98834993 #>"},{"path":"/reference/generate_SN.html","id":null,"dir":"Reference","previous_headings":"","what":"Generate two samples data from skew-normal distributions — generate_SN","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"function generates data skew-normal distributions specified parameters means covariance matrices.","code":""},{"path":"/reference/generate_SN.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"","code":"generate_SN(d, size_x, size_y, mu_x, mu_y, sigma_x, sigma_y, skewness_y)"},{"path":"/reference/generate_SN.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"d number dimensions. size_x number observations sample X size_y number observations sample Y mu_x mean X mu_y mean Y sigma_x standard deviation X sigma_y standard deviation Y skewness_y skewness Y (skewness X set zero).","code":""},{"path":"/reference/generate_SN.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"list containing generated X Y data sets.","code":""},{"path":"/reference/kb.test-class.html","id":null,"dir":"Reference","previous_headings":"","what":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","title":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","text":"class represent results Gaussian kernel-based quadratic distance tests. includes normality test, two-sample test statistics k-sample tests.","code":""},{"path":"/reference/kb.test-class.html","id":"slots","dir":"Reference","previous_headings":"","what":"Slots","title":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","text":"method String indicating kernel-based quadratic distance test performed. Un value test U-statistic. Vn value test V-statistic. H0_Un logical value indicating whether null hypothesis rejected according U-statistic. H0_Vn logical value indicating whether null hypothesis rejected according Vn. data List samples X (Y). CV_Un critical value computed test Un. CV_Vn critical value computed test Vn. cv_method method used estimate critical value (one \"subsampling\", \"permutation\" \"bootstrap\"). h list value bandwidth parameter used Gaussian kernel. function select_h used, also matrix computed power values resulting power plot provided. B Number bootstrap/permutation/subsampling replications. var_Un exact variance kernel-based U-statistic.","code":""},{"path":[]},{"path":"/reference/kb.test-class.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) y <- matrix(rnorm(100),ncol=2) # Normality test kb.test(x, h=0.5) #> #> Kernel-based quadratic distance Normality test #> \t\tU-statistic\tV-statistic #> ------------------------------------------------ #> Test Statistic:\t 1.310854 \t 0.7979308 #> Critical Value:\t 2.4409 \t 6.071062 #> H0 is rejected:\t FALSE \t\t FALSE #> Selected tuning parameter h: 0.5 #> # Two-sample test kb.test(x,y,h=0.5, method=\"subsampling\",b=0.9) #> #> Kernel-based quadratic distance two-sample test #> U-statistic\t Dn \t\t Trace #> ------------------------------------------------ #> Test Statistic:\t 0.1153108 \t 0.133021 #> Critical Value:\t 1.407721 \t 1.625753 #> H0 is rejected:\t FALSE \t\t FALSE #> CV method: subsampling #> Selected tuning parameter h: 0.5 #>"},{"path":"/reference/kb.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"function performs kernel-based quadratic distance goodness--fit tests. includes tests multivariate normality, two-sample tests \\(k\\)-sample tests.","code":""},{"path":"/reference/kb.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"","code":"kb.test( x, y = NULL, h = NULL, method = \"subsampling\", B = 150, b = NULL, Quantile = 0.95, mu_hat = NULL, Sigma_hat = NULL, centeringType = \"Nonparam\", K_threshold = 10, alternative = \"skewness\" ) # S4 method for ANY kb.test( x, y = NULL, h = NULL, method = \"subsampling\", B = 150, b = 0.9, Quantile = 0.95, mu_hat = NULL, Sigma_hat = NULL, centeringType = \"Nonparam\", K_threshold = 10, alternative = \"skewness\" ) # S4 method for kb.test show(object)"},{"path":"/reference/kb.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"x Numeric matrix vector data values. y Numeric matrix vector data values. Depending input y, corresponding test performed. y = NULL, function performs tests normality x y data matrix, dimensions x, function performs two-sample test x y. y numeric factor vector, indicating group memberships observation, function performs k-sample test. h Bandwidth kernel function. value provided, algorithm selection optimal h performed automatically. See function select_h details. method method used critical value estimation (\"subsampling\", \"bootstrap\", \"permutation\")(default: \"subsampling\"). B number iterations use critical value estimation (default: 150). b size subsamples used subsampling algorithm (default: 0.8). Quantile quantile use critical value estimation, 0.95 default value. mu_hat Mean vector reference distribution. Sigma_hat Covariance matrix reference distribution. centeringType String indicating method used centering normal kernel ('Param' 'Nonparam'). K_threshold maximum number groups allowed. Default 10. control parameter. Change case 10 samples. alternative Family alternative chosen selecting h, \"location\", \"scale\" \"skewness\" (h provided). object Object class kb.test","code":""},{"path":"/reference/kb.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"S4 object class kb.test containing results kernel-based quadratic distance tests, based normal kernel. object contains following slots: method: Description kernel-based quadratic distance test performed. x Data list samples X (Y). Un value U-statistic. H0_Un logical value indicating whether null hypothesis rejected according Un. CV_Un critical value computed test Un. Vn value V-statistic (available). H0_Vn logical value indicating whether null hypothesis rejected according Vn (available). CV_Vn critical value computed test Vn (available). h List value bandwidth parameter used normal kernel function. select_h used, matrix computed power values corresponding power plot also provided. B Number bootstrap/permutation/subsampling replications. var_Un exact variance kernel-based U-statistic. cv_method method used estimate critical value (one \"subsampling\", \"permutation\" \"bootstrap\").","code":""},{"path":"/reference/kb.test.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"function kb.test performs kernel-based quadratic distance tests using Gaussian kernel bandwidth parameter h. Depending shape input y function performs tests multivariate normality, non-parametric two-sample tests k-sample tests. quadratic distance two probability distributions \\(F\\) \\(G\\) defined $$d_{K}(F,G)=\\iint K(x,y)d(F-G)(x)d(F-G)(y),$$ \\(G\\) distribution whose goodness fit wish assess \\(K\\) denotes Normal kernel defined $$ K_{{h}}(\\mathbf{s}, \\mathbf{t}) = (2 \\pi)^{-d/2} \\left(\\det{\\mathbf{\\Sigma}_h}\\right)^{-\\frac{1}{2}} \\exp\\left\\{-\\frac{1}{2}(\\mathbf{s} - \\mathbf{t})^\\top \\mathbf{\\Sigma}_h^{-1}(\\mathbf{s} - \\mathbf{t})\\right\\},$$ every \\(\\mathbf{s}, \\mathbf{t} \\\\mathbb{R}^d \\times \\mathbb{R}^d\\), covariance matrix \\(\\mathbf{\\Sigma}_h=h^2 \\) tuning parameter \\(h\\). Test Normality: Let \\(x_1, x_2, ..., x_n\\) random sample empirical distribution function \\(\\hat F\\). test null hypothesis normality, .e. \\(H_0:F=G=\\mathcal{N}_d(\\mu, \\Sigma)\\). consider U-statistic estimate sample KBQD $$U_{n}=\\frac{1}{n(n-1)}\\sum_{=2}^{n}\\sum_{j=1}^{-1} K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ first test statistics $$T_{n}=\\frac{U_{n}}{\\sqrt{Var(U_{n})}},$$ \\(Var(U_n)\\) computed exactly following Lindsay et al.(2014), V-statistic estimate $$V_{n} = \\frac{1}{n}\\sum_{=1}^{n} \\sum_{j=1}^{n}K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ \\(K_{cen}\\) denotes Normal kernel \\(K_h\\) parametric centering respect considered normal distribution \\(G = \\mathcal{N}_d(\\mu, \\Sigma)\\). asymptotic distribution V-statistic infinite combination weighted independent chi-squared random variables one degree freedom. cutoff value obtained using Satterthwaite approximation \\(c \\cdot \\chi_{DOF}^2\\), \\(c\\) \\(DOF\\) computed exactly following formulas Lindsay et al.(2014). \\(U\\)-statistic cutoff determined empirically: Generate data considered normal distribution ; Compute test statistics B Monte Carlo(MC) replications; Compute 95th quantile empirical distribution test statistic. k-sample test: Consider \\(k\\) random samples ..d. observations \\(\\mathbf{x}^{()}_1, \\mathbf{x}^{()}_{2},\\ldots, \\mathbf{x}^{()}_{n_i} \\sim F_i\\), \\(= 1, \\ldots, k\\). test samples generated unknown distribution, \\(H_0: F_1 = F_2 = \\ldots = F_k\\) versus \\(H_1: F_i \\= F_j\\), \\(1 \\le \\= j \\le k\\). construct matrix distance \\(\\hat{\\mathbf{D}}\\), -diagonal elements $$\\hat{D}_{ij} = \\frac{1}{n_i n_j} \\sum_{\\ell=1}^{n_i} \\sum_{r=1}^{n_j}K_{\\bar{F}}(\\mathbf{x}^{()}_\\ell,\\mathbf{x}^{(j)}_r), \\qquad \\mbox{ }\\= j$$ diagonal $$\\hat{D}_{ii} = \\frac{1}{n_i (n_i -1)} \\sum_{\\ell=1}^{n_i} \\sum_{r\\= \\ell}^{n_i} K_{\\bar{F}}(\\mathbf{x}^{()}_\\ell, \\mathbf{x}^{()}_r), \\qquad \\mbox{ }= j,$$ \\(K_{\\bar{F}}\\) denotes Normal kernel \\(K_h\\) centered non-parametrically respect $$\\bar{F} = \\frac{n_1 \\hat{F}_1 + \\ldots + n_k \\hat{F}_k}{n}, \\quad \\mbox{ } n=\\sum_{=1}^k n_i.$$ compute trace statistic $$\\mathrm{trace}(\\hat{\\mathbf{D}}_n) = \\sum_{=1}^{k}\\hat{D}_{ii}$$ \\(D_n\\), derived considering possible pairwise comparisons k-sample null hypothesis, given $$D_n = (k-1) \\mathrm{trace}(\\hat{\\mathbf{D}}_n) - 2 \\sum_{=1}^{k}\\sum_{j> }^{k}\\hat{D}_{ij}.$$ compute empirical critical value employing numerical techniques bootstrap, permutation subsampling algorithms: Generate k-tuples, total size \\(n_B\\), pooled sample following one sampling methods; Compute k-sample test statistic; Repeat B times; Select \\(95^{th}\\) quantile obtained values. Two-sample test: Let \\(x_1, x_2, ..., x_{n_1} \\sim F\\) \\(y_1, y_2, ..., y_{n_2} \\sim G\\) random samples distributions \\(F\\) \\(G\\), respectively. test null hypothesis two samples generated unknown distribution, \\(H_0: F=G\\) vs \\(H_1:F\\=G\\). test statistics coincide \\(k\\)-sample test statistics \\(k=2\\).","code":""},{"path":"/reference/kb.test.html","id":"kernel-centering","dir":"Reference","previous_headings":"","what":"Kernel centering","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"arguments mu_hat Sigma_hat indicate normal model considered normality test, \\(H_0: F = N(\\)mu_hat, Sigma_hat). two-sample \\(k\\)-sample tests, mu_hat Sigma_hat can used parametric centering kernel, case want specify reference distribution, centeringType = \"Param\". default method test normality performed. normal kernel centered respect \\(G \\sim N_d(\\mathbf{\\mu}, \\mathbf{V})\\) can computed $$K_{cen(G)}(\\mathbf{s}, \\mathbf{t}) = K_{\\mathbf{\\Sigma_h}}(\\mathbf{s}, \\mathbf{t}) - K_{\\mathbf{\\Sigma_h} + \\mathbf{V}}(\\mathbf{\\mu}, \\mathbf{t}) - K_{\\mathbf{\\Sigma_h} + \\mathbf{V}}(\\mathbf{s}, \\mathbf{\\mu}) + K_{\\mathbf{\\Sigma_h} + 2\\mathbf{V}}(\\mathbf{\\mu}, \\mathbf{\\mu}).$$ consider non-parametric centering kernel respect \\(\\bar{F}=(n_1 F_1 + \\ldots n_k F_k)/n\\) \\(n=\\sum_{=1}^k n_i\\), centeringType = \"Nonparam\", two- \\(k\\)-sample tests. Let \\(\\mathbf{z}_1,\\ldots, \\mathbf{z}_n\\) denote pooled sample. \\(s,t \\\\{\\mathbf{z}_1,\\ldots, \\mathbf{z}_n\\}\\), given $$K_{cen(\\bar{F})}(\\mathbf{s},\\mathbf{t}) = K(\\mathbf{s},\\mathbf{t}) - \\frac{1}{n}\\sum_{=1}^{n} K(\\mathbf{s},\\mathbf{z}_i) - \\frac{1}{n}\\sum_{=1}^{n} K(\\mathbf{z}_i,\\mathbf{t}) + \\frac{1}{n(n-1)}\\sum_{=1}^{n} \\sum_{j \\=}^{n} K(\\mathbf{z}_i,\\mathbf{z}_j).$$","code":""},{"path":"/reference/kb.test.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"two- \\(k\\)-sample tests, slots Vn, H0_Vn CV_Vn empty, computed statistics reported slots Un, H0_Un CV_Un. U-statistic type statistic used estimate population parameter. based idea averaging possible distinct combinations fixed size sample. V-statistic considers possible tuples certain size, just distinct combinations can used contexts unbiasedness required.","code":""},{"path":"/reference/kb.test.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374 Lindsay, B.G., Markatou, M. Ray, S. (2014) \"Kernels, Degrees Freedom, Power Properties Quadratic Distance Goodness--Fit Tests\", Journal American Statistical Association, 109:505, 395-410, DOI: 10.1080/01621459.2013.836972","code":""},{"path":[]},{"path":"/reference/kb.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) y <- matrix(rnorm(100),ncol=2) # Normality test my_test <- kb.test(x, h=0.5) my_test #> #> Kernel-based quadratic distance Normality test #> \t\tU-statistic\tV-statistic #> ------------------------------------------------ #> Test Statistic:\t 0.3027069 \t 0.6189598 #> Critical Value:\t 1.262023 \t 6.071062 #> H0 is rejected:\t FALSE \t\t FALSE #> Selected tuning parameter h: 0.5 #> # Two-sample test my_test <- kb.test(x,y,h=0.5, method=\"subsampling\",b=0.9, centeringType = \"Nonparam\") my_test #> #> Kernel-based quadratic distance two-sample test #> U-statistic\t Dn \t\t Trace #> ------------------------------------------------ #> Test Statistic:\t 0.239788 \t 0.2819997 #> Critical Value:\t 0.9875162 \t 1.162661 #> H0 is rejected:\t FALSE \t\t FALSE #> CV method: subsampling #> Selected tuning parameter h: 0.5 #> # k-sample test z <- matrix(rnorm(100,2),ncol=2) dat <- rbind(x,y,z) group <- rep(c(1,2,3),each=50) my_test <- kb.test(x=dat,y=group,h=0.5, method=\"subsampling\",b=0.9) my_test #> #> Kernel-based quadratic distance k-sample test #> U-statistic\t Dn \t\t Trace #> ------------------------------------------------ #> Test Statistic:\t 7.325505 \t 11.45482 #> Critical Value:\t 0.7402039 \t 1.158313 #> H0 is rejected:\t TRUE \t\t TRUE #> CV method: subsampling #> Selected tuning parameter h: 0.5 #>"},{"path":"/reference/normal_CV.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"function computes empirical critical value Normality test based KBQD tests using centered Gaussian kernel.","code":""},{"path":"/reference/normal_CV.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"","code":"normal_CV(d, size, h, mu_hat, Sigma_hat, B = 150, Quantile = 0.95)"},{"path":"/reference/normal_CV.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"d dimension generated samples. size number observations generated. h concentration parameter Gaussian kernel. mu_hat Mean vector reference distribution. Sigma_hat Covariance matrix reference distribution. B number replications. Quantile quantile distribution use select critical value","code":""},{"path":"/reference/normal_CV.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"critical value specified dimension, size level.","code":""},{"path":"/reference/normal_CV.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"replication, sample d-dimensional Normal distribution mean vector mu_hat covariance matrix Sigma_hat generated KBQD test U-statistic Normality computed. B iterations, critical value selected Quantile empirical distribution computed test statistics.","code":""},{"path":"/reference/pk.test-class.html","id":null,"dir":"Reference","previous_headings":"","what":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","title":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","text":"class represent results Poisson kernel-based quadratic distance tests Uniformity sphere.","code":""},{"path":"/reference/pk.test-class.html","id":"slots","dir":"Reference","previous_headings":"","what":"Slots","title":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","text":"method Description test. x Matrix data Un value U-statistic. CV_Un critical value Un computed replications. H0_Un logical value indicating whether null hypothesis rejected according Un. Vn value V-statistic. CV_Vn critical value Vn computed following asymptotic distribution. H0_Vn logical value indicating whether null hypothesis rejected according Vn. rho concentration parameter Poisson kernel. B Number replications. var_Un exact variance kernel-based U-statistic.","code":""},{"path":[]},{"path":"/reference/pk.test-class.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","text":"","code":"# create a pk.test object d=3 size=100 x_sp <- sample_hypersphere(d, n_points=size) pk.test(x_sp,rho=0.8) #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Selected consentration parameter rho: 0.8 #> #> U-statistic: #> #> H0 is rejected: FALSE #> Statistic Un: 1.082258 #> Critical value: 1.44412 #> #> V-statistic: #> #> H0 is rejected: FALSE #> Statistic Vn: 49.19885 #> Critical value: 52.23077 #>"},{"path":"/reference/pk.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"function performs kernel-based quadratic distance goodness--fit tests Uniformity multivariate spherical data x \\(\\mathcal{S}^{d-1}\\) using Poisson kernel concentration parameter rho. Poisson kernel-based test uniformity exhibits excellent results especially case multimodal distributions, shown example Uniformity test Sphere vignette.","code":""},{"path":"/reference/pk.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"","code":"pk.test(x, rho, B = 300, Quantile = 0.95) # S4 method for ANY pk.test(x, rho, B = 300, Quantile = 0.95) # S4 method for pk.test show(object)"},{"path":"/reference/pk.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"x numeric \\((n \\times d)\\)-matrix \\(n\\) data points Sphere \\(\\mathcal{S}^(d-1)\\) rows. rho Concentration parameter Poisson kernel function. B Number Monte Carlo iterations critical value estimation Un (default: 300). Quantile quantile use critical value estimation, 0.95 default value. object Object class pk.test","code":""},{"path":"/reference/pk.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"S4 object class pk.test containing results Poisson kernel-based tests. object contains following slots: method: Description test performed. x Data matrix. Un value U-statistic. CV_Un empirical critical value Un. H0_Vn logical value indicating whether null hypothesis rejected according Un. Vn value V-statistic Vn. CV_Vn critical value Vn computed following asymptotic distribution. H0_Vn logical value indicating whether null hypothesis rejected according Vn. rho value concentration parameter used Poisson kernel function. B Number replications critical value U-statistic Un.","code":""},{"path":"/reference/pk.test.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"Let \\(x_1, x_2, ..., x_n\\) random sample empirical distribution function \\(\\hat F\\). test null hypothesis uniformity \\((d-1)\\)-dimensional sphere, .e. \\(H_0:F=G\\), \\(G\\) uniform distribution \\((d-1)\\)-dimensional sphere \\(\\mathcal{S}^{d-1}\\). compute U-statistic estimate sample KBQD (Kernel-Based Quadratic Distance) $$U_{n}=\\frac{1}{n(n-1)}\\sum_{=2}^{n}\\sum_{j=1}^{-1}K_{cen} (\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ first test statistic given $$T_{n}=\\frac{U_{n}}{\\sqrt{Var(U_{n})}},$$ $$Var(U_{n})= \\frac{2}{n(n-1)} \\left[\\frac{1+\\rho^{2}}{(1-\\rho^{2})^{d-1}}-1\\right],$$ V-statistic estimate KBQD $$V_{n} = \\frac{1}{n}\\sum_{=1}^{n}\\sum_{j=1}^{n}K_{cen} (\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ \\(K_{cen}\\) denotes Poisson kernel \\(K_\\rho\\) centered respect uniform distribution \\((d-1)\\)-dimensional sphere, $$K_{cen}(\\mathbf{u}, \\mathbf{v}) = K_\\rho(\\mathbf{u}, \\mathbf{v}) -1$$ $$K_\\rho(\\mathbf{u}, \\mathbf{v}) = \\frac{1-\\rho^{2}}{\\left(1+\\rho^{2}- 2\\rho (\\mathbf{u}\\cdot \\mathbf{v})\\right)^{d/2}},$$ every \\(\\mathbf{u}, \\mathbf{v} \\\\mathcal{S}^{d-1} \\times \\mathcal{S}^{d-1}\\). asymptotic distribution V-statistic infinite combination weighted independent chi-squared random variables one degree freedom. cutoff value obtained using Satterthwaite approximation \\(c \\cdot \\chi_{DOF}^2\\), $$c=\\frac{(1+\\rho^{2})- (1-\\rho^{2})^{d-1}}{(1+\\rho)^{d}-(1-\\rho^{2})^{d-1}}$$ $$DOF(K_{cen} )=\\left(\\frac{1+\\rho}{1-\\rho} \\right)^{d-1}\\left\\{ \\frac{\\left(1+\\rho-(1-\\rho)^{d-1} \\right )^{2}} {1+\\rho^{2}-(1-\\rho^{2})^{d-1}}\\right \\}.$$. \\(U\\)-statistic cutoff determined empirically: Generate data Uniform distribution d-dimensional sphere; Compute test statistics B Monte Carlo(MC) replications; Compute 95th quantile empirical distribution test statistic.","code":""},{"path":"/reference/pk.test.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"U-statistic type statistic used estimate population parameter. based idea averaging possible distinct combinations fixed size sample. V-statistic considers possible tuples certain size, just distinct combinations can used contexts unbiasedness required.","code":""},{"path":"/reference/pk.test.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi:10.5705/ss.202022.0347","code":""},{"path":[]},{"path":"/reference/pk.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"","code":"# create a pk.test object x_sp <- sample_hypersphere(3, n_points=100) unif_test <- pk.test(x_sp,rho=0.8) unif_test #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Selected consentration parameter rho: 0.8 #> #> U-statistic: #> #> H0 is rejected: FALSE #> Statistic Un: 0.4735996 #> Critical value: 1.543174 #> #> V-statistic: #> #> H0 is rejected: FALSE #> Statistic Vn: 46.27503 #> Critical value: 52.23077 #>"},{"path":"/reference/pkbc-class.html","id":null,"dir":"Reference","previous_headings":"","what":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","title":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","text":"class represent results Poisson kernel-based clustering procedure spherical observations.","code":""},{"path":"/reference/pkbc-class.html","id":"slots","dir":"Reference","previous_headings":"","what":"Slots","title":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","text":"res_k List objects results clustering algorithm value possible number clusters considered. input List input data","code":""},{"path":[]},{"path":"/reference/pkbc-class.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","text":"","code":"data(\"wireless\") res <- pkbc(as.matrix(wireless[,-8]),4)"},{"path":"/reference/pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Poisson kernel-based clustering on the sphere — pkbc","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"function pkbc() performs Poisson kernel-based clustering algorithm sphere proposed Golzy Markatou (2020). proposed algorithm based mixture, \\(M\\) components, Poisson kernel-based densities hypersphere \\(\\mathcal{S}^{d-1}\\) given $$f(x|\\Theta) = \\sum_{j=1}^M \\alpha_j f_j(x|\\rho_j, \\mu_j)$$ \\(\\alpha_j\\)'s mixing proportions \\(f_j(x|\\rho_j, \\mu_j)\\)'s denote probability density function \\(d\\)-variate Poisson kernel-based density given $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d}.$$ parameters \\(\\alpha_j, \\mu_j, \\rho_j\\) estimated iterative reweighted EM algorithm. proposed clustering algorithm exhibits excellent results (1) clusters well separated; (2) data points fairly well concentrated around vectors \\(\\mu_j\\) cluster; (3) percentage noise data increases.","code":""},{"path":"/reference/pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"","code":"pkbc( dat, nClust, maxIter = 300, stoppingRule = \"loglik\", initMethod = \"sampleData\", numInit = 10 ) # S4 method for ANY pkbc( dat, nClust, maxIter = 300, stoppingRule = \"loglik\", initMethod = \"sampleData\", numInit = 10 ) # S4 method for pkbc show(object)"},{"path":"/reference/pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"dat \\((n \\times d)\\)-data matrix data.frame data points sphere clustered. observations dat normalized dividing length vector ensure lie \\(d\\)-dimensional sphere. Note \\(d > 1\\). nClust Number clusters. can single value numeric vector. maxIter maximum number iterations run terminated. stoppingRule String describing stopping rule used within run. Currently must either 'max', 'membership', 'loglik'. initMethod String describing initialization method used. Currently must 'sampleData'. numInit Number initialization. object Object class pkbc","code":""},{"path":"/reference/pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"S4 object class pkbc containing results clustering procedure based Poisson kernel-based distributions. object contains following slots: res_k: List results Poisson kernel-based clustering algorithm value number clusters specified nClust. object list contains: postProbs Posterior probabilities observation indicated clusters. LogLik Maximum value log-likelihood function wcss Values within-cluster sum squares computed Euclidean distance cosine similarity, respectively. params List estimated parameters mixture model mu estimated centroids rho estimated concentration parameters rho alpha estimated mixing proportions finalMemb Vector final memberships runInfo List information EM algorithm iterations lokLikVec vector log-likelihood values numIterPerRun number E-M iterations per run input: List input information.","code":""},{"path":"/reference/pkbc.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"set concentration parameters equal 0.5 mixing proportions equal. initialization method 'sampleData' indicates observation points randomly chosen initializers centroids \\(\\mu_j\\). random starts strategy chance obtaining initial representatives underlying clusters, clustering performed numInit times random start highest likelihood chosen final estimate parameters. possible stoppingRule iteration : 'loglik' run algorithm change log-likelihood one iteration next less given threshold (1e-7) 'membership' run algorithm membership unchanged points one iteration next 'max' reach maximum number iterations maxIter obtained estimates used assigning final memberships, identifying nClust clusters, according following rule $$P(x_i, \\Theta) = \\arg\\max_{j \\\\{1, \\ldots, k\\}} \\{ \\frac{\\alpha_j f_j(x_i|\\mu_j, \\rho_j)}{f(x_i, \\Theta)}\\}.$$ number clusters nClust must provided input clustering algorithm.","code":""},{"path":"/reference/pkbc.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"clustering algorithm tailored data points sphere \\(\\mathcal{S}^{d-1}\\), can also performed spherically transformed observations, .e. data points Euclidean space \\(\\mathbb{R}^d\\) normalized lie corresponding \\((d-1)\\)-dimensional sphere \\(\\mathcal{S}^{d-1}\\).","code":""},{"path":"/reference/pkbc.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713.","code":""},{"path":[]},{"path":"/reference/pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions size<-100 groups<-c(rep(1, size), rep(2, size),rep(3,size)) rho<-0.8 set.seed(081423) data1<-rpkb(size, c(1,0,0),rho) data2<-rpkb(size, c(0,1,0),rho) data3<-rpkb(size, c(0,0,1),rho) dat<-rbind(data1,data2, data3) #Perform the clustering algorithm with number of clusters k=3. pkbd<- pkbc(dat=dat, nClust=3) show(pkbd) #> Poisson Kernel-Based Clustering on the Sphere (pkbc) Object #> ------------------------------------------------------------ #> #> Available components: #> Input Parameters: #> [1] \"dat\" \"nClust\" \"maxIter\" \"stoppingRule\" \"initMethod\" #> [6] \"numInit\" #> #> #> Considered possible number of clusters: 3 #> #> Available components for each value of number of clusters: #> [1] \"postProbs\" \"LogLik\" \"wcss\" \"params\" \"finalMemb\" \"runInfo\""},{"path":"/reference/pkbc_validation.html","id":null,"dir":"Reference","previous_headings":"","what":"Validation of Poisson kernel-based clustering results — pkbc_validation","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"Method objects class pkbc computes evaluation measures clustering results. following evaluation measures computed: -Group Proportion (Kapp Tibshirani (2007)). true label provided, ARI, Average Silhouette Width (Rousseeuw (1987)), Macro-Precision Macro-Recall computed.","code":""},{"path":"/reference/pkbc_validation.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"","code":"pkbc_validation(object, true_label = NULL)"},{"path":"/reference/pkbc_validation.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"object Object class pkbc true_label factor vector true membership clusters (available). must length final memberships.","code":""},{"path":"/reference/pkbc_validation.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"List following components: metrics Table computed evaluation measures value number clusters pkbc object. number cluster indicated column name. IGP List -group proportions value number clusters specified.","code":""},{"path":"/reference/pkbc_validation.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"IGP statistical measure quantifies proportion observations within group belong predefined category class. often used assess homogeneity group evaluating many members share label. higher IGP indicates group cohesive, lower proportion suggests greater diversity misclassification within group (Kapp Tibshirani 2007). Adjusted Rand Index (ARI) statistical measure used data clustering analysis. quantifies similarity two partitions dataset comparing assignments data points clusters. ARI value ranges 0 1, value 1 indicates perfect match partitions value close 0 indicates random assignment data points clusters. average silhouette width quantifies quality clustering measuring well object fits within assigned cluster. mean silhouette values, compare tightness object within cluster separation clusters. Higher values indicate well-separated, cohesive clusters, making useful selecting appropriate number clusters (Rousseeuw 1987). Macro Precision metric used multi-class classification calculates precision class independently takes average values. Precision class defined proportion true positive predictions predictions made class. Macro Recall similar Macro Precision focuses recall. Recall class proportion true positive predictions actual instances class. Macro Recall average recall values computed class.","code":""},{"path":"/reference/pkbc_validation.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"Note Macro Precision Macro Recall depend assigned labels, ARI measures similarity partition label switching. required packages (mclust ARI, clusterRepro IGP, cluster ASW) installed, function display message asking user install missing package(s).","code":""},{"path":"/reference/pkbc_validation.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"Kapp, .V. Tibshirani, R. (2007) \"clusters found one dataset present another dataset?\", Biostatistics, 8(1), 9–31, https://doi.org/10.1093/biostatistics/kxj029 Rousseeuw, P.J. (1987) Silhouettes: graphical aid interpretation validation cluster analysis. Journal Computational Applied Mathematics, 20, 53–65.","code":""},{"path":[]},{"path":"/reference/pkbc_validation.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions size<-20 groups<-c(rep(1, size), rep(2, size),rep(3,size)) rho<-0.8 set.seed(081423) data1<-rpkb(size, c(1,0,0),rho,method='rejvmf') data2<-rpkb(size, c(0,1,0),rho,method='rejvmf') data3<-rpkb(size, c(1,0,0),rho,method='rejvmf') data<-rbind(data1,data2, data3) #Perform the clustering algorithm pkbc_res<- pkbc(data, 3) pkbc_validation(pkbc_res) #> $metrics #> 3 #> ASW 0.03602451 #> #> $IGP #> $IGP[[1]] #> NULL #> #> $IGP[[2]] #> NULL #> #> $IGP[[3]] #> [1] 0.952381 1.000000 1.000000 #> #>"},{"path":"/reference/plot.pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Plotting method for Poisson kernel-based clustering — plot.pkbc","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"Plots pkbc object.","code":""},{"path":"/reference/plot.pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"","code":"# S4 method for pkbc,ANY plot(x, k = NULL, true_label = NULL, pca_res = FALSE, ...)"},{"path":"/reference/plot.pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"x Object class pkbc k number considered clusters. provided scatter plot displayed value number clusters present x object true_label factor vector true membership clusters (available). must length final memberships. pca_res Logical. TRUE results PCALocantore also reported (dimension greater 3). ... Additional arguments can passed plot function","code":""},{"path":"/reference/plot.pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"scatter-plot(s) elbow plot.","code":""},{"path":"/reference/plot.pkbc.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"scatterplot: dimension equal 2 3, points displayed circle sphere, respectively. dimension greater 3, spherical Principal Component procedure proposed Locantore et al. (1999), applied dimensionality reduction first three principal components normalized displayed sphere. d > 3, complete results PcaLocantore function (package rrcov) returned pca_res=TRUE. elbow plot: within cluster sum squares (wcss) computed using Euclidean distance (left) cosine similarity (right).","code":""},{"path":"/reference/plot.pkbc.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"elbow plot commonly used graphical method choosing appropriate number clusters. Specifically, plotting wcss versus number clusters, suggested number clusters correspond point plotted line greatest change slope, showing elbow.","code":""},{"path":"/reference/plot.pkbc.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"Locantore, N., Marron, J.S., Simpson, D.G. et al. (1999) \"Robust principal component analysis functional data.\" Test 8, 1–73. https://doi.org/10.1007/BF02595862","code":""},{"path":[]},{"path":"/reference/plot.pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"","code":"dat<-matrix(rnorm(300),ncol=3) pkbc_res<- pkbc(dat, 3) plot(pkbc_res, 3) #> `geom_line()`: Each group consists of only one observation. #> ℹ Do you need to adjust the group aesthetic? #> `geom_line()`: Each group consists of only one observation. #> ℹ Do you need to adjust the group aesthetic?"},{"path":"/reference/poisson_CV.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"function computes empirical critical value U-statistics testing uniformity sphere based centered poisson kernel.","code":""},{"path":"/reference/poisson_CV.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"","code":"poisson_CV(d, size, rho, B, Quantile)"},{"path":"/reference/poisson_CV.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"d dimension generated samples. size number observations generated. rho concentration parameter Poisson kernel. B number replications. Quantile quantile distribution use select critical value.","code":""},{"path":"/reference/poisson_CV.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"critical value specified dimension, size level.","code":""},{"path":"/reference/poisson_CV.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"replication, sample d-dimensional observations uniform distribution Sphere generated Poisson kernel-based U-statistic computed. B iterations, critical value selected Quantile empirical distribution computed test statistics.","code":""},{"path":"/reference/poisson_CV.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"Ding Yuxin, Markatou Marianthi, Saraceno Giovanni (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: doi:10.5705/ss.202022.0347","code":""},{"path":"/reference/predict.pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"Obtain predictions membership spherical observations based mixture Poisson kernel-based densities estimated pkbc","code":""},{"path":"/reference/predict.pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"","code":"# S4 method for pkbc predict(object, k, newdata = NULL)"},{"path":"/reference/predict.pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"object Object class pkbc k Number clusters used. newdata data.frame matrix data. missing clustering data obtained pkbc object classified.","code":""},{"path":"/reference/predict.pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"Returns list following components Memb: vector predicted memberships newdata Probs: matrix entry (,j) denotes probability observation belongs k-th cluster.","code":""},{"path":[]},{"path":"/reference/predict.pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"","code":"# generate data dat <- rbind(matrix(rnorm(100),ncol=2),matrix(rnorm(100,5),ncol=2)) res <- pkbc(dat,2) # extract membership of dat predict(res,k=2) #> [1] 2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 2 2 #> [38] 2 2 2 1 2 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 #> [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 # predict membership of new data newdat <- rbind(matrix(rnorm(10),ncol=2),matrix(rnorm(10,5),ncol=2)) predict(res, k=2, newdat) #> $Memb #> [1] 1 2 2 2 2 1 1 1 1 1 #> #> $Probs #> [,1] [,2] #> [1,] 0.79214887 0.20785113 #> [2,] 0.09155787 0.90844213 #> [3,] 0.10633003 0.89366997 #> [4,] 0.05939363 0.94060637 #> [5,] 0.11831316 0.88168684 #> [6,] 0.95577420 0.04422580 #> [7,] 0.97150510 0.02849490 #> [8,] 0.96104005 0.03895995 #> [9,] 0.84807303 0.15192697 #> [10,] 0.95487786 0.04512214 #>"},{"path":"/reference/QuadratiK-package.html","id":null,"dir":"Reference","previous_headings":"","what":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"Collection Methods Constructed using Kernel-Based Quadratic Distances QuadratiK provides first implementation, R Python, comprehensive set goodness--fit tests clustering technique \\(d\\)-dimensional spherical data \\(d \\ge 2\\) using kernel-based quadratic distances. includes: Goodness--Fit Tests: software implements one, two, k-sample tests goodness fit, offering efficient mathematically sound way assess fit probability distributions. tests particularly useful large, high dimensional data sets assessment fit probability models interest. Specifically, offer tests normality, well two- k-sample tests, testing equality two distributions interest, \\(H_0: F_1 = F_2\\) \\(H_0: F_1 = \\ldots = F_k\\) respectively. proposed tests perform well terms level power contiguous alternatives, heavy tailed distributions higher dimensions. Expanded capabilities include supporting tests uniformity d-dimensional Sphere based Poisson kernel, exhibiting excellent results especially case multimodal distributions. Poisson kernel-based distribution (PKBD): package offers functions computing density value generating random samples PKBD. Poisson kernel-based densities based normalized Poisson kernel defined \\(d\\)-dimensional unit sphere. Given vector \\(\\mu \\\\mathcal{S}^{d-1}\\), parameter \\(\\rho\\) \\(0 < \\rho < 1\\), probability density function \\(d\\)-variate Poisson kernel-based density defined : $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d},$$ \\(\\mu\\) vector orienting center distribution, \\(\\rho\\) parameter control concentration distribution around vector \\(\\mu\\) related variance distribution. Furthermore, \\(\\omega_d = 2\\pi^{d/2} [\\Gamma(d/2)]^{-1}\\) surface area unit sphere \\(\\mathbb{R}^d\\) (see Golzy Markatou, 2020). Clustering Algorithm Spherical Data: package incorporates unique clustering algorithm specifically tailored \\(d\\)-dimensional spherical data especially useful presence noise data presence non-negligible overlap clusters. algorithm leverages mixture Poisson kernel-based densities Sphere, enabling effective clustering spherical data data spherically transformed. Additional Features: Alongside functionalities, software includes additional graphical functions, aiding users validating representing cluster results well enhancing interpretability usability analysis. introduction QuadratiK see vignette Introduction QuadratiK Package.","code":""},{"path":"/reference/QuadratiK-package.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"work supported Kaleida Health Foundation National Science Foundation.","code":""},{"path":"/reference/QuadratiK-package.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"QuadratiK package also available Python PyPI https://pypi.org/project/QuadratiK/ also Dashboard application. Usage instruction Dashboard can found .","code":""},{"path":"/reference/QuadratiK-package.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. arXiv preprint arXiv:2402.02290. Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: doi:10.5705/ss.202022.0347. Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713. Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":[]},{"path":"/reference/QuadratiK-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"Giovanni Saraceno, Marianthi Markatou, Raktim Mukhopadhyay, Mojgan Golzy Maintainer: Giovanni Saraceno gsaracen@buffalo.edu","code":""},{"path":"/reference/sample_hypersphere.html","id":null,"dir":"Reference","previous_headings":"","what":"Generate random sample from the hypersphere — sample_hypersphere","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"Generate random sample uniform distribution hypersphere.","code":""},{"path":"/reference/sample_hypersphere.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"","code":"sample_hypersphere(d, n_points = 1)"},{"path":"/reference/sample_hypersphere.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"d Number dimensions. n_points Number sampled observations.","code":""},{"path":"/reference/sample_hypersphere.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"Data matrix sampled observations.","code":""},{"path":"/reference/sample_hypersphere.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"","code":"x_sp <- sample_hypersphere(3,100)"},{"path":"/reference/select_h.html","id":null,"dir":"Reference","previous_headings":"","what":"Select the value of the kernel tuning parameter — select_h","title":"Select the value of the kernel tuning parameter — select_h","text":"function computes kernel bandwidth Gaussian kernel normality, two-sample k-sample kernel-based quadratic distance (KBQD) tests.","code":""},{"path":"/reference/select_h.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Select the value of the kernel tuning parameter — select_h","text":"","code":"select_h( x, y = NULL, alternative = NULL, method = \"subsampling\", b = 0.8, B = 100, delta_dim = 1, delta = NULL, h_values = NULL, Nrep = 50, n_cores = 2, Quantile = 0.95, power.plot = TRUE )"},{"path":"/reference/select_h.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Select the value of the kernel tuning parameter — select_h","text":"x Data set observations X. y Numeric matrix vector data values. Depending input y, selection h performed corresponding test. y = NULL, function performs tests normality x. y data matrix, dimensions x, function performs two-sample test x y. y numeric factor vector, indicating group memberships observation, function performs k-sample test. alternative Family alternative chosen selecting h, \"location\", \"scale\" \"skewness\". method method used critical value estimation (\"subsampling\", \"bootstrap\", \"permutation\"). b size subsamples used subsampling algorithm . B number iterations use critical value estimation, B = 150 default. delta_dim Vector coefficient alternative respect dimension delta Vector parameter values indicating chosen alternatives h_values Values tuning parameter used selection Nrep Number bootstrap/permutation/subsampling replications. n_cores Number cores used parallel h selection algorithm. provided, function detect available cores. Quantile quantile use critical value estimation, 0.95 default value. power.plot Logical. TRUE, displayed plot power values h_values delta.","code":""},{"path":"/reference/select_h.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Select the value of the kernel tuning parameter — select_h","text":"list following attributes: h_sel selected value tuning parameter h; power matrix power values computed considered values delta h_values; power.plot power plots (power.plot TRUE).","code":""},{"path":"/reference/select_h.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Select the value of the kernel tuning parameter — select_h","text":"function performs selection optimal value tuning parameter \\(h\\) normal kernel function, normality test, two-sample k-sample KBQD tests. performs small simulation study, generating samples according family alternative specified, chosen values h_values delta. consider target alternatives \\(F_\\delta(\\hat{\\mathbf{\\mu}}, \\hat{\\mathbf{\\Sigma}}, \\hat{\\mathbf{\\lambda}})\\), \\(\\hat{\\mathbf{\\mu}}, \\hat{\\mathbf{\\Sigma}}\\) \\(\\hat{\\mathbf{\\lambda}}\\) indicate location, covariance skewness parameter estimates pooled sample. Compute estimates mean \\(\\hat{\\mu}\\), covariance matrix \\(\\hat{\\Sigma}\\) skewness \\(\\hat{\\lambda}\\) pooled sample. Choose family alternatives \\(F_\\delta = F_\\delta(\\hat{\\mu} ,\\hat{\\Sigma}, \\hat{\\lambda})\\). value \\(\\delta\\) \\(h\\): Generate \\(\\mathbf{X}_1,\\ldots,\\mathbf{X}_{k-1} \\sim F_0\\), \\(\\delta=0\\); Generate \\(\\mathbf{X}_k \\sim F_\\delta\\); Compute \\(k\\)-sample test statistic \\(\\mathbf{X}_1, \\mathbf{X}_2, \\ldots, \\mathbf{X}_k\\) kernel parameter \\(h\\); Compute power test. greater 0.5, select \\(h\\) optimal value. optimal value selected, choose \\(h\\) corresponds maximum power. available alternative location alternatives, \\(F_\\delta = SN_d(\\hat{\\mu} + \\delta,\\hat{\\Sigma}, \\hat{\\lambda})\\),\\(\\delta = 0.2, 0.3, 0.4\\); scale alternatives, \\(F_\\delta = SN_d(\\hat{\\mu} ,\\hat{\\Sigma}*\\delta, \\hat{\\lambda})\\), \\(\\delta = 0.1, 0.3, 0.5\\); skewness alternatives, \\(F_\\delta = SN_d(\\hat{\\mu} ,\\hat{\\Sigma}, \\hat{\\lambda} + \\delta)\\), \\(\\delta = 0.2, 0.3, 0.6\\). values \\(h = 0.6, 1, 1.4, 1.8, 2.2\\) \\(N=50\\) set default values. function select_h() allows user set values \\(\\delta\\) \\(h\\) extensive grid search. suggest set extensive grid search computational resources permit.","code":""},{"path":"/reference/select_h.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Select the value of the kernel tuning parameter — select_h","text":"Please aware select_h() function may take significant amount time run, especially larger datasets using larger number parameters h_values delta. Consider applying function large complex data.","code":""},{"path":"/reference/select_h.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Select the value of the kernel tuning parameter — select_h","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374 Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. https://arxiv.org/abs/2402.02290.","code":""},{"path":[]},{"path":"/reference/select_h.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Select the value of the kernel tuning parameter — select_h","text":"","code":"# Select the value of h using the mid-power algorithm # \\donttest{ x <- matrix(rnorm(100),ncol=2) y <- matrix(rnorm(100),ncol=2) h_sel <- select_h(x,y,\"skewness\") h_sel #> $h_sel #> [1] 1.6 #> #> $power #> h delta power #> 20 1.6 0.6 0.46 #> 24 3.2 0.6 0.44 #> 18 0.8 0.6 0.36 #> 22 2.4 0.6 0.36 #> 21 2.0 0.6 0.34 #> 23 2.8 0.6 0.34 #> 19 1.2 0.6 0.30 #> 17 0.4 0.6 0.18 #> 14 2.4 0.3 0.18 #> 10 0.8 0.3 0.16 #> 12 1.6 0.3 0.16 #> 13 2.0 0.3 0.12 #> 15 2.8 0.3 0.12 #> 16 3.2 0.3 0.12 #> 3 1.2 0.2 0.10 #> 6 2.4 0.2 0.10 #> 2 0.8 0.2 0.08 #> 8 3.2 0.2 0.08 #> 1 0.4 0.2 0.06 #> 9 0.4 0.3 0.06 #> 11 1.2 0.3 0.06 #> 5 2.0 0.2 0.06 #> 7 2.8 0.2 0.06 #> 4 1.6 0.2 0.02 #> #> $power.plot #> # }"},{"path":"/reference/stats_clusters.html","id":null,"dir":"Reference","previous_headings":"","what":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"Method objects class pkbc computes descriptive variable respect detected groups. Method objects class pkbc computes descriptive statistics variable respect detected groups.","code":""},{"path":"/reference/stats_clusters.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"","code":"stats_clusters(object, ...) # S4 method for pkbc stats_clusters(object, k)"},{"path":"/reference/stats_clusters.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"object Object class pkbc. ... possible additional inputs k Number clusters used.","code":""},{"path":"/reference/stats_clusters.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"List computed descriptive statistics dimension.","code":""},{"path":"/reference/stats_clusters.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"function computes mean, standard deviation, median, inter-quantile range, minimum maximum variable data set given final membership assigned clustering algorithm.","code":""},{"path":[]},{"path":"/reference/stats_clusters.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions dat<-matrix(rnorm(300),ncol=3) #Perform the clustering algorithm pkbc_res<- pkbc(dat, 3) stats_clusters(pkbc_res, 3) #> [[1]] #> Group 1 Group 2 Group 3 Overall #> mean -0.53360502 0.2185515 0.007405148 0.04344402 #> sd 0.34258917 0.5269849 NA 0.58084656 #> median -0.55364837 0.2151044 0.007405148 0.07318718 #> IQR 0.53206272 0.9127813 0.000000000 0.93759633 #> min -0.99487956 -0.9642025 0.007405148 -0.99487956 #> max 0.07466077 0.9873736 0.007405148 0.98737364 #> #> [[2]] #> Group 1 Group 2 Group 3 Overall #> mean 0.1608868 -0.1893987 0.8593866 -0.0983452 #> sd 0.3330494 0.6198886 NA 0.5890662 #> median 0.1009154 -0.3009186 0.8593866 -0.1183249 #> IQR 0.5036676 0.9810143 0.0000000 1.0017535 #> min -0.4511650 -0.9888086 0.8593866 -0.9888086 #> max 0.7908170 0.9928995 0.8593866 0.9928995 #> #> [[3]] #> Group 1 Group 2 Group 3 Overall #> mean -0.62082799 0.09072061 -0.5112727 -0.0789555 #> sd 0.29913303 0.50819936 NA 0.5547490 #> median -0.65141402 0.08038502 -0.5112727 -0.1405382 #> IQR 0.47845791 0.77660008 0.0000000 0.8655454 #> min -0.95185464 -0.99561644 -0.5112727 -0.9956164 #> max -0.09110684 0.96905979 -0.5112727 0.9690598 #>"},{"path":"/reference/summary-kb.test-method.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"summary method class kb.test","code":""},{"path":"/reference/summary-kb.test-method.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"","code":"# S4 method for kb.test summary(object)"},{"path":"/reference/summary-kb.test-method.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"object Object class kb.test","code":""},{"path":"/reference/summary-kb.test-method.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"List following components: summary_tables Table computed descriptive statistics per variable (per group available). test_results Data frame results performed kernel-based quadratic distance test. qqplots Figure qq-plots variable.","code":""},{"path":[]},{"path":"/reference/summary-kb.test-method.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) # Normality test my_test <- kb.test(x, h=0.5) summary(my_test) #> #> Kernel-based quadratic distance Normality test #> Statistic Test_Statistic Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> $summary_tables #> V1 V2 #> mean 0.2225601 -0.01276311 #> sd 1.0933236 1.14937193 #> median 0.2061474 0.14471308 #> IQR 1.3443250 1.40042674 #> min -1.5643294 -4.11865455 #> max 3.5340310 2.55373421 #> #> $test_results #> Statistic Test_Statistic Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> #> $qqplots #>"},{"path":"/reference/summary-pk.test-method.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"summary method class pk.test","code":""},{"path":"/reference/summary-pk.test-method.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"","code":"# S4 method for pk.test summary(object)"},{"path":"/reference/summary-pk.test-method.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"object Object class pk.test","code":""},{"path":"/reference/summary-pk.test-method.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"List following components: summary_tables Table computed descriptive statistics per variable. test_results Data frame results performed Poisson kernel-based test. qqplots Figure qq-plots variable uniform distribution.","code":""},{"path":[]},{"path":"/reference/summary-pk.test-method.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"","code":"# create a pk.test object x_sp <- sample_hypersphere(3, n_points=100) unif_test <- pk.test(x_sp,rho=0.8) summary(unif_test) #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> $summary_tables #> V1 V2 V3 #> mean -0.09616741 0.08709887 -0.01460631 #> sd 0.57359035 0.57642804 0.57585233 #> median -0.17876519 0.20832080 -0.04974959 #> IQR 0.95429937 0.89814932 1.00133903 #> min -0.99451952 -0.97609712 -0.92478198 #> max 0.94196356 0.98234388 0.93822530 #> #> $test_results #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> #> $qqplots #>"},{"path":"/reference/summary.kb.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary.kb.test","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"summary method class kb.test","code":""},{"path":"/reference/summary.kb.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"","code":"# S4 method for kb.test summary(object)"},{"path":"/reference/summary.kb.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"object Object class kb.test","code":""},{"path":"/reference/summary.kb.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"List following components: summary_tables Table computed descriptive statistics per variable (per group available). test_results Data frame results performed kernel-based quadratic distance test. qqplots Figure qq-plots variable.","code":""},{"path":[]},{"path":"/reference/summary.kb.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) # Normality test my_test <- kb.test(x, h=0.5) summary(my_test) #> #> Kernel-based quadratic distance Normality test #> Statistic Value Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> $summary_tables #> V1 V2 #> mean 0.2225601 -0.01276311 #> sd 1.0933236 1.14937193 #> median 0.2061474 0.14471308 #> IQR 1.3443250 1.40042674 #> min -1.5643294 -4.11865455 #> max 3.5340310 2.55373421 #> #> $test_results #> Statistic Value Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> #> $qqplots #>"},{"path":"/reference/summary.pk.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary.pk.test","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"summary method class pk.test","code":""},{"path":"/reference/summary.pk.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"","code":"# S4 method for pk.test summary(object)"},{"path":"/reference/summary.pk.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"object Object class pk.test","code":""},{"path":"/reference/summary.pk.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"List following components: summary_tables Table computed descriptive statistics per variable. test_results Data frame results performed Poisson kernel-based test. qqplots Figure qq-plots variable uniform distribution.","code":""},{"path":[]},{"path":"/reference/summary.pk.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"","code":"# create a pk.test object x_sp <- sample_hypersphere(3, n_points=100) unif_test <- pk.test(x_sp,rho=0.8) summary(unif_test) #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> $summary_tables #> V1 V2 V3 #> mean -0.09616741 0.08709887 -0.01460631 #> sd 0.57359035 0.57642804 0.57585233 #> median -0.17876519 0.20832080 -0.04974959 #> IQR 0.95429937 0.89814932 1.00133903 #> min -0.99451952 -0.97609712 -0.92478198 #> max 0.94196356 0.98234388 0.93822530 #> #> $test_results #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> #> $qqplots #>"},{"path":"/reference/summary.pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing PKBD mixture Fits — summary.pkbc","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"Summary method class \"pkbc\"","code":""},{"path":"/reference/summary.pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"","code":"# S4 method for pkbc summary(object)"},{"path":"/reference/summary.pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"object Object class pkbc","code":""},{"path":"/reference/summary.pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"Display logLikelihood values within cluster sum squares (wcss) values number clusters provided. values estimated mixing proportions showed together table assigned memberships.","code":""},{"path":[]},{"path":"/reference/summary.pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"","code":"dat <- rbind(matrix(rnorm(100),2),matrix(rnorm(100,5),2)) res <- pkbc(dat,2:4) summary(res) #> Poisson Kernel-Based Clustering on the Sphere (pkbc) Results #> ------------------------------------------------------------ #> #> Summary: #> nClust LogLik WCSS #> [1,] 2 744.1161 4.978247 #> [2,] 3 1104.1246 4.049000 #> [3,] 4 1391.2432 4.000000 #> #> Results for 2 clusters: #> Estimated Mixing Proportions (alpha): #> [1] 0.2879668 0.7120332 #> #> Clustering table: #> #> 1 2 #> 1 3 #> #> #> Results for 3 clusters: #> Estimated Mixing Proportions (alpha): #> [1] 0.25 0.25 0.50 #> #> Clustering table: #> #> 1 2 3 #> 1 1 2 #> #> #> Results for 4 clusters: #> Estimated Mixing Proportions (alpha): #> [1] 0.25 0.25 0.25 0.25 #> #> Clustering table: #> #> 1 2 3 4 #> 1 1 1 1 #> #>"},{"path":"/reference/var_k.html","id":null,"dir":"Reference","previous_headings":"","what":"Exact variance of k-sample test — var_k","title":"Exact variance of k-sample test — var_k","text":"Compute exact variance kernel test k-sample problem null hypothesis F1=...=Fk.","code":""},{"path":"/reference/var_k.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Exact variance of k-sample test — var_k","text":"","code":"var_k(Kcen, sizes, cum_size)"},{"path":"/reference/var_k.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Exact variance of k-sample test — var_k","text":"Kcen matrix centered kernel values sizes vector indicating sample's size. cum_size vector indicating sample's cumulative sizes.","code":""},{"path":"/reference/var_k.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Exact variance of k-sample test — var_k","text":"value computed variance.","code":""},{"path":"/reference/var_norm.html","id":null,"dir":"Reference","previous_headings":"","what":"Exact variance of normality test — var_norm","title":"Exact variance of normality test — var_norm","text":"Compute exact variance kernel test normality null hypothesis G=N(0,).","code":""},{"path":"/reference/var_norm.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Exact variance of normality test — var_norm","text":"","code":"var_norm(Sigma_h, V, n)"},{"path":"/reference/var_norm.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Exact variance of normality test — var_norm","text":"Sigma_h covariance matrix gaussian kernel V Covariance matrix tested distribution G n sample size","code":""},{"path":"/reference/var_norm.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Exact variance of normality test — var_norm","text":"value computed variance","code":""},{"path":"/reference/var_two.html","id":null,"dir":"Reference","previous_headings":"","what":"Exact variance of two-sample test — var_two","title":"Exact variance of two-sample test — var_two","text":"Compute exact variance kernel test two-sample problem null hypothesis F=G.","code":""},{"path":"/reference/var_two.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Exact variance of two-sample test — var_two","text":"","code":"var_two(Kcen, nsamples)"},{"path":"/reference/var_two.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Exact variance of two-sample test — var_two","text":"Kcen matrix centered kernel values nsamples vector indicating sample's membership.","code":""},{"path":"/reference/var_two.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Exact variance of two-sample test — var_two","text":"value computed variance.","code":""},{"path":"/reference/wine.html","id":null,"dir":"Reference","previous_headings":"","what":"Wine data set — wine","title":"Wine data set — wine","text":"wine data frame 178 rows 14 columns. first 13 variables report 13 constituents found three types wines. last column indicates class labels (1,2 3).","code":""},{"path":"/reference/wine.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Wine data set — wine","text":"","code":"wine"},{"path":"/reference/wine.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Wine data set — wine","text":"data frame containing following columns: Alcohol Malic acid Ash Alcalinity ash Magnesium Total phenols Flavanoids Nonflavanoid phenols Proanthocyanins Color intensity Hue OD280/OD315 diluted wines Proline y: class membership","code":""},{"path":"/reference/wine.html","id":"source","dir":"Reference","previous_headings":"","what":"Source","title":"Wine data set — wine","text":"Aeberhard, S. Forina, M. (1991). Wine. UCI Machine Learning Repository. https://doi.org/10.24432/C5PC7J.","code":""},{"path":"/reference/wine.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Wine data set — wine","text":"data results chemical analysis wines grown region Italy derived three different cultivars. analysis determined quantities 13 constituents found three types wines.","code":""},{"path":"/reference/wine.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Wine data set — wine","text":"Aeberhard, S., Coomans, D. De Vel, O. (1994). Comparative analysis statistical pattern recognition methods high dimensional settings. Pattern Recognition, 27(8), 1065-1077.","code":""},{"path":"/reference/wine.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Wine data set — wine","text":"","code":"data(wine) summary(wine) #> Alcohol Malicacid Ash Alcalinity_of_ash #> Min. :11.03 Min. :0.740 Min. :1.360 Min. :10.60 #> 1st Qu.:12.36 1st Qu.:1.603 1st Qu.:2.210 1st Qu.:17.20 #> Median :13.05 Median :1.865 Median :2.360 Median :19.50 #> Mean :13.00 Mean :2.336 Mean :2.367 Mean :19.49 #> 3rd Qu.:13.68 3rd Qu.:3.083 3rd Qu.:2.558 3rd Qu.:21.50 #> Max. :14.83 Max. :5.800 Max. :3.230 Max. :30.00 #> Magnesium Total_phenols Flavanoids Nonflavanoid_phenols #> Min. : 70.00 Min. :0.980 Min. :0.340 Min. :0.1300 #> 1st Qu.: 88.00 1st Qu.:1.742 1st Qu.:1.205 1st Qu.:0.2700 #> Median : 98.00 Median :2.355 Median :2.135 Median :0.3400 #> Mean : 99.74 Mean :2.295 Mean :2.029 Mean :0.3619 #> 3rd Qu.:107.00 3rd Qu.:2.800 3rd Qu.:2.875 3rd Qu.:0.4375 #> Max. :162.00 Max. :3.880 Max. :5.080 Max. :0.6600 #> Proanthocyanins Color_intensity Hue #> Min. :0.410 Min. : 1.280 Min. :0.4800 #> 1st Qu.:1.250 1st Qu.: 3.220 1st Qu.:0.7825 #> Median :1.555 Median : 4.690 Median :0.9650 #> Mean :1.591 Mean : 5.058 Mean :0.9574 #> 3rd Qu.:1.950 3rd Qu.: 6.200 3rd Qu.:1.1200 #> Max. :3.580 Max. :13.000 Max. :1.7100 #> X0D280_0D315_of_diluted_wines Proline y #> Min. :1.270 Min. : 278.0 Min. :1.000 #> 1st Qu.:1.938 1st Qu.: 500.5 1st Qu.:1.000 #> Median :2.780 Median : 673.5 Median :2.000 #> Mean :2.612 Mean : 746.9 Mean :1.938 #> 3rd Qu.:3.170 3rd Qu.: 985.0 3rd Qu.:3.000 #> Max. :4.000 Max. :1680.0 Max. :3.000"},{"path":"/reference/wireless.html","id":null,"dir":"Reference","previous_headings":"","what":"Wireless Indoor Localization — wireless","title":"Wireless Indoor Localization — wireless","text":"wireless data frame 2000 rows 8 columns. first 7 variables report measurements Wi-Fi signal strength received 7 Wi-Fi routers office location Pittsburgh (USA). last column indicates class labels.","code":""},{"path":"/reference/wireless.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Wireless Indoor Localization — wireless","text":"","code":"wireless"},{"path":"/reference/wireless.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Wireless Indoor Localization — wireless","text":"data frame containing following columns: V1 Signal strength router 1. V2 Signal strength router 2. V3 Signal strength router 3. V4 Signal strength router 4. V5 Signal strength router 5. V6 Signal strength router 6. V7 Signal strength router 7. V8 Group memberships, 1 4.","code":""},{"path":"/reference/wireless.html","id":"source","dir":"Reference","previous_headings":"","what":"Source","title":"Wireless Indoor Localization — wireless","text":"Bhatt, R. (2017). Wireless Indoor Localization. UCI Machine Learning Repository. https://doi.org/10.24432/C51880.","code":""},{"path":"/reference/wireless.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Wireless Indoor Localization — wireless","text":"Wi-Fi signal strength measured dBm, decibel milliwatts, expressed negative value ranging -100 0. labels correspond 4 different rooms. total, 4 groups 500 observations .","code":""},{"path":"/reference/wireless.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Wireless Indoor Localization — wireless","text":"Rohra, J.G., Perumal, B., Narayanan, S.J., Thakur, P. Bhatt, R.B. (2017). \"User Localization Indoor Environment Using Fuzzy Hybrid Particle Swarm Optimization & Gravitational Search Algorithm Neural Networks\". : Deep, K., et al. Proceedings Sixth International Conference Soft Computing Problem Solving. Advances Intelligent Systems Computing, vol 546. Springer, Singapore. https://doi.org/10.1007/978-981-10-3322-3_27","code":""},{"path":"/reference/wireless.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Wireless Indoor Localization — wireless","text":"","code":"data(wireless) summary(wireless) #> V1 V2 V3 V4 #> Min. :-74.00 Min. :-74.00 Min. :-73.00 Min. :-77.00 #> 1st Qu.:-61.00 1st Qu.:-58.00 1st Qu.:-58.00 1st Qu.:-63.00 #> Median :-55.00 Median :-56.00 Median :-55.00 Median :-56.00 #> Mean :-52.33 Mean :-55.62 Mean :-54.96 Mean :-53.57 #> 3rd Qu.:-46.00 3rd Qu.:-53.00 3rd Qu.:-51.00 3rd Qu.:-46.00 #> Max. :-10.00 Max. :-45.00 Max. :-40.00 Max. :-11.00 #> V5 V6 V7 V8 #> Min. :-89.00 Min. :-97.00 Min. :-98.00 Min. :1.00 #> 1st Qu.:-69.00 1st Qu.:-86.00 1st Qu.:-87.00 1st Qu.:1.75 #> Median :-64.00 Median :-82.00 Median :-83.00 Median :2.50 #> Mean :-62.64 Mean :-80.98 Mean :-81.73 Mean :2.50 #> 3rd Qu.:-56.00 3rd Qu.:-77.00 3rd Qu.:-78.00 3rd Qu.:3.25 #> Max. :-36.00 Max. :-61.00 Max. :-63.00 Max. :4.00"},{"path":[]},{"path":"/news/index.html","id":"new-features-1-1-3","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.3 (Development version)","text":"function rpkb() returns directly generated data set list","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-3","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.3 (Development version)","text":"packages ggplot2 foreach called full packages using importFrom nClust argument pkbc function default value usage standardGeneric (see https://adv-r.hadley.nz/s4.html#s4-generics)","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-3","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.3 (Development version)","text":"goodpractice sugestions","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-3","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.3 (Development version)","text":"Uniform notation: d dimension Eucldean space spherical data lying (d-1)-dimensional sphere Specify dimension input matrix x clustering functions. Improve explaination Average Silhouette Width","code":""},{"path":"/news/index.html","id":"quadratik-112-2024-10-29","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.1.2 (2024-10-29)","title":"QuadratiK 1.1.2 (2024-10-29)","text":"CRAN release: 2024-10-29","code":""},{"path":"/news/index.html","id":"new-features-1-1-2","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.2 (2024-10-29)","text":"Added NEWS.md file detailed information README file documentation (methods usable clearer broader audience).","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-2","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.2 (2024-10-29)","text":"Added badges rOpenSci review status, test coverage, R CMD CHECK, CRAN downloads Addition test codes testthat package. particular, testing random sampling PKBD wrapped Cauchy distribution equivalent.","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-2","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.2 (2024-10-29)","text":"scatter-plots related clustering algorithm results. computation evaluation measures (pkdc_validation function) true labels provided.","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-2","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.2 (2024-10-29)","text":"help documentation methods defined classes package can accessed directly (addition roxygen tag name aliases)","code":""},{"path":"/news/index.html","id":"quadratik-111-2024-06-05","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.1.1 (2024-06-05)","title":"QuadratiK 1.1.1 (2024-06-05)","text":"CRAN release: 2024-06-05","code":""},{"path":"/news/index.html","id":"new-features-1-1-1","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.1 (2024-06-05)","text":"Improve computation variance k-sample test Add breast_cancer wine data sets","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-1","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.1 (2024-06-05)","text":"print() summary() methods kb.test object two k-sample tests","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-1","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.1 (2024-06-05)","text":"Computation test statistics normality test","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-1","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.1 (2024-06-05)","text":"Correction typos","code":""},{"path":"/news/index.html","id":"quadratik-110-2024-05-14","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.1.0 (2024-05-14)","title":"QuadratiK 1.1.0 (2024-05-14)","text":"CRAN release: 2024-05-14","code":""},{"path":"/news/index.html","id":"new-features-1-1-0","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.0 (2024-05-14)","text":"Add GitHub links DESCRIPTION file Add unit tests Add CONTRIBUTING.md Add srr standards categories: General, Clustering Probability Distributions Add methods clustering object pkbc: show(), summary(), estract_stats(), plot() predict() Add vignette generate_rpkb.Rmd Add CODE_OF_CONDUCT.md Create GitHub page using pkgdown Add V-statistic Normality test, corresponding Critical Value Degrees Freedom function DOF_norm() Update kb.test S4 class slots, kb.test() function, unit-test codes, summary() print() method including V-statistic Add Trace statistic two k-sample tests Add computation Variance two k-sample tests","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-0","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.0 (2024-05-14)","text":"Addition GitHub workflows R-CMD-check, test-coverage pkgcheck Update codes check arguments function pk.test() Update codes check arguments function kb.test() select_h() Modify indentation assignments (using ‘<-’) following goodpractice standards Update references README.md file","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-0","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.0 (2024-05-14)","text":"Add codemeta.json .Rbuildignore Add CONTRIBUTING.md .Rbuildignore Fix warning internal function compare_qq() Add doc folder .gitignore Modify returned test statistics multiplying sample size Remove “Nonparam” centering option Normality test Critical value V-statistic divided variance function pk.test()","code":""},{"path":"/news/index.html","id":"deprecated-and-defunct-1-1-0","dir":"Changelog","previous_headings":"","what":"DEPRECATED AND DEFUNCT","title":"QuadratiK 1.1.0 (2024-05-14)","text":"summary_stat() validation() substituted estract_stats() pkbc_validation()","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-0","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.0 (2024-05-14)","text":"C++ functions set internal functions Correction typos Remove Date DESCRIPTION Update References DESCRIPTION file","code":""},{"path":"/news/index.html","id":"quadratik-100-2024-02-23","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.0.0 (2024-02-23)","title":"QuadratiK 1.0.0 (2024-02-23)","text":"CRAN release: 2024-02-23","code":""},{"path":"/news/index.html","id":"new-features-1-0-0","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.0.0 (2024-02-23)","text":"released CRAN","code":""}] +[{"path":"/articles/generate_rpkb.html","id":"note","dir":"Articles","previous_headings":"","what":"Note","title":"Random sampling from the Poisson kernel-based density","text":"limitation rejvmf method ensure computational feasibility sampler \\(\\rho\\) approaching 1.","code":""},{"path":"/articles/generate_rpkb.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Random sampling from the Poisson kernel-based density","text":"Golzy, M. Markatou, M. (2020). Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29(4), 758-770. DOI: 10.1080/10618600.2020.1740713. Sablica, L., Hornik, K. Leydold, J. (2023). “Efficient sampling PKBD distribution”, Electronic Journal Statistics, 17(2), 2180-2209. DOI: 10.1214/23-EJS2149","code":""},{"path":"/articles/Introduction.html","id":"overview","dir":"Articles","previous_headings":"","what":"Overview","title":"Introduction to the QuadratiK Package","text":"QuadratiK package provides first implementation, R Python, comprehensive set goodness--fit tests clustering technique spherical data using kernel-based quadratic distances. primary goal QuadratiK offer flexible tools testing multivariate high-dimensional data uniformity, normality, comparing two samples. package includes several novel algorithms designed handle spherical data, often encountered fields like directional statistics, geospatial data analysis, signal processing. particular, offers functions clustering spherical data efficiently, computing density value generating random samples Poisson kernel-based density.","code":""},{"path":"/articles/Introduction.html","id":"installation","dir":"Articles","previous_headings":"Overview","what":"Installation","title":"Introduction to the QuadratiK Package","text":"can install version published CRAN QuadratiK: development version GitHub: QuadratiK package also available Python PyPI Dashboard application. Usage instruction Dashboard can found https://quadratik.readthedocs.io/en/latest/user_guide/dashboard_application_usage.html.","code":"install.packages(\"QuadratiK\") library(devtools) devtools::install_github('giovsaraceno/QuadratiK-package')"},{"path":"/articles/Introduction.html","id":"citation","dir":"Articles","previous_headings":"Overview","what":"Citation","title":"Introduction to the QuadratiK Package","text":"use package research work, please cite follows: Saraceno G, Markatou M, Mukhopadhyay R, Golzy M (2024). QuadratiK: Collection Methods Constructed using Kernel-Based Quadratic Distances. https://cran.r-project.org/package=QuadratiK associated paper: Saraceno Giovanni, Markatou Marianthi, Mukhopadhyay Raktim, Golzy Mojgan (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. arXiv preprint arXiv:2402.02290.","code":"@Manual{saraceno2024QuadratiK, title = {QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances}, author = {Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year = {2024}, note = {, , } } @misc{saraceno2024package, title={Goodness-of-Fit and Clustering of Spherical Data: the QuadratiK package in R and Python}, author={Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year={2024}, eprint={2402.02290}, archivePrefix={arXiv}, primaryClass={stat.CO}, url={} }"},{"path":[]},{"path":"/articles/Introduction.html","id":"goodness-of-fit-tests","dir":"Articles","previous_headings":"Key features and basic usage","what":"Goodness-of-Fit Tests","title":"Introduction to the QuadratiK Package","text":"software implements one, two, k-sample tests goodness fit, offering efficient mathematically sound way assess fit probability distributions. tests particularly useful large, high dimensional data sets assessment fit probability models interest. provided goodness--fit tests can performed using kb.test() function. kernel-based quadratic distance tests constructed using normal kernel depends tuning parameter \\(h\\). value \\(h\\) provided, function perform select_h() algorithm searching optimal value. details please visit relative help documentations. proposed tests perform well terms level power contiguous alternatives, heavy tailed distributions higher dimensions. Test normality test null hypothesis normality \\(H_0:F=\\mathcal{N}_d(\\mu, \\Sigma)\\) needed, can specify \\(\\mu\\) \\(\\Sigma\\), otherwise standard normal distribution considered. Two-sample test case want compare two samples \\(X \\sim F\\) \\(Y \\sim G\\) null hypothesis \\(H_0:F=G\\) vs \\(H_1:F\\=G\\). k-sample test case want compare \\(k\\) samples, \\(k>2\\), \\(H_0:F_1=F_2=\\ldots=F_k\\) vs \\(H_1:F_i\\=F_j\\) \\(\\= j\\).","code":"library(QuadratiK) ?kb.test ?select_h x <- matrix(rnorm(100), ncol = 2) # Does x come from a multivariate standard normal distribution? kb.test(x, h=0.4) ## ## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ ## Test Statistic: 3.298167 1.714125 ## Critical Value: 1.815007 8.901682 ## H0 is rejected: TRUE FALSE ## Selected tuning parameter h: 0.4 x <- matrix(rnorm(100,4), ncol = 2) # Does x come from the specified multivariate normal distribution? kb.test(x, mu_hat = c(4,4), Sigma_hat = diag(2), h = 0.4) ## ## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ ## Test Statistic: -0.8648377 0.7204634 ## Critical Value: 1.803737 8.901682 ## H0 is rejected: FALSE FALSE ## Selected tuning parameter h: 0.4 x <- matrix(rnorm(100), ncol = 2) y <- matrix(rnorm(100,mean = 5), ncol = 2) # Do x and y come from the same distribution? kb.test(x, y, h = 0.4) ## ## Kernel-based quadratic distance two-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 5.668556 10.60736 ## Critical Value: 0.7071142 1.324683 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4 x1 <- matrix(rnorm(100), ncol = 2) x2 <- matrix(rnorm(100), ncol = 2) x3 <- matrix(rnorm(100, mean = 5), ncol = 2) y <- rep(c(1, 2, 3), each = 50) # Do x1, x2 and x3 come from the same distribution? x <- rbind(x1, x2, x3) kb.test(x, y, h = 0.4) ## ## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 7.540686 10.45173 ## Critical Value: 0.7273207 1.008851 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4"},{"path":"/articles/Introduction.html","id":"test-for-uniformity-on-the-sphere","dir":"Articles","previous_headings":"Key features and basic usage > Goodness-of-Fit Tests","what":"Test for uniformity on the sphere","title":"Introduction to the QuadratiK Package","text":"Expanded capabilities include supporting tests uniformity (d-1)-dimensional Sphere based Poisson kernel. Poisson kernel depends concentration parameter \\(\\rho\\) location vector \\(\\mu\\). details please visit help documentation pk.test() function. test null hypothesis uniformity \\((d-1)\\)-dimensional sphere \\(\\mathcal{S}^{d-1} = \\{x \\\\mathbb{R}^d : ||x||=1 \\}\\)","code":"?pk.test # Generate points on the sphere from the uniform ditribution x <- sample_hypersphere(d = 3, n_points = 100) # Does x come from the uniform distribution on the sphere? pk.test(x, rho = 0.7) ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Selected consentration parameter rho: 0.7 ## ## U-statistic: ## ## H0 is rejected: FALSE ## Statistic Un: -0.04448323 ## Critical value: 1.814438 ## ## V-statistic: ## ## H0 is rejected: FALSE ## Statistic Vn: 17.75278 ## Critical value: 23.22949"},{"path":"/articles/Introduction.html","id":"poisson-kernel-based-distribution-pkbd","dir":"Articles","previous_headings":"Key features and basic usage","what":"Poisson kernel-based distribution (PKBD)","title":"Introduction to the QuadratiK Package","text":"package offers functions computing density value generating random samples PKBD. Poisson kernel-based densities based normalized Poisson kernel defined \\((d-1)\\)-dimensional unit sphere. details please visit help documentation dpkb() rpkb() functions. Example","code":"?dpkb ?rpkb mu <- c(1,0,0) rho <- 0.9 x <- rpkb(n = 100, mu = mu, rho = rho) head(x) ## [,1] [,2] [,3] ## [1,] 0.9889098 0.144026221 0.03624804 ## [2,] 0.9979113 -0.059248435 0.02574338 ## [3,] 0.9996806 0.010460509 -0.02300656 ## [4,] 0.9963903 -0.001169547 0.08488232 ## [5,] 0.9933974 0.113643034 0.01570940 ## [6,] 0.9756830 -0.079833339 0.20413065 dens_x <- dpkb(x, mu = mu, rho = rho) head(dens_x) ## [,1] ## [1,] 2.915268 ## [2,] 9.367626 ## [3,] 13.903580 ## [4,] 7.135393 ## [5,] 4.670202 ## [6,] 1.212625"},{"path":"/articles/Introduction.html","id":"clustering-algorithm-for-spherical-data","dir":"Articles","previous_headings":"Key features and basic usage","what":"Clustering Algorithm for Spherical Data","title":"Introduction to the QuadratiK Package","text":"package incorporates unique clustering algorithm specifically tailored spherical data especially useful presence noise data presence non-negligible overlap clusters. algorithm leverages mixture Poisson kernel-based densities Sphere, enabling effective clustering spherical data data spherically transformed. details please visit help documentation pkbc() function. Example software includes additional graphical functions, aiding users validating representing cluster results well enhancing interpretability usability analysis.","code":"?pkbc # Generate 3 samples from the PKBD with different location directions x1 <- rpkb(n = 100, mu = c(1,0,0), rho = rho) x2 <- rpkb(n = 100, mu = c(-1,0,0), rho = rho) x3 <- rpkb(n = 100, mu = c(0,0,1), rho = rho) x <- rbind(x1, x2, x3) # Perform the clustering algorithm # Serch for 2, 3 or 4 clusters cluster_res <- pkbc(dat = x, nClust = c(2, 3, 4)) summary(cluster_res) ## Poisson Kernel-Based Clustering on the Sphere (pkbc) Results ## ------------------------------------------------------------ ## ## Summary: ## nClust LogLik WCSS ## [1,] 2 -657.1994 398.0559 ## [2,] 3 -392.3710 328.9308 ## [3,] 4 -380.0957 328.7125 ## ## Results for 2 clusters: ## Estimated Mixing Proportions (alpha): ## [1] 0.2724978 0.7275022 ## ## Clustering table: ## ## 1 2 ## 85 215 ## ## ## Results for 3 clusters: ## Estimated Mixing Proportions (alpha): ## [1] 0.3252225 0.3426152 0.3321622 ## ## Clustering table: ## ## 1 2 3 ## 94 106 100 ## ## ## Results for 4 clusters: ## Estimated Mixing Proportions (alpha): ## [1] 0.332925621 0.008442903 0.343746968 0.314884508 ## ## Clustering table: ## ## 1 2 3 4 ## 100 3 106 91 # Predict the membership of new data with respect to the clustering results x_new <- rpkb(n = 10, mu = c(1,0,0), rho = rho) memb_mew <- predict(cluster_res, k = 3, newdata = x_new) memb_mew$Memb ## [1] 3 3 3 3 3 3 3 3 1 3 # Compute measures for evaluating the clustering results val_res <- pkbc_validation(cluster_res) val_res ## $metrics ## 2 3 4 ## ASW 0.4763106 0.650476 0.4810613 ## ## $IGP ## $IGP[[1]] ## NULL ## ## $IGP[[2]] ## [1] 1.0000000 0.9948454 ## ## $IGP[[3]] ## [1] 1.0000000 1.0000000 0.9803922 ## ## $IGP[[4]] ## [1] 0.9803922 1.0000000 1.0000000 1.0000000 # Plot method for the pkbc object: # - scatter plot of data points on the sphere # - elbow plot for helping the choice of the number of clusters plot(cluster_res)"},{"path":"/articles/Introduction.html","id":"additional-resources","dir":"Articles","previous_headings":"","what":"Additional Resources","title":"Introduction to the QuadratiK Package","text":"detailed information QuadratiK package, can explore following resources: Package Documentation CRAN – Official package documentation CRAN. GitHub Repository – GitHub repository development version, issues, community discussions. QuadratiK Package Website – dedicated website additional tutorials examples. ’re new package, recommend starting available vignettes: Two-sample test k-sample test Test uniformity Clustering sphere Generate PKBD","code":""},{"path":"/articles/Introduction.html","id":"references","dir":"Articles","previous_headings":"Additional Resources","what":"References","title":"Introduction to the QuadratiK Package","text":"information methods implemented package, refer associated research papers: Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” arXiv:2407.16374 Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: 10.5705/ss.202022.0347. Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713.","code":""},{"path":"/articles/kSample_test.html","id":"note","dir":"Articles","previous_headings":"","what":"Note","title":"k-sample test","text":"value \\(h\\) provided kb.test(), function performs function select_h automatic search optimal value \\(h\\) use. . following code shows usage, executed since obtain results. details visit help documentation select_h() function. kb.test() function, critical value can computed subsampling, bootstrap permutation algorithm. default method set subsampling since needs less computational time. details sampling algorithm see documentation kb.test() function following reference. proposed tests exhibit high power asymmetric alternatives close null hypothesis small sample size, well \\(k \\ge 3\\) sample comparison, dimension \\(d>2\\) sample sizes. details, see extensive simulation study reported following reference.","code":"k_test_h <- kb.test(x=x, y=y) help(select_h)"},{"path":"/articles/kSample_test.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"k-sample test","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.”https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":"/articles/TwoSample_test.html","id":"non-parametric-two-sample-test","dir":"Articles","previous_headings":"","what":"Non-parametric two-sample test","title":"Non-parametric Two-sample test","text":"Let \\(x_1, x_2, \\ldots, x_{n_1} \\sim F\\) \\(y_1, y_2, \\ldots, y_{n_2} \\sim G\\) random samples distributions \\(F\\) \\(G\\), respectively. test null hypothesis two samples generated unknown distribution \\(\\bar{F}\\), : \\[ H_0: F = G = \\bar{F} \\] versus alternative hypothesis two distributions different, \\[ H_1: F \\= G. \\] compute kernel-based quadratic distance (KBQD) tests \\[ \\mathrm{trace}_n = \\frac{1}{n_1(n_1-1)}\\sum_{=1}^{n_1} \\sum_{j \\=}^{n_1} K_{\\bar{F}}(\\mathbf{x}_i,\\mathbf{x}_j) + \\frac{1}{n_2(n_2-1)}\\sum_{=1}^{n_2} \\sum_{j \\=}^{n_2} K_{\\bar{F}}(\\mathbf{y}_i,\\mathbf{y}_j), \\] \\[ D_{n} = \\frac{1}{n_1(n_1-1)}\\sum_{=1}^{n_1} \\sum_{j \\=}^{n_2} K_{\\bar{F}}(\\mathbf{x}_i,\\mathbf{x}_j) - \\frac{2}{n_1 n_2}\\sum_{=1}^{n_1} \\sum_{j =1}^{n_2} K_{\\bar{F}}(\\mathbf{x}_i,\\mathbf{y}_j) + \\frac{1}{n_2(n_2-1)}\\sum_{=1}^{n_2} \\sum_{j \\=}^{n_2} K_{\\bar{F}}(\\mathbf{y}_i,\\mathbf{y}_j). \\nonumber \\] \\(K_{\\bar{F}}\\) denotes Normal kernel \\(K\\) defined \\[ K(\\mathbf{s}, \\mathbf{t}) = (2 \\pi)^{-d/2} \\left(\\det{\\mathbf{\\Sigma}_h}\\right)^{-\\frac{1}{2}} \\exp\\left\\{-\\frac{1}{2}(\\mathbf{s} - \\mathbf{t})^\\top \\mathbf{\\Sigma}_h^{-1}(\\mathbf{s} - \\mathbf{t})\\right\\}, \\] every \\(\\mathbf{s}, \\mathbf{t} \\\\mathbb{R}^d \\times \\mathbb{R}^d\\), covariance matrix \\(\\mathbf{\\Sigma}_h = h^2 \\) tuning parameter \\(h\\), centered respect \\(\\bar{F} = \\frac{n_1F + n_2G}{n_1 + n_2}\\). information centering kernel, see documentation kb.test() function. KBQD tests exhibit high power asymmetric alternatives close null hypothesis small sample size. consider example scenario. generate samples \\(x = (x_1, \\ldots,x_n)\\) standard normal distribution \\(N_d(0,I_d)\\) \\(y = (y_1, \\ldots,y_n)\\) skew-normal distribution \\(SN_d(0,I_d, \\lambda)\\), \\(d=4\\), \\(n=100\\) \\(\\lambda= (0.5,\\ldots,0.5)\\). two-sample test can performed providing two samples compared x y kb.test() function. value \\(h\\) provided, function automatically performs function select_h. can display chosen optimal value \\(h\\) together power plot obtained versus considered \\(h\\), alternatives \\(\\delta\\) select_h() function. details visit help documentation select_h() function. two-sample case, summary function provides results test list tables standard descriptive statistics variable, computed per group overall. Additionally, generates qq-plots comparing quantiles two groups variable.","code":"help(kb.test) library(sn) library(mvtnorm) library(QuadratiK) n <- 100 d <- 4 skewness_y <- 0.5 set.seed(2468) x_2 <- rmvnorm(n, mean = rep(0,d)) y_2 <- rmsn(n=n, xi=0, Omega = diag(d), alpha=rep(skewness_y,d)) set.seed(2468) two_test <- kb.test(x=x_2, y=y_2) two_test ## ## Kernel-based quadratic distance two-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 1.679763 2.312427 ## Critical Value: 1.087455 1.497868 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 1.6 two_test@h$h_sel ## [1] 1.6 two_test@h$power.plot help(select_h) summary_two <- summary(two_test) ## ## Kernel-based quadratic distance two-sample test ## Statistic Value Critical_Value Reject_H0 ## 1 Dn 1.679763 1.087455 TRUE ## 2 Trace 2.312427 1.497868 TRUE summary_two$summary_tables ## [[1]] ## Group 1 Group 2 Overall ## mean -0.005393522 0.3197861 0.1571963 ## sd 1.039119207 0.9094193 0.9875137 ## median -0.019317321 0.4448058 0.1601955 ## IQR 1.562613453 1.3612937 1.4292426 ## min -2.675477796 -1.4256211 -2.6754778 ## max 2.151784802 2.9375947 2.9375947 ## ## [[2]] ## Group 1 Group 2 Overall ## mean -0.10005083 0.1936138 0.04678149 ## sd 1.10476260 1.0556439 1.08777010 ## median -0.07955849 0.2235325 0.10130247 ## IQR 1.48816630 1.4716179 1.41498342 ## min -3.22222061 -2.4336333 -3.22222061 ## max 2.50192633 2.6879362 2.68793623 ## ## [[3]] ## Group 1 Group 2 Overall ## mean -0.006524772 0.1701261 0.08180065 ## sd 0.958942739 0.9524916 0.95742170 ## median -0.039301279 0.1887394 0.11877637 ## IQR 1.329868172 1.4657107 1.40312077 ## min -2.860006689 -2.1762183 -2.86000669 ## max 1.923763114 2.4237195 2.42371949 ## ## [[4]] ## Group 1 Group 2 Overall ## mean -0.06757686 0.2236458 0.07803449 ## sd 0.98684958 0.9862135 0.99481815 ## median -0.03258747 0.1097711 0.05517931 ## IQR 1.30933016 1.4088334 1.39890664 ## min -2.29625537 -2.1827156 -2.29625537 ## max 2.40795077 2.9929942 2.99299420"},{"path":"/articles/TwoSample_test.html","id":"select-h","dir":"Articles","previous_headings":"Non-parametric two-sample test","what":"Select h","title":"Non-parametric Two-sample test","text":"search optimal value tuning parameter \\(h\\) can performed independently test computation using select_h function. requires two samples, provided x y, considered family alternatives. code evaluated since obtain results.","code":"set.seed(2468) two_test_h <- select_h(x=x_2, y=y_2, alternative = \"skewness\")"},{"path":"/articles/TwoSample_test.html","id":"note","dir":"Articles","previous_headings":"Non-parametric two-sample test","what":"Note","title":"Non-parametric Two-sample test","text":"Notice test statistics two-sample testing coincide \\(k\\)-sample test statistics \\(k=2\\). Hence, alternatively two sample tests can performed providing two samples together x indicating membership groups argument y. See k-sample test vignette details. kb.test() function, critical value can computed subsampling, bootstrap permutation algorithm. default method set subsampling since needs less computational time. details sampling algorithm see documentation kb.test() function. details level power performance considered two-sample tests, see extensive simulation study reported following reference.","code":"x_pool <- rbind(x_2,y_2) y_memb <- rep(c(1,2),each=n) h <- two_test@h$h_sel set.seed(2468) kb.test(x=x_pool, y=y_memb, h=h) ## ## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ ## Test Statistic: 1.679763 2.312427 ## Critical Value: 1.087455 1.497868 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 1.6"},{"path":"/articles/TwoSample_test.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Non-parametric Two-sample test","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.”https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":"/articles/uniformity.html","id":"uniformity-test-on-the-sphere","dir":"Articles","previous_headings":"","what":"Uniformity test on the Sphere","title":"Uniformity test on the Sphere","text":"Let \\(x_1, x_2, \\ldots, x_n \\sim F\\) random sample empirical distribution function \\(\\hat{F}\\). test null hypothesis uniformity \\(d\\)-dimensional sphere, .e., \\(H_0: F = G\\) \\(G\\) uniform distribution \\(d\\)-dimensional sphere \\(\\mathcal{S}^{d-1}\\). compute U-statistic estimate sample KBQD (Kernel-Based Quadratic Distance) \\[ U_{n}=\\frac{2}{n(n-1)}\\sum_{=2}^{n}\\sum_{j=1}^{-1}K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}), \\] first test statistic given \\[T_{n}=\\frac{U_{n}}{\\sqrt{Var(U_{n})}},\\] \\[ Var(U_{n})= \\frac{2}{n(n-1)}\\left[\\frac{1+\\rho^{2}}{(1-\\rho^{2})^{d-1}}-1\\right], \\] V-statistic estimate KBQD\\[ V_{n} = \\frac{1}{n}\\sum_{=1}^{n}\\sum_{j=1}^{n}K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}), \\] \\(K_{cen}\\) denotes Poisson kernel \\(K_\\rho\\) centered respect uniform distribution \\(d\\)-dimensional sphere, \\[ K_{cen}(\\mathbf{u}, \\mathbf{v}) = K_\\rho(\\mathbf{u}, \\mathbf{v}) -1 \\] \\[ K_\\rho(\\mathbf{u}, \\mathbf{v}) = \\frac{1-\\rho^{2}}{\\left(1+\\rho^{2}-2\\rho (\\mathbf{u}\\cdot \\mathbf{v})\\right)^{d/2}}, \\] every \\(\\mathbf{u}, \\mathbf{v} \\\\mathcal{S}^{d-1} \\times \\mathcal{S}^{d-1}\\). generated \\(n=200\\) observations uniform distribution \\(S^{d-1}\\), \\(d=3\\). pk.test used testing uniformity generated sample, providing data matrix x value concentration parameter rho. pk.test() function returns object class pk.test. show function displays computed statistics corresponding critical values. test correctly reject null hypothesis uniformity. summary function pk.test output object provides results performed test, generates figure showing qq-plots uniform distribution variable table standard descriptive statistics. figure automatically generated summary function result test uniformity displays qq-plots given samples uniform distribution table standard descriptive statistics variable.","code":"library(QuadratiK) n <- 200 d <- 3 set.seed(2468) z <- matrix(rnorm(n * d), n, d) dat_sphere <- z/sqrt(rowSums(z^2)) rho <- 0.7 set.seed(2468) res_unif <- pk.test(x=dat_sphere, rho=rho) show(res_unif) ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Selected consentration parameter rho: 0.7 ## ## U-statistic: ## ## H0 is rejected: FALSE ## Statistic Un: -0.9756673 ## Critical value: 1.725683 ## ## V-statistic: ## ## H0 is rejected: FALSE ## Statistic Vn: 14.89598 ## Critical value: 23.22949 summary_unif <- summary(res_unif) ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Test_Statistics Critical_Value Reject_H0 ## 1 -0.9756673 1.725683 FALSE ## 2 14.8959834 23.229487 FALSE"},{"path":"/articles/uniformity.html","id":"multimodal-example","dir":"Articles","previous_headings":"Uniformity test on the Sphere","what":"Multimodal example","title":"Uniformity test on the Sphere","text":"Poisson kernel-based test uniformity exhibits excellent results especially case multimodal distributions. example, generate data points mixture 4 von Mises-Fisher distributions 2 dimensions. direction mean vectors distributions set (1, 0), (0, 1), (-1, 0), (0, -1), concentration parameter \\(\\kappa=5\\). now compare results pk.test function Ajne Bingham tests using sphunif package. example, Poisson kernel-based test statistics reject null hypothesis, Bingham test Ajne test obtain p-values equal 0.0532 0.3156, respectively.","code":"# Load necessary libraries library(movMF) library(sphunif) set.seed(2468) # Define the mean directions of the 4 von Mises-Fisher distributions means <- rbind( c(1, 0), c(0, 1), c(-1,0), c(0, -1) ) # Define the concentration parameter (kappa) kappa <- 5 # Generate 100 samples from a mixture of 4 von Mises-Fisher distributions samples <- matrix(rmovMF(100, theta = kappa * means), ncol=2) # Run the pk.test from the QuadratiK package to test the data pk_test_result <- pk.test(samples, rho=0.8) # Run the Bingham and Ajne tests from the sphunif package other_test_result <- unif_test(samples, type = c(\"Bingham\", \"Ajne\")) pk_test_result ## ## Poisson Kernel-based quadratic distance test of ## Uniformity on the Sphere ## Selected consentration parameter rho: 0.8 ## ## U-statistic: ## ## H0 is rejected: TRUE ## Statistic Un: 2.692594 ## Critical value: 1.612663 ## ## V-statistic: ## ## H0 is rejected: TRUE ## Statistic Vn: 15.14426 ## Critical value: 12.8308 other_test_result ## $Bingham ## ## Bingham test of circular uniformity ## ## data: samples ## statistic = 5.8674, p-value = 0.0532 ## alternative hypothesis: scatter matrix different from constant ## ## ## $Ajne ## ## Ajne test of circular uniformity ## ## data: samples ## statistic = 0.28268, p-value = 0.3156 ## alternative hypothesis: any non-axial alternative to circular uniformity"},{"path":"/articles/uniformity.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Uniformity test on the Sphere","text":"Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi:10.5705/ss.202022.0347.","code":""},{"path":"/articles/wireless_clustering.html","id":"the-wireless-indoor-localization-data","dir":"Articles","previous_headings":"","what":"The Wireless Indoor Localization Data","title":"Clustering algorithm on the Wireless data","text":"consider Wireless Indoor Localization Data Set, publicly available UCI Machine Learning Repository’s website. data set used study performance different indoor localization algorithms. available within QuadratiK package wireless. Wireless Indoor Localization data set contains measurements Wi-Fi signal strength different indoor rooms. consists data frame 2000 rows 8 columns. first 7 variables report values Wi-Fi signal strength received 7 different Wi-Fi routers office location Pittsburgh (USA). last column indicates class labels, 1 4, indicating different rooms. Notice , Wi-Fi signal strength measured dBm, decibel milliwatts, expressed negative value ranging -100 0. total, 500 observations room. many wireless applications, relative signal strengths across routers relevant underlying spatial patterns device positioning absolute magnitudes. Additionally, absolute signal strength can affected noise, device orientation environmental factors. case, reasonable consider spherically transformed data points using L2 normalization. transformation maps data onto surface 6-dimensional sphere, ensuring observation uniform length. Given absolute signal strength critical research question, spherical representation provides meaningful interpretable framework studying data set. general, appropriate consider spherically transformed data points : () absolute length measurements irrelevant noisy; (ii) interested relative distributions (angular relationships) data points. perform clustering algorithm wireless data set. consider \\(K= 3, 4, 5\\) possible values number clusters. pkbc function creates object class pkbc containing clustering results value number clusters considered. guide choice number clusters, function pkbc_validation provides cluster validation measures graphical tools. Specifically, returns object InGroup Proportion (IGP), metrics, table computed evaluation measures. table includes Average Silhouette Width (ASW) , true labels provided, measures adjusted rand index (ARI), Macro-Precision Macro-Recall. clusters identified \\(k=4\\) achieve high values ARI, Macro Precision Macro Recall. brief description reported evaluation measures, corresponding references, please visit help documentation pkbc_validation function. plot method pkbc class can used display scatter plot data points Elbow plot computed within-cluster sum squares values. scatter plot, \\(d=2\\) \\(d=3\\), observations displayed unit circle unit sphere, respectively. \\(d>3\\), spherical PCA applied data set, first 3 principal components used visualizing data points sphere. generated scatter plot specified number clusters, data points colored assigned membership. Elbow plots reported metrics suggest \\(K=4\\) number clusters. accordance known ground truth. Additionally, true labels available provided plot method, scatter plot display data points colored respect true labels assigned memberships two adjacent plots. plot points using principal components also shows identified cluster follows initial labels. number clusters selected, method summary_stat can used obtain additional summary information respect clustering results. particular, function provides mean, standard deviation, median, inter-quantile range, minimum maximum computed variable, overall assigned membership.","code":"library(QuadratiK) head(wireless) ## V1 V2 V3 V4 V5 V6 V7 V8 ## 1 -64 -56 -61 -66 -71 -82 -81 1 ## 2 -68 -57 -61 -65 -71 -85 -85 1 ## 3 -63 -60 -60 -67 -76 -85 -84 1 ## 4 -61 -60 -68 -62 -77 -90 -80 1 ## 5 -63 -65 -60 -63 -77 -81 -87 1 ## 6 -64 -55 -63 -66 -76 -88 -83 1 wire <- wireless[,-8] labels <- wireless[,8] wire_norm <- wire/sqrt(rowSums(wire^2)) set.seed(2468) res_pk <- pkbc(as.matrix(wire_norm),3:5) set.seed(2468) res_validation <- pkbc_validation(res_pk, true_label = labels) res_validation$IGP ## [[1]] ## NULL ## ## [[2]] ## NULL ## ## [[3]] ## [1] 0.9860558 0.9706215 0.9429038 ## ## [[4]] ## [1] 0.9662698 0.9733607 0.9526627 0.9880240 ## ## [[5]] ## [1] 0.9713701 0.7727273 0.9880240 0.9639831 0.9433498 round(res_validation$metrics, 5) ## 3 4 5 ## ASW 0.35326 0.38031 0.30240 ## ARI 0.69526 0.94031 0.91409 ## Macro_Precision 0.18389 0.97719 0.00120 ## Macro_Recall 0.26000 0.97700 0.00150 help(pkbc_validation) plot(res_pk, k = 4) plot(res_pk, k=4, true_label = labels) summary_clust <- stats_clusters(res_pk,4) summary_clust ## [[1]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.33976246 -0.23423419 -0.29470203 -0.34377323 -0.30363137 ## sd 0.01283014 0.05324085 0.01581506 0.01298448 0.05251967 ## median -0.33952107 -0.24604104 -0.29625176 -0.34242063 -0.31847692 ## IQR 0.01674475 0.03014941 0.02182651 0.01835030 0.06412536 ## min -0.37455424 -0.30643954 -0.34994496 -0.39357081 -0.39357081 ## max -0.29339739 -0.06308050 -0.23847076 -0.31226867 -0.06308050 ## ## [[2]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.30636324 -0.35918738 -0.32636022 -0.31540450 -0.32655905 ## sd 0.01392354 0.02052666 0.01842337 0.01608549 0.02635941 ## median -0.30600005 -0.35809184 -0.32487137 -0.31538238 -0.32221067 ## IQR 0.01958541 0.02460137 0.02426854 0.02263003 0.03624468 ## min -0.35600342 -0.45578394 -0.40026324 -0.36631517 -0.45578394 ## max -0.26903743 -0.30605235 -0.27775661 -0.27586207 -0.26903743 ## ## [[3]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.32905221 -0.35786158 -0.31417147 -0.28929074 -0.32231793 ## sd 0.01563476 0.02459888 0.01710971 0.02094437 0.03163402 ## median -0.32915518 -0.35575396 -0.31237754 -0.29072716 -0.32209343 ## IQR 0.01957377 0.03437142 0.02422092 0.02851547 0.03973413 ## min -0.38248214 -0.43623816 -0.38676345 -0.33639128 -0.43623816 ## max -0.28256746 -0.28237524 -0.26460827 -0.23485570 -0.23485570 ## ## [[4]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.34918727 -0.24111635 -0.30079691 -0.35013402 -0.31082318 ## sd 0.01475930 0.04839750 0.01973759 0.01679759 0.05251032 ## median -0.34858664 -0.24898566 -0.30107248 -0.35004522 -0.32519736 ## IQR 0.01898212 0.03356955 0.02672633 0.02286452 0.06907237 ## min -0.40129910 -0.31690470 -0.35542814 -0.41273961 -0.41273961 ## max -0.30779911 -0.07399736 -0.23119131 -0.30906423 -0.07399736 ## ## [[5]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.38226045 -0.43319403 -0.37583631 -0.28257085 -0.36807675 ## sd 0.01770626 0.02787579 0.01814674 0.02110273 0.05820777 ## median -0.37991233 -0.43009295 -0.37549563 -0.28337199 -0.37738801 ## IQR 0.02531565 0.04108696 0.02541214 0.02710382 0.06966093 ## min -0.45385132 -0.52580164 -0.42808192 -0.34353824 -0.52580164 ## max -0.34027255 -0.36916034 -0.33333333 -0.21464345 -0.21464345 ## ## [[6]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.45125600 -0.46327257 -0.48366352 -0.49713921 -0.47391075 ## sd 0.01446455 0.02195263 0.01814601 0.01412842 0.02488735 ## median -0.45081811 -0.46519225 -0.48319020 -0.49649227 -0.47411000 ## IQR 0.01833254 0.02770315 0.02503768 0.01895624 0.03825696 ## min -0.51475369 -0.53938031 -0.53130759 -0.53840040 -0.53938031 ## max -0.41652096 -0.40313011 -0.44107352 -0.45551482 -0.40313011 ## ## [[7]] ## Group 1 Group 2 Group 3 Group 4 Overall ## mean -0.45728346 -0.46863418 -0.48984740 -0.49701329 -0.47827795 ## sd 0.01710219 0.02396734 0.01993870 0.01587742 0.02515570 ## median -0.45703017 -0.46920446 -0.49013407 -0.49711582 -0.47914650 ## IQR 0.02311965 0.03050579 0.02784124 0.02030659 0.03643335 ## min -0.50728615 -0.56835661 -0.54571977 -0.53648350 -0.56835661 ## max -0.41326597 -0.38995633 -0.43815927 -0.44423198 -0.38995633"},{"path":"/articles/wireless_clustering.html","id":"note","dir":"Articles","previous_headings":"","what":"Note","title":"Clustering algorithm on the Wireless data","text":"number cluster k provided plot function, one scatter plot displayed possible number clusters available object class pkbc.","code":""},{"path":"/articles/wireless_clustering.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"Clustering algorithm on the Wireless data","text":"Golzy, M. Markatou, M. (2020). “Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling,” Journal Computational Graphical Statistics, 29(4), 758-770. DOI: 10.1080/10618600.2020.1740713.","code":""},{"path":"/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Giovanni Saraceno. Author, maintainer. ORCID 000-0002-1753-2367 Marianthi Markatou. Author. Raktim Mukhopadhyay. Author. Mojgan Golzy. Author. Hingee Kassel. Reviewer.","code":""},{"path":"/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Saraceno G, Markatou M, Mukhopadhyay R, Golzy M (2024). QuadratiK: Collection Methods Constructed using Kernel-Based Quadratic Distances. Version 1.1.2, https://github.com/giovsaraceno/QuadratiK-package. Markatou M, Saraceno G (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” arXiv preprint. https://doi.org/10.48550/arXiv.2407.16374.","code":"@Manual{, title = {QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances}, author = {Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year = {2024}, note = {Version 1.1.2}, url = {https://github.com/giovsaraceno/QuadratiK-package}, } @Article{, title = {A Unified Framework for Multivariate Two- and k-Sample Kernel-based Quadratic Distance Goodness-of-Fit Tests}, author = {Marianthi Markatou and Giovanni Saraceno}, journal = {arXiv preprint}, year = {2024}, note = {https://doi.org/10.48550/arXiv.2407.16374}, }"},{"path":[]},{"path":"/CODE_OF_CONDUCT.html","id":"our-pledge","dir":"","previous_headings":"","what":"Our Pledge","title":"Contributor Covenant Code of Conduct","text":"interest fostering open welcoming environment, contributors maintainers pledge make participation project community harassment-free experience everyone, regardless age, body size, disability, ethnicity, sex characteristics, gender identity expression, level experience, education, socio-economic status, nationality, personal appearance, race, religion, sexual identity orientation.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"our-standards","dir":"","previous_headings":"","what":"Our Standards","title":"Contributor Covenant Code of Conduct","text":"Examples behavior contributes creating positive environment include: Using welcoming inclusive language respectful differing viewpoints experiences Gracefully accepting constructive criticism Focusing best community Showing empathy towards community members Examples unacceptable behavior participants include: use sexualized language imagery unwelcome sexual attention advances Trolling, insulting/derogatory comments, personal political attacks Public private harassment Publishing others’ private information, physical electronic address, without explicit permission conduct reasonably considered inappropriate professional setting","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"our-responsibilities","dir":"","previous_headings":"","what":"Our Responsibilities","title":"Contributor Covenant Code of Conduct","text":"Project maintainers responsible clarifying standards acceptable behavior expected take appropriate fair corrective action response instances unacceptable behavior. Project maintainers right responsibility remove, edit, reject comments, commits, code, wiki edits, issues, contributions aligned Code Conduct, ban temporarily permanently contributor behaviors deem inappropriate, threatening, offensive, harmful.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"scope","dir":"","previous_headings":"","what":"Scope","title":"Contributor Covenant Code of Conduct","text":"Code Conduct applies within project spaces, also applies individual representing project community public spaces. Examples representing project community include using official project e-mail address, posting via official social media account, acting appointed representative online offline event. Representation project may defined clarified project maintainers.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"enforcement","dir":"","previous_headings":"","what":"Enforcement","title":"Contributor Covenant Code of Conduct","text":"Instances abusive, harassing, otherwise unacceptable behavior may reported contacting project team gsaracen buffalo.edu. complaints reviewed investigated result response deemed necessary appropriate circumstances. project team obligated maintain confidentiality regard reporter incident. details specific enforcement policies may posted separately. Project maintainers follow enforce Code Conduct good faith may face temporary permanent repercussions determined members project’s leadership.","code":""},{"path":"/CODE_OF_CONDUCT.html","id":"attribution","dir":"","previous_headings":"","what":"Attribution","title":"Contributor Covenant Code of Conduct","text":"Code Conduct adapted Contributor Covenant, version 1.4, available https://www.contributor-covenant.org/version/1/4/code--conduct.html answers common questions code conduct, see https://www.contributor-covenant.org/faq","code":""},{"path":"/CONTRIBUTING.html","id":null,"dir":"","previous_headings":"","what":"Contributing to QuadratiK package","title":"Contributing to QuadratiK package","text":"Version: 1.1.2 First , thanks considering contributing QuadratiK! 👍 QuadratiK comprehensive statistical analysis package providing comprehensive set goodness--fit tests clustering technique spherical data using kernel-based quadratic distances.","code":""},{"path":"/CONTRIBUTING.html","id":"code-of-conduct","dir":"","previous_headings":"","what":"Code of conduct","title":"Contributing to QuadratiK package","text":"Please note project released Contributor Code Conduct. participating project agree abide terms.","code":""},{"path":"/CONTRIBUTING.html","id":"share-the-love-️","dir":"","previous_headings":"Code of conduct","what":"Share the love ❤️","title":"Contributing to QuadratiK package","text":"Think QuadratiK useful? Let others discover , telling person, via Twitter, ResearchGate blog post. Using QuadratiK paper writing? Cite .","code":""},{"path":"/CONTRIBUTING.html","id":"ask-a-question","dir":"","previous_headings":"Code of conduct","what":"Ask a question","title":"Contributing to QuadratiK package","text":"Using QuadratiK got stuck? Browse documentation see can find solution. Post questions requests issue GitHub. ’ll try best address , questions often lead better documentation discovery bugs. Want ask question private? Contact package maintainer gsaracen@buffalo.edu.","code":""},{"path":"/CONTRIBUTING.html","id":"propose-an-idea-","dir":"","previous_headings":"Code of conduct","what":"Propose an idea 💡","title":"Contributing to QuadratiK package","text":"idea new QuadratiK feature? Take look documentation issue list see isn’t included suggested yet. , suggest idea issue GitHub. Please Explain detail work. Keep scope narrow possible. Functions described comments near code translated documentation using roxygen2. Write documentation using roxygen2 package. Test changes goodpractice submitting. try follow GitHub flow development. Fork repo clone computer. learn process, see guide. forked cloned project since worked , pull changes original repo clone using git pull upstream master. Open RStudio project file (.Rproj). Write code. Test code (bonus points adding unit tests). Document code (see function documentation ). Check code devtools::check() aim 0 errors warnings. Commit push changes. Submit pull request.","code":""},{"path":"/CONTRIBUTING.html","id":"report-a-bug","dir":"","previous_headings":"Code of conduct","what":"Report a bug","title":"Contributing to QuadratiK package","text":"Using QuadratiK discovered bug? ’s annoying! Don’t let others experience report issue GitHub can fix . good bug report makes easier us , please include: operating system name version (e.g. Mac OS 10.13.6). details local setup might helpful troubleshooting. Detailed steps reproduce bug.","code":""},{"path":"/CONTRIBUTING.html","id":"the-website","dir":"","previous_headings":"Code of conduct > Report a bug","what":"The website","title":"Contributing to QuadratiK package","text":"website generated pkgdown. means don’t write html: content pulled together documentation code, vignettes, Markdown files, package DESCRIPTION _pkgdown.yml settings. know way around pkgdown, can propose file change improve documentation. , report issue can point right direction.","code":""},{"path":"/CONTRIBUTING.html","id":"workflow","dir":"","previous_headings":"","what":"Workflow","title":"Contributing to QuadratiK package","text":"QuadratiK package engineered combination R C++ code balance usability high-speed computation, catering complex data analysis needs. package undergone extensive testing various examples. Continuous integration practices employed maintain high standards reliability functionality. QuadratiK first released CRAN February 23, 2024. Regular updates bug fixes planned continually enhance package’s functionality user experience. actively planning include additional methods based kernel-based quadratic-distance. One primary goals make QuadratiK increasingly user-friendly, improvements user experience layout outputs. User feedback highly valued key driver future development. Life Cycle Statement subject periodic review updated reflect evolving nature QuadratiK.","code":""},{"path":"/index.html","id":"collection-of-methods-constructed-using-the-kernel-based-quadratic-distances","dir":"","previous_headings":"","what":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"QuadratiK provides first implementation, R Python, comprehensive set goodness--fit tests clustering technique d-dimensional spherical data d ≥ 2 using kernel-based quadratic distances. includes: Goodness--Fit Tests: software implements one, two, k-sample tests goodness fit, offering efficient mathematically sound way assess fit probability distributions. tests particularly useful large, high dimensional data sets assessment fit probability models interest. Specifically, offer tests multivariate normality, well two- k-sample tests, testing equality two distributions interest, H0 : F1 = F2 H0 : F1 = … = Fk respectively. proposed tests perform well terms level power contiguous alternatives, heavy tailed distributions higher dimensions. Expanded capabilities include supporting tests uniformity d-dimensional Sphere based Poisson kernel, exhibiting excellent results especially case multimodal distributions. Poisson kernel-based distribution (PKBD): package offers functions computing density value generating random samples PKBD. Poisson kernel-based densities based normalized Poisson kernel defined d-dimensional unit sphere. Given vector μ ∈ 𝒮d − 1, parameter ρ 0 < ρ < 1, probability density function d-variate Poisson kernel-based density defined : $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d},$$ μ vector orienting center distribution, ρ parameter control concentration distribution around vector μ related variance distribution. Furthermore, ωd = 2πd/2[Γ(d/2)]−1 surface area unit sphere ℝd (see Golzy Markatou, 2020). Clustering Algorithm Spherical Data: package incorporates unique clustering algorithm specifically tailored d-dimensional spherical data especially useful presence noise data presence non-negligible overlap clusters. algorithm leverages mixture Poisson kernel-based densities d-dimensional Sphere, enabling effective clustering spherical data data spherically transformed. Additional Features: Alongside functionalities, software includes additional graphical functions, aiding users validating representing cluster results well enhancing interpretability usability analysis. introduction usage QuadratiK see vignette Introduction QuadratiK Package.","code":""},{"path":"/index.html","id":"installation","dir":"","previous_headings":"","what":"Installation","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"can install version published CRAN QuadratiK development version GitHub QuadratiK package also available Python PyPI https://pypi.org/project/QuadratiK/ also Dashboard application. Usage instruction Dashboard can found https://quadratik.readthedocs.io/en/latest/user_guide/dashboard_application_usage.html.","code":"install.packages(\"QuadratiK\") library(devtools) install_github('giovsaraceno/QuadratiK-package')"},{"path":"/index.html","id":"authors","dir":"","previous_headings":"","what":"Authors","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"Giovanni Saraceno, Marianthi Markatou, Raktim Mukhopadhyay, Mojgan Golzy Maintainer: Giovanni Saraceno ","code":""},{"path":"/index.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"use package research work, please cite follows: Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). QuadratiK: Collection Methods Constructed using Kernel-Based Quadratic Distances. https://cran.r-project.org/package=QuadratiK, https://github.com/giovsaraceno/QuadratiK-package, https://giovsaraceno.github.io/QuadratiK-package/. associated paper: Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. arXiv preprint arXiv:2402.02290v2.","code":"@Manual{saraceno2024QuadratiK, title = {QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances}, author = {Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year = {2024}, note = {, , }, } @misc{saraceno2024package, title={Goodness-of-Fit and Clustering of Spherical Data: the QuadratiK package in R and Python}, author={Giovanni Saraceno and Marianthi Markatou and Raktim Mukhopadhyay and Mojgan Golzy}, year={2024}, eprint={2402.02290}, archivePrefix={arXiv}, primaryClass={stat.CO}, url={https://arxiv.org/abs/2402.02290} }"},{"path":"/index.html","id":"references","dir":"","previous_headings":"","what":"References","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: 10.5705/ss.202022.0347. Golzy, M. & Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713. Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” arXiv:2407.16374","code":""},{"path":"/index.html","id":"details","dir":"","previous_headings":"","what":"Details","title":"Collection of Methods Constructed using Kernel-Based Quadratic Distances","text":"work supported Kaleida Health Foundation National Science Foundation.","code":""},{"path":"/LICENSE.html","id":null,"dir":"","previous_headings":"","what":"GNU General Public License","title":"GNU General Public License","text":"Version 3, 29 June 2007Copyright © 2007 Free Software Foundation, Inc.  Everyone permitted copy distribute verbatim copies license document, changing allowed.","code":""},{"path":"/LICENSE.html","id":"preamble","dir":"","previous_headings":"","what":"Preamble","title":"GNU General Public License","text":"GNU General Public License free, copyleft license software kinds works. licenses software practical works designed take away freedom share change works. contrast, GNU General Public License intended guarantee freedom share change versions program–make sure remains free software users. , Free Software Foundation, use GNU General Public License software; applies also work released way authors. can apply programs, . speak free software, referring freedom, price. 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This is free software, and you are welcome to redistribute it under certain conditions; type 'show c' for details."},{"path":"/reference/breast_cancer.html","id":null,"dir":"Reference","previous_headings":"","what":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"breast_cancer Wisconsin data 569 rows 31 columns. first 30 variables report features computed digitized image fine needle aspirate (FNA) breast mass. describe characteristics cell nuclei present image. last column indicates class labels (Benign = 0 Malignant = 1).","code":""},{"path":"/reference/breast_cancer.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"","code":"breast_cancer"},{"path":"/reference/breast_cancer.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"data frame 569 observations 31 variables.","code":""},{"path":"/reference/breast_cancer.html","id":"source","dir":"Reference","previous_headings":"","what":"Source","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"Wolberg, W., Mangasarian, O., Street, N., & Street, W. (1993). Breast Cancer Wisconsin (Diagnostic). UCI Machine Learning Repository. https://doi.org/10.24432/C5DW2B.","code":""},{"path":"/reference/breast_cancer.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"Street, W. N., Wolberg, W. H., & Mangasarian, O. L. (1993, July). Nuclear feature extraction breast tumor diagnosis. Biomedical image processing biomedical visualization (Vol. 1905, pp. 861-870). SPIE.","code":""},{"path":"/reference/breast_cancer.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Breast Cancer Wisconsin (Diagnostic) — breast_cancer","text":"","code":"data(breast_cancer) summary(breast_cancer) #> radius1 texture1 perimeter1 area1 #> Min. : 6.981 Min. : 9.71 Min. : 43.79 Min. : 143.5 #> 1st Qu.:11.700 1st Qu.:16.17 1st Qu.: 75.17 1st Qu.: 420.3 #> Median :13.370 Median :18.84 Median : 86.24 Median : 551.1 #> Mean :14.127 Mean :19.29 Mean : 91.97 Mean : 654.9 #> 3rd Qu.:15.780 3rd Qu.:21.80 3rd Qu.:104.10 3rd Qu.: 782.7 #> Max. :28.110 Max. :39.28 Max. :188.50 Max. :2501.0 #> smoothness1 compactness1 concavity1 concave_points1 #> Min. :0.05263 Min. :0.01938 Min. :0.00000 Min. :0.00000 #> 1st Qu.:0.08637 1st Qu.:0.06492 1st Qu.:0.02956 1st Qu.:0.02031 #> Median :0.09587 Median :0.09263 Median :0.06154 Median :0.03350 #> Mean :0.09636 Mean :0.10434 Mean :0.08880 Mean :0.04892 #> 3rd Qu.:0.10530 3rd Qu.:0.13040 3rd Qu.:0.13070 3rd Qu.:0.07400 #> Max. :0.16340 Max. :0.34540 Max. :0.42680 Max. :0.20120 #> symmetry1 fractal_dimension1 radius2 texture2 #> Min. :0.1060 Min. :0.04996 Min. :0.1115 Min. :0.3602 #> 1st Qu.:0.1619 1st Qu.:0.05770 1st Qu.:0.2324 1st Qu.:0.8339 #> Median :0.1792 Median :0.06154 Median :0.3242 Median :1.1080 #> Mean :0.1812 Mean :0.06280 Mean :0.4052 Mean :1.2169 #> 3rd Qu.:0.1957 3rd Qu.:0.06612 3rd Qu.:0.4789 3rd Qu.:1.4740 #> Max. :0.3040 Max. :0.09744 Max. :2.8730 Max. :4.8850 #> perimeter2 area2 smoothness2 compactness2 #> Min. : 0.757 Min. : 6.802 Min. :0.001713 Min. :0.002252 #> 1st Qu.: 1.606 1st Qu.: 17.850 1st Qu.:0.005169 1st Qu.:0.013080 #> Median : 2.287 Median : 24.530 Median :0.006380 Median :0.020450 #> Mean : 2.866 Mean : 40.337 Mean :0.007041 Mean :0.025478 #> 3rd Qu.: 3.357 3rd Qu.: 45.190 3rd Qu.:0.008146 3rd Qu.:0.032450 #> Max. :21.980 Max. :542.200 Max. :0.031130 Max. :0.135400 #> concavity2 concave_points2 symmetry2 fractal_dimension2 #> Min. :0.00000 Min. :0.000000 Min. :0.007882 Min. :0.0008948 #> 1st Qu.:0.01509 1st Qu.:0.007638 1st Qu.:0.015160 1st Qu.:0.0022480 #> Median :0.02589 Median :0.010930 Median :0.018730 Median :0.0031870 #> Mean :0.03189 Mean :0.011796 Mean :0.020542 Mean :0.0037949 #> 3rd Qu.:0.04205 3rd Qu.:0.014710 3rd Qu.:0.023480 3rd Qu.:0.0045580 #> Max. :0.39600 Max. :0.052790 Max. :0.078950 Max. :0.0298400 #> radius3 texture3 perimeter3 area3 #> Min. : 7.93 Min. :12.02 Min. : 50.41 Min. : 185.2 #> 1st Qu.:13.01 1st Qu.:21.08 1st Qu.: 84.11 1st Qu.: 515.3 #> Median :14.97 Median :25.41 Median : 97.66 Median : 686.5 #> Mean :16.27 Mean :25.68 Mean :107.26 Mean : 880.6 #> 3rd Qu.:18.79 3rd Qu.:29.72 3rd Qu.:125.40 3rd Qu.:1084.0 #> Max. :36.04 Max. :49.54 Max. :251.20 Max. :4254.0 #> smoothness3 compactness3 concavity3 concave_points3 #> Min. :0.07117 Min. :0.02729 Min. :0.0000 Min. :0.00000 #> 1st Qu.:0.11660 1st Qu.:0.14720 1st Qu.:0.1145 1st Qu.:0.06493 #> Median :0.13130 Median :0.21190 Median :0.2267 Median :0.09993 #> Mean :0.13237 Mean :0.25427 Mean :0.2722 Mean :0.11461 #> 3rd Qu.:0.14600 3rd Qu.:0.33910 3rd Qu.:0.3829 3rd Qu.:0.16140 #> Max. :0.22260 Max. :1.05800 Max. :1.2520 Max. :0.29100 #> symmetry3 fractal_dimension3 y #> Min. :0.1565 Min. :0.05504 Length:569 #> 1st Qu.:0.2504 1st Qu.:0.07146 Class :character #> Median :0.2822 Median :0.08004 Mode :character #> Mean :0.2901 Mean :0.08395 #> 3rd Qu.:0.3179 3rd Qu.:0.09208 #> Max. :0.6638 Max. :0.20750"},{"path":"/reference/compare_qq.html","id":null,"dir":"Reference","previous_headings":"","what":"QQ-plot of given two samples using ggplot2 — compare_qq","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"QQ-plot given two samples using ggplot2","code":""},{"path":"/reference/compare_qq.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"","code":"compare_qq(sample1, sample2, main_title)"},{"path":"/reference/compare_qq.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"sample1 matrix observations sample 1 sample2 matrix observations sample 2 main_title title generated plot","code":""},{"path":"/reference/compare_qq.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"QQ-plot of given two samples using ggplot2 — compare_qq","text":"QQ-plot given samples","code":""},{"path":"/reference/compute_CV.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for two-sample KBQD tests — compute_CV","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"function computes critical value two-sample kernel tests centered Gaussian kernel using one three methods: bootstrap, permutation, subsampling.","code":""},{"path":"/reference/compute_CV.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"","code":"compute_CV( B, Quantile, data_pool, size_x, size_y, h, method, b = 1, compute_variance )"},{"path":"/reference/compute_CV.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"B number bootstrap/permutation/subsampling samples generate. Quantile quantile bootstrap/permutation/subsampling distribution use critical value. data_pool matrix containing data used test. size_x number rows data_pool matrix corresponding group X. size_y number rows data_pool matrix corresponding group Y. h tuning parameter kernel test. method method use computing critical value (one \"bootstrap\", \"permutation\", \"subsampling\"). b subsampling block size (used method \"subsampling\"). compute_variance indicates nonparametric variance computed. Default TRUE.","code":""},{"path":"/reference/compute_CV.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"critical value specified method significance level.","code":""},{"path":"/reference/compute_CV.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Compute the critical value for two-sample KBQD tests — compute_CV","text":"Markatou Marianthi & Saraceno Giovanni (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":"/reference/compute_stats.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute and display some descriptive statistics for the two sample tests — compute_stats","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"Compute display descriptive statistics two sample tests","code":""},{"path":"/reference/compute_stats.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"","code":"compute_stats(var1, var2, var_name, eps = 3)"},{"path":"/reference/compute_stats.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"var1 vector observations given variable sample 1 var2 vector observations given variable sample 2 var_name Name variable displayed eps precision displayed statistics","code":""},{"path":"/reference/compute_stats.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute and display some descriptive statistics for the two sample tests — compute_stats","text":"Computed statistics plot","code":""},{"path":"/reference/cv_ksample.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for the KBQD k-sample tests — cv_ksample","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"function computes empirical critical value k-sample KBQD tests using centered Gaussian kernel, bootstrap, permutation, subsampling.","code":""},{"path":"/reference/cv_ksample.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"","code":"cv_ksample( x, y, h, B = 150, b = 0.9, Quantile = 0.95, method = \"subsampling\", compute_variance = TRUE )"},{"path":"/reference/cv_ksample.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"x matrix containing observations used k-sample test y vector indicating sample observation h tuning parameter test using Gaussian kernel B number bootstrap/permutation/subsampling samples generate b subsampling block size (used method \"subsampling\") Quantile quantile bootstrap/permutation/subsampling distribution use critical value method method use computing critical value (one \"bootstrap\", \"permutation\") compute_variance indicates nonparametric variance computed. Default TRUE.","code":""},{"path":"/reference/cv_ksample.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for the KBQD k-sample tests — cv_ksample","text":"vector two critical values corresponding different formulation k-sample test statistics.","code":""},{"path":"/reference/DOF.html","id":null,"dir":"Reference","previous_headings":"","what":"Degrees of freedom (DOF) for the Poisson kernel — DOF","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"Compute Degrees Freedom (DOF) Poisson Kernel given dimension d concentration parameter rho","code":""},{"path":"/reference/DOF.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"","code":"DOF(d, rho)"},{"path":"/reference/DOF.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"d number dimensions rho concentration parameter","code":""},{"path":"/reference/DOF.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Degrees of freedom (DOF) for the Poisson kernel — DOF","text":"list containing DOF coefficient c asymptotic distribution","code":""},{"path":"/reference/DOF_norm.html","id":null,"dir":"Reference","previous_headings":"","what":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"Compute Degrees Freedom (DOF) normal Kernel centered respect standard normal distribution, given dimension d bandwidth parameter h.","code":""},{"path":"/reference/DOF_norm.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"","code":"DOF_norm(Sigma_h, V)"},{"path":"/reference/DOF_norm.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"Sigma_h covariance matrix gaussian kernel V Covariance matrix tested distribution G","code":""},{"path":"/reference/DOF_norm.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Degrees of freedom (DOF) for the Normal kernel — DOF_norm","text":"list containing DOF coefficient c asymptotic distribution","code":""},{"path":"/reference/dpkb.html","id":null,"dir":"Reference","previous_headings":"","what":"The Poisson kernel-based Distribution (PKBD) — dpkb","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"Poisson kernel-based densities based normalized Poisson kernel defined \\((d-1)\\)-dimensional unit sphere. Given vector \\(\\mathbf{\\mu} \\\\mathcal{S}^{d-1}\\), \\(\\mathcal{S}^{d-1}= \\{x \\\\mathbb{R}^d : ||x|| = 1\\}\\), parameter \\(\\rho\\) \\(0 < \\rho < 1\\), probability density function \\(d\\)-variate Poisson kernel-based density defined : $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d},$$ \\(\\mu\\) vector orienting center distribution, \\(\\rho\\) parameter control concentration distribution around vector \\(\\mu\\) related variance distribution. Recall , \\(x = (x_1, \\ldots, x_d) \\\\mathbb{R}^d\\), \\(||x|| = \\sqrt{x_1^2 + \\ldots + x_d^2}\\). Furthermore, \\(\\omega_d = 2\\pi^{d/2} [\\Gamma(d/2)]^{-1}\\) surface area unit sphere \\(\\mathbb{R}^d\\) (see Golzy Markatou, 2020). \\(\\rho \\0\\), Poisson kernel-based density tends uniform density sphere. Connections PKBDs distributions discussed detail Golzy Markatou (2020). note \\(d=2\\), PKBDs reduce wrapped Cauchy distribution. Additionally, precise choice parameters \\(\\rho\\) \\(\\mu\\) two-dimensional PKBD becomes two-dimensional projected normal distribution. However, connection \\(d\\)-dimensional projected normal distributions carry beyond \\(d=2\\). Golzy Markatou (2020) proposed acceptance-rejection method simulating data PKBD using von Mises-Fisher envelops (rejvmf method). Furthermore Sablica, Hornik Leydold (2023) proposed new ways simulating PKBD, using angular central Gaussian envelops (rejacg) using projected Saw distributions (rejpsaw).","code":""},{"path":"/reference/dpkb.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"","code":"dpkb(x, mu, rho, logdens = FALSE) rpkb( n, mu, rho, method = \"rejacg\", tol.eps = .Machine$double.eps^0.25, max.iter = 1000 )"},{"path":"/reference/dpkb.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"x \\(n \\times d\\)-matrix (data.frame) \\(n\\) data point sphere \\(\\mathcal{S}^{d-1}\\), \\(d \\ge 2\\). mu location vector parameter length indicating dimension generated points. rho Concentration parameter, \\(0 \\le\\) rho \\(< 1\\). logdens Logical; 'TRUE', densities returned logarithmic scale. n number observations. method string indicates method used sampling observations. available methods 'rejvmf' acceptance-rejection algorithm using von Mises-Fisher envelops (Algorithm Table 2 Golzy Markatou 2020); 'rejacg' using angular central Gaussian envelops (Algorithm Table 1 Sablica et al. 2023); 'rejpsaw' using projected Saw distributions (Algorithm Table 2 Sablica et al. 2023). tol.eps desired accuracy convergence tolerance ('rejacg' method). max.iter maximum number iterations ('rejacg' method).","code":""},{"path":"/reference/dpkb.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"dpkb gives density value; rpkb generates random observations PKBD.","code":""},{"path":"/reference/dpkb.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"function dpkb() computes density value given point x Poisson kernel-based distribution mean direction vector mu concentration parameter rho. number observations generated determined n rpkb(). function returns \\((n \\times d)\\)-matrix generated \\(n\\) observations \\(\\mathcal{S}^{(d-1)}\\). limitation rejvmf method ensure computational feasibility sampler \\(\\rho\\) approaching 1. chosen method 'rejacg', function uniroot, stat package, used estimate beta parameter. case, complete results provided output.","code":""},{"path":"/reference/dpkb.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"required packages (movMF rejvmf method, Tinflex rejpsaw) installed, function display message asking user install missing package(s).","code":""},{"path":"/reference/dpkb.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713. Sablica L., Hornik K. Leydold J. (2023) \"Efficient sampling PKBD distribution\", Electronic Journal Statistics, 17(2), 2180-2209.","code":""},{"path":"/reference/dpkb.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"The Poisson kernel-based Distribution (PKBD) — dpkb","text":"","code":"# Generate some data from pkbd density pkbd_dat <- rpkb(10, c(0.5,0), 0.5) # Calculate the PKBD density values dens_val <- dpkb(pkbd_dat, c(0.5,0.5),0.5)"},{"path":"/reference/extract_stats.html","id":null,"dir":"Reference","previous_headings":"","what":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"Method objects class pkbc computes descriptive variable respect detected groups. Method objects class pkbc computes descriptive variable respect detected groups.","code":""},{"path":"/reference/extract_stats.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"","code":"extract_stats(object, ...) # S4 method for pkbc extract_stats(object, k)"},{"path":"/reference/extract_stats.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"object Object class pkbc. ... possible additional inputs k Number clusters used.","code":""},{"path":"/reference/extract_stats.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"List computed descriptive statistics variable.","code":""},{"path":"/reference/extract_stats.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"function computes mean, standard deviation, median, inter-quantile range, minimum maximum variable data set given final membership assigned clustering algorithm.","code":""},{"path":"/reference/extract_stats.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — extract_stats","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions dat<-matrix(rnorm(300),ncol=3) #Perform the clustering algorithm pkbc_res<- pkbc(dat, 3) extract_stats(pkbc_res, 3) #> [[1]] #> Group 1 Group 2 Group 3 Overall #> mean 0.08920104 -0.91948554 -0.730958308 0.03245039 #> sd 0.56580990 0.02016632 0.005881618 0.59346106 #> median 0.04157567 -0.92107038 -0.730958308 0.01132900 #> IQR 0.96291884 0.01823508 0.004158932 1.02074711 #> min -0.97048825 -0.94215818 -0.735117240 -0.97048825 #> max 0.99916163 -0.89364323 -0.726799376 0.99916163 #> #> [[2]] #> Group 1 Group 2 Group 3 Overall #> mean -0.04517405 0.26661864 0.130846021 -0.02918194 #> sd 0.57552347 0.09128171 0.005606366 0.56187290 #> median -0.09048645 0.30696292 0.130846021 -0.05715153 #> IQR 0.94679985 0.05778760 0.003964299 0.86955159 #> min -0.98329467 0.13035633 0.126881722 -0.98329467 #> max 0.93041624 0.32219239 0.134810320 0.93041624 #> #> [[3]] #> Group 1 Group 2 Group 3 Overall #> mean -0.02838592 -0.25722878 0.669725381 -0.02357741 #> sd 0.59030936 0.11966575 0.005324043 0.58285373 #> median -0.00847352 -0.29000844 0.669725381 -0.02467391 #> IQR 1.05195509 0.13113221 0.003764667 1.03465610 #> min -0.95321117 -0.35654011 0.665960714 -0.95321117 #> max 0.98834993 -0.09235811 0.673490048 0.98834993 #>"},{"path":"/reference/generate_SN.html","id":null,"dir":"Reference","previous_headings":"","what":"Generate two samples data from skew-normal distributions — generate_SN","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"function generates data skew-normal distributions specified parameters means covariance matrices.","code":""},{"path":"/reference/generate_SN.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"","code":"generate_SN(d, size_x, size_y, mu_x, mu_y, sigma_x, sigma_y, skewness_y)"},{"path":"/reference/generate_SN.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"d number dimensions. size_x number observations sample X size_y number observations sample Y mu_x mean X mu_y mean Y sigma_x standard deviation X sigma_y standard deviation Y skewness_y skewness Y (skewness X set zero).","code":""},{"path":"/reference/generate_SN.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Generate two samples data from skew-normal distributions — generate_SN","text":"list containing generated X Y data sets.","code":""},{"path":"/reference/kb.test-class.html","id":null,"dir":"Reference","previous_headings":"","what":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","title":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","text":"class represent results Gaussian kernel-based quadratic distance tests. includes normality test, two-sample test statistics k-sample tests.","code":""},{"path":"/reference/kb.test-class.html","id":"slots","dir":"Reference","previous_headings":"","what":"Slots","title":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","text":"method String indicating kernel-based quadratic distance test performed. Un value test U-statistic. Vn value test V-statistic. H0_Un logical value indicating whether null hypothesis rejected according U-statistic. H0_Vn logical value indicating whether null hypothesis rejected according Vn. data List samples X (Y). CV_Un critical value computed test Un. CV_Vn critical value computed test Vn. cv_method method used estimate critical value (one \"subsampling\", \"permutation\" \"bootstrap\"). h list value bandwidth parameter used Gaussian kernel. function select_h used, also matrix computed power values resulting power plot provided. B Number bootstrap/permutation/subsampling replications. var_Un exact variance kernel-based U-statistic.","code":""},{"path":[]},{"path":"/reference/kb.test-class.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"An S4 class for kernel-based distance tests with normal kernel — kb.test-class","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) y <- matrix(rnorm(100),ncol=2) # Normality test kb.test(x, h=0.5) #> #> Kernel-based quadratic distance Normality test #> \t\tU-statistic\tV-statistic #> ------------------------------------------------ #> Test Statistic:\t 1.310854 \t 0.7979308 #> Critical Value:\t 2.4409 \t 6.071062 #> H0 is rejected:\t FALSE \t\t FALSE #> Selected tuning parameter h: 0.5 #> # Two-sample test kb.test(x,y,h=0.5, method=\"subsampling\",b=0.9) #> #> Kernel-based quadratic distance two-sample test #> U-statistic\t Dn \t\t Trace #> ------------------------------------------------ #> Test Statistic:\t 0.1153108 \t 0.133021 #> Critical Value:\t 1.407721 \t 1.625753 #> H0 is rejected:\t FALSE \t\t FALSE #> CV method: subsampling #> Selected tuning parameter h: 0.5 #>"},{"path":"/reference/kb.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"function performs kernel-based quadratic distance goodness--fit tests. includes tests multivariate normality, two-sample tests \\(k\\)-sample tests.","code":""},{"path":"/reference/kb.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"","code":"kb.test( x, y = NULL, h = NULL, method = \"subsampling\", B = 150, b = NULL, Quantile = 0.95, mu_hat = NULL, Sigma_hat = NULL, centeringType = \"Nonparam\", K_threshold = 10, alternative = \"skewness\" ) # S4 method for ANY kb.test( x, y = NULL, h = NULL, method = \"subsampling\", B = 150, b = 0.9, Quantile = 0.95, mu_hat = NULL, Sigma_hat = NULL, centeringType = \"Nonparam\", K_threshold = 10, alternative = \"skewness\" ) # S4 method for kb.test show(object)"},{"path":"/reference/kb.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"x Numeric matrix vector data values. y Numeric matrix vector data values. Depending input y, corresponding test performed. y = NULL, function performs tests normality x y data matrix, dimensions x, function performs two-sample test x y. y numeric factor vector, indicating group memberships observation, function performs k-sample test. h Bandwidth kernel function. value provided, algorithm selection optimal h performed automatically. See function select_h details. method method used critical value estimation (\"subsampling\", \"bootstrap\", \"permutation\")(default: \"subsampling\"). B number iterations use critical value estimation (default: 150). b size subsamples used subsampling algorithm (default: 0.8). Quantile quantile use critical value estimation, 0.95 default value. mu_hat Mean vector reference distribution. Sigma_hat Covariance matrix reference distribution. centeringType String indicating method used centering normal kernel ('Param' 'Nonparam'). K_threshold maximum number groups allowed. Default 10. control parameter. Change case 10 samples. alternative Family alternative chosen selecting h, \"location\", \"scale\" \"skewness\" (h provided). object Object class kb.test","code":""},{"path":"/reference/kb.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"S4 object class kb.test containing results kernel-based quadratic distance tests, based normal kernel. object contains following slots: method: Description kernel-based quadratic distance test performed. x Data list samples X (Y). Un value U-statistic. H0_Un logical value indicating whether null hypothesis rejected according Un. CV_Un critical value computed test Un. Vn value V-statistic (available). H0_Vn logical value indicating whether null hypothesis rejected according Vn (available). CV_Vn critical value computed test Vn (available). h List value bandwidth parameter used normal kernel function. select_h used, matrix computed power values corresponding power plot also provided. B Number bootstrap/permutation/subsampling replications. var_Un exact variance kernel-based U-statistic. cv_method method used estimate critical value (one \"subsampling\", \"permutation\" \"bootstrap\").","code":""},{"path":"/reference/kb.test.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"function kb.test performs kernel-based quadratic distance tests using Gaussian kernel bandwidth parameter h. Depending shape input y function performs tests multivariate normality, non-parametric two-sample tests k-sample tests. quadratic distance two probability distributions \\(F\\) \\(G\\) defined $$d_{K}(F,G)=\\iint K(x,y)d(F-G)(x)d(F-G)(y),$$ \\(G\\) distribution whose goodness fit wish assess \\(K\\) denotes Normal kernel defined $$ K_{{h}}(\\mathbf{s}, \\mathbf{t}) = (2 \\pi)^{-d/2} \\left(\\det{\\mathbf{\\Sigma}_h}\\right)^{-\\frac{1}{2}} \\exp\\left\\{-\\frac{1}{2}(\\mathbf{s} - \\mathbf{t})^\\top \\mathbf{\\Sigma}_h^{-1}(\\mathbf{s} - \\mathbf{t})\\right\\},$$ every \\(\\mathbf{s}, \\mathbf{t} \\\\mathbb{R}^d \\times \\mathbb{R}^d\\), covariance matrix \\(\\mathbf{\\Sigma}_h=h^2 \\) tuning parameter \\(h\\). Test Normality: Let \\(x_1, x_2, ..., x_n\\) random sample empirical distribution function \\(\\hat F\\). test null hypothesis normality, .e. \\(H_0:F=G=\\mathcal{N}_d(\\mu, \\Sigma)\\). consider U-statistic estimate sample KBQD $$U_{n}=\\frac{1}{n(n-1)}\\sum_{=2}^{n}\\sum_{j=1}^{-1} K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ first test statistics $$T_{n}=\\frac{U_{n}}{\\sqrt{Var(U_{n})}},$$ \\(Var(U_n)\\) computed exactly following Lindsay et al.(2014), V-statistic estimate $$V_{n} = \\frac{1}{n}\\sum_{=1}^{n} \\sum_{j=1}^{n}K_{cen}(\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ \\(K_{cen}\\) denotes Normal kernel \\(K_h\\) parametric centering respect considered normal distribution \\(G = \\mathcal{N}_d(\\mu, \\Sigma)\\). asymptotic distribution V-statistic infinite combination weighted independent chi-squared random variables one degree freedom. cutoff value obtained using Satterthwaite approximation \\(c \\cdot \\chi_{DOF}^2\\), \\(c\\) \\(DOF\\) computed exactly following formulas Lindsay et al.(2014). \\(U\\)-statistic cutoff determined empirically: Generate data considered normal distribution ; Compute test statistics B Monte Carlo(MC) replications; Compute 95th quantile empirical distribution test statistic. k-sample test: Consider \\(k\\) random samples ..d. observations \\(\\mathbf{x}^{()}_1, \\mathbf{x}^{()}_{2},\\ldots, \\mathbf{x}^{()}_{n_i} \\sim F_i\\), \\(= 1, \\ldots, k\\). test samples generated unknown distribution, \\(H_0: F_1 = F_2 = \\ldots = F_k\\) versus \\(H_1: F_i \\= F_j\\), \\(1 \\le \\= j \\le k\\). construct matrix distance \\(\\hat{\\mathbf{D}}\\), -diagonal elements $$\\hat{D}_{ij} = \\frac{1}{n_i n_j} \\sum_{\\ell=1}^{n_i} \\sum_{r=1}^{n_j}K_{\\bar{F}}(\\mathbf{x}^{()}_\\ell,\\mathbf{x}^{(j)}_r), \\qquad \\mbox{ }\\= j$$ diagonal $$\\hat{D}_{ii} = \\frac{1}{n_i (n_i -1)} \\sum_{\\ell=1}^{n_i} \\sum_{r\\= \\ell}^{n_i} K_{\\bar{F}}(\\mathbf{x}^{()}_\\ell, \\mathbf{x}^{()}_r), \\qquad \\mbox{ }= j,$$ \\(K_{\\bar{F}}\\) denotes Normal kernel \\(K_h\\) centered non-parametrically respect $$\\bar{F} = \\frac{n_1 \\hat{F}_1 + \\ldots + n_k \\hat{F}_k}{n}, \\quad \\mbox{ } n=\\sum_{=1}^k n_i.$$ compute trace statistic $$\\mathrm{trace}(\\hat{\\mathbf{D}}_n) = \\sum_{=1}^{k}\\hat{D}_{ii}$$ \\(D_n\\), derived considering possible pairwise comparisons k-sample null hypothesis, given $$D_n = (k-1) \\mathrm{trace}(\\hat{\\mathbf{D}}_n) - 2 \\sum_{=1}^{k}\\sum_{j> }^{k}\\hat{D}_{ij}.$$ compute empirical critical value employing numerical techniques bootstrap, permutation subsampling algorithms: Generate k-tuples, total size \\(n_B\\), pooled sample following one sampling methods; Compute k-sample test statistic; Repeat B times; Select \\(95^{th}\\) quantile obtained values. Two-sample test: Let \\(x_1, x_2, ..., x_{n_1} \\sim F\\) \\(y_1, y_2, ..., y_{n_2} \\sim G\\) random samples distributions \\(F\\) \\(G\\), respectively. test null hypothesis two samples generated unknown distribution, \\(H_0: F=G\\) vs \\(H_1:F\\=G\\). test statistics coincide \\(k\\)-sample test statistics \\(k=2\\).","code":""},{"path":"/reference/kb.test.html","id":"kernel-centering","dir":"Reference","previous_headings":"","what":"Kernel centering","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"arguments mu_hat Sigma_hat indicate normal model considered normality test, \\(H_0: F = N(\\)mu_hat, Sigma_hat). two-sample \\(k\\)-sample tests, mu_hat Sigma_hat can used parametric centering kernel, case want specify reference distribution, centeringType = \"Param\". default method test normality performed. normal kernel centered respect \\(G \\sim N_d(\\mathbf{\\mu}, \\mathbf{V})\\) can computed $$K_{cen(G)}(\\mathbf{s}, \\mathbf{t}) = K_{\\mathbf{\\Sigma_h}}(\\mathbf{s}, \\mathbf{t}) - K_{\\mathbf{\\Sigma_h} + \\mathbf{V}}(\\mathbf{\\mu}, \\mathbf{t}) - K_{\\mathbf{\\Sigma_h} + \\mathbf{V}}(\\mathbf{s}, \\mathbf{\\mu}) + K_{\\mathbf{\\Sigma_h} + 2\\mathbf{V}}(\\mathbf{\\mu}, \\mathbf{\\mu}).$$ consider non-parametric centering kernel respect \\(\\bar{F}=(n_1 F_1 + \\ldots n_k F_k)/n\\) \\(n=\\sum_{=1}^k n_i\\), centeringType = \"Nonparam\", two- \\(k\\)-sample tests. Let \\(\\mathbf{z}_1,\\ldots, \\mathbf{z}_n\\) denote pooled sample. \\(s,t \\\\{\\mathbf{z}_1,\\ldots, \\mathbf{z}_n\\}\\), given $$K_{cen(\\bar{F})}(\\mathbf{s},\\mathbf{t}) = K(\\mathbf{s},\\mathbf{t}) - \\frac{1}{n}\\sum_{=1}^{n} K(\\mathbf{s},\\mathbf{z}_i) - \\frac{1}{n}\\sum_{=1}^{n} K(\\mathbf{z}_i,\\mathbf{t}) + \\frac{1}{n(n-1)}\\sum_{=1}^{n} \\sum_{j \\=}^{n} K(\\mathbf{z}_i,\\mathbf{z}_j).$$","code":""},{"path":"/reference/kb.test.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"two- \\(k\\)-sample tests, slots Vn, H0_Vn CV_Vn empty, computed statistics reported slots Un, H0_Un CV_Un. U-statistic type statistic used estimate population parameter. based idea averaging possible distinct combinations fixed size sample. V-statistic considers possible tuples certain size, just distinct combinations can used contexts unbiasedness required.","code":""},{"path":"/reference/kb.test.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374 Lindsay, B.G., Markatou, M. Ray, S. (2014) \"Kernels, Degrees Freedom, Power Properties Quadratic Distance Goodness--Fit Tests\", Journal American Statistical Association, 109:505, 395-410, DOI: 10.1080/01621459.2013.836972","code":""},{"path":[]},{"path":"/reference/kb.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests — kb.test","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) y <- matrix(rnorm(100),ncol=2) # Normality test my_test <- kb.test(x, h=0.5) my_test #> #> Kernel-based quadratic distance Normality test #> \t\tU-statistic\tV-statistic #> ------------------------------------------------ #> Test Statistic:\t 0.3027069 \t 0.6189598 #> Critical Value:\t 1.262023 \t 6.071062 #> H0 is rejected:\t FALSE \t\t FALSE #> Selected tuning parameter h: 0.5 #> # Two-sample test my_test <- kb.test(x,y,h=0.5, method=\"subsampling\",b=0.9, centeringType = \"Nonparam\") my_test #> #> Kernel-based quadratic distance two-sample test #> U-statistic\t Dn \t\t Trace #> ------------------------------------------------ #> Test Statistic:\t 0.239788 \t 0.2819997 #> Critical Value:\t 0.9875162 \t 1.162661 #> H0 is rejected:\t FALSE \t\t FALSE #> CV method: subsampling #> Selected tuning parameter h: 0.5 #> # k-sample test z <- matrix(rnorm(100,2),ncol=2) dat <- rbind(x,y,z) group <- rep(c(1,2,3),each=50) my_test <- kb.test(x=dat,y=group,h=0.5, method=\"subsampling\",b=0.9) my_test #> #> Kernel-based quadratic distance k-sample test #> U-statistic\t Dn \t\t Trace #> ------------------------------------------------ #> Test Statistic:\t 7.325505 \t 11.45482 #> Critical Value:\t 0.7402039 \t 1.158313 #> H0 is rejected:\t TRUE \t\t TRUE #> CV method: subsampling #> Selected tuning parameter h: 0.5 #>"},{"path":"/reference/normal_CV.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"function computes empirical critical value Normality test based KBQD tests using centered Gaussian kernel.","code":""},{"path":"/reference/normal_CV.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"","code":"normal_CV(d, size, h, mu_hat, Sigma_hat, B = 150, Quantile = 0.95)"},{"path":"/reference/normal_CV.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"d dimension generated samples. size number observations generated. h concentration parameter Gaussian kernel. mu_hat Mean vector reference distribution. Sigma_hat Covariance matrix reference distribution. B number replications. Quantile quantile distribution use select critical value","code":""},{"path":"/reference/normal_CV.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"critical value specified dimension, size level.","code":""},{"path":"/reference/normal_CV.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Compute the critical value for the KBQD tests for multivariate Normality — normal_CV","text":"replication, sample d-dimensional Normal distribution mean vector mu_hat covariance matrix Sigma_hat generated KBQD test U-statistic Normality computed. B iterations, critical value selected Quantile empirical distribution computed test statistics.","code":""},{"path":"/reference/pk.test-class.html","id":null,"dir":"Reference","previous_headings":"","what":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","title":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","text":"class represent results Poisson kernel-based quadratic distance tests Uniformity sphere.","code":""},{"path":"/reference/pk.test-class.html","id":"slots","dir":"Reference","previous_headings":"","what":"Slots","title":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","text":"method Description test. x Matrix data Un value U-statistic. CV_Un critical value Un computed replications. H0_Un logical value indicating whether null hypothesis rejected according Un. Vn value V-statistic. CV_Vn critical value Vn computed following asymptotic distribution. H0_Vn logical value indicating whether null hypothesis rejected according Vn. rho concentration parameter Poisson kernel. B Number replications. var_Un exact variance kernel-based U-statistic.","code":""},{"path":[]},{"path":"/reference/pk.test-class.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"An S4 class for Poisson kernel-based quadratic distance tests. — pk.test-class","text":"","code":"# create a pk.test object d=3 size=100 x_sp <- sample_hypersphere(d, n_points=size) pk.test(x_sp,rho=0.8) #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Selected consentration parameter rho: 0.8 #> #> U-statistic: #> #> H0 is rejected: FALSE #> Statistic Un: 1.082258 #> Critical value: 1.44412 #> #> V-statistic: #> #> H0 is rejected: FALSE #> Statistic Vn: 49.19885 #> Critical value: 52.23077 #>"},{"path":"/reference/pk.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"function performs kernel-based quadratic distance goodness--fit tests Uniformity multivariate spherical data x \\(\\mathcal{S}^{d-1}\\) using Poisson kernel concentration parameter rho. Poisson kernel-based test uniformity exhibits excellent results especially case multimodal distributions, shown example Uniformity test Sphere vignette.","code":""},{"path":"/reference/pk.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"","code":"pk.test(x, rho, B = 300, Quantile = 0.95) # S4 method for ANY pk.test(x, rho, B = 300, Quantile = 0.95) # S4 method for pk.test show(object)"},{"path":"/reference/pk.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"x numeric \\((n \\times d)\\)-matrix \\(n\\) data points Sphere \\(\\mathcal{S}^(d-1)\\) rows. rho Concentration parameter Poisson kernel function. B Number Monte Carlo iterations critical value estimation Un (default: 300). Quantile quantile use critical value estimation, 0.95 default value. object Object class pk.test","code":""},{"path":"/reference/pk.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"S4 object class pk.test containing results Poisson kernel-based tests. object contains following slots: method: Description test performed. x Data matrix. Un value U-statistic. CV_Un empirical critical value Un. H0_Vn logical value indicating whether null hypothesis rejected according Un. Vn value V-statistic Vn. CV_Vn critical value Vn computed following asymptotic distribution. H0_Vn logical value indicating whether null hypothesis rejected according Vn. rho value concentration parameter used Poisson kernel function. B Number replications critical value U-statistic Un.","code":""},{"path":"/reference/pk.test.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"Let \\(x_1, x_2, ..., x_n\\) random sample empirical distribution function \\(\\hat F\\). test null hypothesis uniformity \\((d-1)\\)-dimensional sphere, .e. \\(H_0:F=G\\), \\(G\\) uniform distribution \\((d-1)\\)-dimensional sphere \\(\\mathcal{S}^{d-1}\\). compute U-statistic estimate sample KBQD (Kernel-Based Quadratic Distance) $$U_{n}=\\frac{1}{n(n-1)}\\sum_{=2}^{n}\\sum_{j=1}^{-1}K_{cen} (\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ first test statistic given $$T_{n}=\\frac{U_{n}}{\\sqrt{Var(U_{n})}},$$ $$Var(U_{n})= \\frac{2}{n(n-1)} \\left[\\frac{1+\\rho^{2}}{(1-\\rho^{2})^{d-1}}-1\\right],$$ V-statistic estimate KBQD $$V_{n} = \\frac{1}{n}\\sum_{=1}^{n}\\sum_{j=1}^{n}K_{cen} (\\mathbf{x}_{}, \\mathbf{x}_{j}),$$ \\(K_{cen}\\) denotes Poisson kernel \\(K_\\rho\\) centered respect uniform distribution \\((d-1)\\)-dimensional sphere, $$K_{cen}(\\mathbf{u}, \\mathbf{v}) = K_\\rho(\\mathbf{u}, \\mathbf{v}) -1$$ $$K_\\rho(\\mathbf{u}, \\mathbf{v}) = \\frac{1-\\rho^{2}}{\\left(1+\\rho^{2}- 2\\rho (\\mathbf{u}\\cdot \\mathbf{v})\\right)^{d/2}},$$ every \\(\\mathbf{u}, \\mathbf{v} \\\\mathcal{S}^{d-1} \\times \\mathcal{S}^{d-1}\\). asymptotic distribution V-statistic infinite combination weighted independent chi-squared random variables one degree freedom. cutoff value obtained using Satterthwaite approximation \\(c \\cdot \\chi_{DOF}^2\\), $$c=\\frac{(1+\\rho^{2})- (1-\\rho^{2})^{d-1}}{(1+\\rho)^{d}-(1-\\rho^{2})^{d-1}}$$ $$DOF(K_{cen} )=\\left(\\frac{1+\\rho}{1-\\rho} \\right)^{d-1}\\left\\{ \\frac{\\left(1+\\rho-(1-\\rho)^{d-1} \\right )^{2}} {1+\\rho^{2}-(1-\\rho^{2})^{d-1}}\\right \\}.$$. \\(U\\)-statistic cutoff determined empirically: Generate data Uniform distribution d-dimensional sphere; Compute test statistics B Monte Carlo(MC) replications; Compute 95th quantile empirical distribution test statistic.","code":""},{"path":"/reference/pk.test.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"U-statistic type statistic used estimate population parameter. based idea averaging possible distinct combinations fixed size sample. V-statistic considers possible tuples certain size, just distinct combinations can used contexts unbiasedness required.","code":""},{"path":"/reference/pk.test.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi:10.5705/ss.202022.0347","code":""},{"path":[]},{"path":"/reference/pk.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Poisson kernel-based quadratic distance test of Uniformity on the sphere — pk.test","text":"","code":"# create a pk.test object x_sp <- sample_hypersphere(3, n_points=100) unif_test <- pk.test(x_sp,rho=0.8) unif_test #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Selected consentration parameter rho: 0.8 #> #> U-statistic: #> #> H0 is rejected: FALSE #> Statistic Un: 0.4735996 #> Critical value: 1.543174 #> #> V-statistic: #> #> H0 is rejected: FALSE #> Statistic Vn: 46.27503 #> Critical value: 52.23077 #>"},{"path":"/reference/pkbc-class.html","id":null,"dir":"Reference","previous_headings":"","what":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","title":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","text":"class represent results Poisson kernel-based clustering procedure spherical observations.","code":""},{"path":"/reference/pkbc-class.html","id":"slots","dir":"Reference","previous_headings":"","what":"Slots","title":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","text":"res_k List objects results clustering algorithm value possible number clusters considered. input List input data","code":""},{"path":[]},{"path":"/reference/pkbc-class.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"A S4 class for the clustering algorithm on the sphere based on\r\nPoisson kernel-based distributions. — pkbc-class","text":"","code":"data(\"wireless\") res <- pkbc(as.matrix(wireless[,-8]),4)"},{"path":"/reference/pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Poisson kernel-based clustering on the sphere — pkbc","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"function pkbc() performs Poisson kernel-based clustering algorithm sphere proposed Golzy Markatou (2020). proposed algorithm based mixture, \\(M\\) components, Poisson kernel-based densities hypersphere \\(\\mathcal{S}^{d-1}\\) given $$f(x|\\Theta) = \\sum_{j=1}^M \\alpha_j f_j(x|\\rho_j, \\mu_j)$$ \\(\\alpha_j\\)'s mixing proportions \\(f_j(x|\\rho_j, \\mu_j)\\)'s denote probability density function \\(d\\)-variate Poisson kernel-based density given $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d}.$$ parameters \\(\\alpha_j, \\mu_j, \\rho_j\\) estimated iterative reweighted EM algorithm. proposed clustering algorithm exhibits excellent results (1) clusters well separated; (2) data points fairly well concentrated around vectors \\(\\mu_j\\) cluster; (3) percentage noise data increases.","code":""},{"path":"/reference/pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"","code":"pkbc( dat, nClust, maxIter = 300, stoppingRule = \"loglik\", initMethod = \"sampleData\", numInit = 10 ) # S4 method for ANY pkbc( dat, nClust, maxIter = 300, stoppingRule = \"loglik\", initMethod = \"sampleData\", numInit = 10 ) # S4 method for pkbc show(object)"},{"path":"/reference/pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"dat \\((n \\times d)\\)-data matrix data.frame data points sphere clustered. observations dat normalized dividing length vector ensure lie \\(d\\)-dimensional sphere. Note \\(d > 1\\). nClust Number clusters. can single value numeric vector. maxIter maximum number iterations run terminated. stoppingRule String describing stopping rule used within run. Currently must either 'max', 'membership', 'loglik'. initMethod String describing initialization method used. Currently must 'sampleData'. numInit Number initialization. object Object class pkbc","code":""},{"path":"/reference/pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"S4 object class pkbc containing results clustering procedure based Poisson kernel-based distributions. object contains following slots: res_k: List results Poisson kernel-based clustering algorithm value number clusters specified nClust. object list contains: postProbs Posterior probabilities observation indicated clusters. LogLik Maximum value log-likelihood function wcss Values within-cluster sum squares computed Euclidean distance cosine similarity, respectively. params List estimated parameters mixture model mu estimated centroids rho estimated concentration parameters rho alpha estimated mixing proportions finalMemb Vector final memberships runInfo List information EM algorithm iterations lokLikVec vector log-likelihood values numIterPerRun number E-M iterations per run input: List input information.","code":""},{"path":"/reference/pkbc.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"set concentration parameters equal 0.5 mixing proportions equal. initialization method 'sampleData' indicates observation points randomly chosen initializers centroids \\(\\mu_j\\). random starts strategy chance obtaining initial representatives underlying clusters, clustering performed numInit times random start highest likelihood chosen final estimate parameters. possible stoppingRule iteration : 'loglik' run algorithm change log-likelihood one iteration next less given threshold (1e-7) 'membership' run algorithm membership unchanged points one iteration next 'max' reach maximum number iterations maxIter obtained estimates used assigning final memberships, identifying nClust clusters, according following rule $$P(x_i, \\Theta) = \\arg\\max_{j \\\\{1, \\ldots, k\\}} \\{ \\frac{\\alpha_j f_j(x_i|\\mu_j, \\rho_j)}{f(x_i, \\Theta)}\\}.$$ number clusters nClust must provided input clustering algorithm.","code":""},{"path":"/reference/pkbc.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"clustering algorithm tailored data points sphere \\(\\mathcal{S}^{d-1}\\), can also performed spherically transformed observations, .e. data points Euclidean space \\(\\mathbb{R}^d\\) normalized lie corresponding \\((d-1)\\)-dimensional sphere \\(\\mathcal{S}^{d-1}\\).","code":""},{"path":"/reference/pkbc.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713.","code":""},{"path":[]},{"path":"/reference/pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Poisson kernel-based clustering on the sphere — pkbc","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions size<-100 groups<-c(rep(1, size), rep(2, size),rep(3,size)) rho<-0.8 set.seed(081423) data1<-rpkb(size, c(1,0,0),rho) data2<-rpkb(size, c(0,1,0),rho) data3<-rpkb(size, c(0,0,1),rho) dat<-rbind(data1,data2, data3) #Perform the clustering algorithm with number of clusters k=3. pkbd<- pkbc(dat=dat, nClust=3) show(pkbd) #> Poisson Kernel-Based Clustering on the Sphere (pkbc) Object #> ------------------------------------------------------------ #> #> Available components: #> Input Parameters: #> [1] \"dat\" \"nClust\" \"maxIter\" \"stoppingRule\" \"initMethod\" #> [6] \"numInit\" #> #> #> Considered possible number of clusters: 3 #> #> Available components for each value of number of clusters: #> [1] \"postProbs\" \"LogLik\" \"wcss\" \"params\" \"finalMemb\" \"runInfo\""},{"path":"/reference/pkbc_validation.html","id":null,"dir":"Reference","previous_headings":"","what":"Validation of Poisson kernel-based clustering results — pkbc_validation","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"Method objects class pkbc computes evaluation measures clustering results. following evaluation measures computed: -Group Proportion (Kapp Tibshirani (2007)). true label provided, ARI, Average Silhouette Width (Rousseeuw (1987)), Macro-Precision Macro-Recall computed.","code":""},{"path":"/reference/pkbc_validation.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"","code":"pkbc_validation(object, true_label = NULL)"},{"path":"/reference/pkbc_validation.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"object Object class pkbc true_label factor vector true membership clusters (available). must length final memberships.","code":""},{"path":"/reference/pkbc_validation.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"List following components: metrics Table computed evaluation measures value number clusters pkbc object. number cluster indicated column name. IGP List -group proportions value number clusters specified.","code":""},{"path":"/reference/pkbc_validation.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"IGP statistical measure quantifies proportion observations within group belong predefined category class. often used assess homogeneity group evaluating many members share label. higher IGP indicates group cohesive, lower proportion suggests greater diversity misclassification within group (Kapp Tibshirani 2007). Adjusted Rand Index (ARI) statistical measure used data clustering analysis. quantifies similarity two partitions dataset comparing assignments data points clusters. ARI value ranges 0 1, value 1 indicates perfect match partitions value close 0 indicates random assignment data points clusters. average silhouette width quantifies quality clustering measuring well object fits within assigned cluster. mean silhouette values, compare tightness object within cluster separation clusters. Higher values indicate well-separated, cohesive clusters, making useful selecting appropriate number clusters (Rousseeuw 1987). Macro Precision metric used multi-class classification calculates precision class independently takes average values. Precision class defined proportion true positive predictions predictions made class. Macro Recall similar Macro Precision focuses recall. Recall class proportion true positive predictions actual instances class. Macro Recall average recall values computed class.","code":""},{"path":"/reference/pkbc_validation.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"Note Macro Precision Macro Recall depend assigned labels, ARI measures similarity partition label switching. required packages (mclust ARI, clusterRepro IGP, cluster ASW) installed, function display message asking user install missing package(s).","code":""},{"path":"/reference/pkbc_validation.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"Kapp, .V. Tibshirani, R. (2007) \"clusters found one dataset present another dataset?\", Biostatistics, 8(1), 9–31, https://doi.org/10.1093/biostatistics/kxj029 Rousseeuw, P.J. (1987) Silhouettes: graphical aid interpretation validation cluster analysis. Journal Computational Applied Mathematics, 20, 53–65.","code":""},{"path":[]},{"path":"/reference/pkbc_validation.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Validation of Poisson kernel-based clustering results — pkbc_validation","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions size<-20 groups<-c(rep(1, size), rep(2, size),rep(3,size)) rho<-0.8 set.seed(081423) data1<-rpkb(size, c(1,0,0),rho,method='rejvmf') data2<-rpkb(size, c(0,1,0),rho,method='rejvmf') data3<-rpkb(size, c(1,0,0),rho,method='rejvmf') data<-rbind(data1,data2, data3) #Perform the clustering algorithm pkbc_res<- pkbc(data, 3) pkbc_validation(pkbc_res) #> $metrics #> 3 #> ASW 0.03602451 #> #> $IGP #> $IGP[[1]] #> NULL #> #> $IGP[[2]] #> NULL #> #> $IGP[[3]] #> [1] 0.952381 1.000000 1.000000 #> #>"},{"path":"/reference/plot.pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Plotting method for Poisson kernel-based clustering — plot.pkbc","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"Plots pkbc object.","code":""},{"path":"/reference/plot.pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"","code":"# S4 method for pkbc,ANY plot(x, k = NULL, true_label = NULL, pca_res = FALSE, ...)"},{"path":"/reference/plot.pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"x Object class pkbc k number considered clusters. provided scatter plot displayed value number clusters present x object true_label factor vector true membership clusters (available). must length final memberships. pca_res Logical. TRUE results PCALocantore also reported (dimension greater 3). ... Additional arguments can passed plot function","code":""},{"path":"/reference/plot.pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"scatter-plot(s) elbow plot.","code":""},{"path":"/reference/plot.pkbc.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"scatterplot: dimension equal 2 3, points displayed circle sphere, respectively. dimension greater 3, spherical Principal Component procedure proposed Locantore et al. (1999), applied dimensionality reduction first three principal components normalized displayed sphere. d > 3, complete results PcaLocantore function (package rrcov) returned pca_res=TRUE. elbow plot: within cluster sum squares (wcss) computed using Euclidean distance (left) cosine similarity (right).","code":""},{"path":"/reference/plot.pkbc.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"elbow plot commonly used graphical method choosing appropriate number clusters. Specifically, plotting wcss versus number clusters, suggested number clusters correspond point plotted line greatest change slope, showing elbow.","code":""},{"path":"/reference/plot.pkbc.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"Locantore, N., Marron, J.S., Simpson, D.G. et al. (1999) \"Robust principal component analysis functional data.\" Test 8, 1–73. https://doi.org/10.1007/BF02595862","code":""},{"path":[]},{"path":"/reference/plot.pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Plotting method for Poisson kernel-based clustering — plot.pkbc","text":"","code":"dat<-matrix(rnorm(300),ncol=3) pkbc_res<- pkbc(dat, 3) plot(pkbc_res, 3) #> `geom_line()`: Each group consists of only one observation. #> ℹ Do you need to adjust the group aesthetic? #> `geom_line()`: Each group consists of only one observation. #> ℹ Do you need to adjust the group aesthetic?"},{"path":"/reference/poisson_CV.html","id":null,"dir":"Reference","previous_headings":"","what":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"function computes empirical critical value U-statistics testing uniformity sphere based centered poisson kernel.","code":""},{"path":"/reference/poisson_CV.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"","code":"poisson_CV(d, size, rho, B, Quantile)"},{"path":"/reference/poisson_CV.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"d dimension generated samples. size number observations generated. rho concentration parameter Poisson kernel. B number replications. Quantile quantile distribution use select critical value.","code":""},{"path":"/reference/poisson_CV.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"critical value specified dimension, size level.","code":""},{"path":"/reference/poisson_CV.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"replication, sample d-dimensional observations uniform distribution Sphere generated Poisson kernel-based U-statistic computed. B iterations, critical value selected Quantile empirical distribution computed test statistics.","code":""},{"path":"/reference/poisson_CV.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Compute the critical value for the Poisson KBQD tests for Uniformity — poisson_CV","text":"Ding Yuxin, Markatou Marianthi, Saraceno Giovanni (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: doi:10.5705/ss.202022.0347","code":""},{"path":"/reference/predict.pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"Obtain predictions membership spherical observations based mixture Poisson kernel-based densities estimated pkbc","code":""},{"path":"/reference/predict.pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"","code":"# S4 method for pkbc predict(object, k, newdata = NULL)"},{"path":"/reference/predict.pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"object Object class pkbc k Number clusters used. newdata data.frame matrix data. missing clustering data obtained pkbc object classified.","code":""},{"path":"/reference/predict.pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"Returns list following components Memb: vector predicted memberships newdata Probs: matrix entry (,j) denotes probability observation belongs k-th cluster.","code":""},{"path":[]},{"path":"/reference/predict.pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Cluster spherical observations using a mixture of Poisson kernel-based\r\ndensities — predict.pkbc","text":"","code":"# generate data dat <- rbind(matrix(rnorm(100),ncol=2),matrix(rnorm(100,5),ncol=2)) res <- pkbc(dat,2) # extract membership of dat predict(res,k=2) #> [1] 2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 2 2 #> [38] 2 2 2 1 2 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 #> [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 # predict membership of new data newdat <- rbind(matrix(rnorm(10),ncol=2),matrix(rnorm(10,5),ncol=2)) predict(res, k=2, newdat) #> $Memb #> [1] 1 2 2 2 2 1 1 1 1 1 #> #> $Probs #> [,1] [,2] #> [1,] 0.79214887 0.20785113 #> [2,] 0.09155787 0.90844213 #> [3,] 0.10633003 0.89366997 #> [4,] 0.05939363 0.94060637 #> [5,] 0.11831316 0.88168684 #> [6,] 0.95577420 0.04422580 #> [7,] 0.97150510 0.02849490 #> [8,] 0.96104005 0.03895995 #> [9,] 0.84807303 0.15192697 #> [10,] 0.95487786 0.04512214 #>"},{"path":"/reference/QuadratiK-package.html","id":null,"dir":"Reference","previous_headings":"","what":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"Collection Methods Constructed using Kernel-Based Quadratic Distances QuadratiK provides first implementation, R Python, comprehensive set goodness--fit tests clustering technique \\(d\\)-dimensional spherical data \\(d \\ge 2\\) using kernel-based quadratic distances. includes: Goodness--Fit Tests: software implements one, two, k-sample tests goodness fit, offering efficient mathematically sound way assess fit probability distributions. tests particularly useful large, high dimensional data sets assessment fit probability models interest. Specifically, offer tests normality, well two- k-sample tests, testing equality two distributions interest, \\(H_0: F_1 = F_2\\) \\(H_0: F_1 = \\ldots = F_k\\) respectively. proposed tests perform well terms level power contiguous alternatives, heavy tailed distributions higher dimensions. Expanded capabilities include supporting tests uniformity d-dimensional Sphere based Poisson kernel, exhibiting excellent results especially case multimodal distributions. Poisson kernel-based distribution (PKBD): package offers functions computing density value generating random samples PKBD. Poisson kernel-based densities based normalized Poisson kernel defined \\(d\\)-dimensional unit sphere. Given vector \\(\\mu \\\\mathcal{S}^{d-1}\\), parameter \\(\\rho\\) \\(0 < \\rho < 1\\), probability density function \\(d\\)-variate Poisson kernel-based density defined : $$f(\\mathbf{x}|\\rho, \\mathbf{\\mu}) = \\frac{1-\\rho^2}{\\omega_d ||\\mathbf{x} - \\rho \\mathbf{\\mu}||^d},$$ \\(\\mu\\) vector orienting center distribution, \\(\\rho\\) parameter control concentration distribution around vector \\(\\mu\\) related variance distribution. Furthermore, \\(\\omega_d = 2\\pi^{d/2} [\\Gamma(d/2)]^{-1}\\) surface area unit sphere \\(\\mathbb{R}^d\\) (see Golzy Markatou, 2020). Clustering Algorithm Spherical Data: package incorporates unique clustering algorithm specifically tailored \\(d\\)-dimensional spherical data especially useful presence noise data presence non-negligible overlap clusters. algorithm leverages mixture Poisson kernel-based densities Sphere, enabling effective clustering spherical data data spherically transformed. Additional Features: Alongside functionalities, software includes additional graphical functions, aiding users validating representing cluster results well enhancing interpretability usability analysis. introduction QuadratiK see vignette Introduction QuadratiK Package.","code":""},{"path":"/reference/QuadratiK-package.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"work supported Kaleida Health Foundation National Science Foundation.","code":""},{"path":"/reference/QuadratiK-package.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"QuadratiK package also available Python PyPI https://pypi.org/project/QuadratiK/ also Dashboard application. Usage instruction Dashboard can found .","code":""},{"path":"/reference/QuadratiK-package.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. arXiv preprint arXiv:2402.02290. Ding, Y., Markatou, M. Saraceno, G. (2023). “Poisson Kernel-Based Tests Uniformity d-Dimensional Sphere.” Statistica Sinica. doi: doi:10.5705/ss.202022.0347. Golzy, M. Markatou, M. (2020) Poisson Kernel-Based Clustering Sphere: Convergence Properties, Identifiability, Method Sampling, Journal Computational Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713. Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374","code":""},{"path":[]},{"path":"/reference/QuadratiK-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"Collection of Methods Constructed using the Kernel-Based Quadratic\r\nDistances — QuadratiK-package","text":"Giovanni Saraceno, Marianthi Markatou, Raktim Mukhopadhyay, Mojgan Golzy Maintainer: Giovanni Saraceno gsaracen@buffalo.edu","code":""},{"path":"/reference/sample_hypersphere.html","id":null,"dir":"Reference","previous_headings":"","what":"Generate random sample from the hypersphere — sample_hypersphere","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"Generate random sample uniform distribution hypersphere.","code":""},{"path":"/reference/sample_hypersphere.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"","code":"sample_hypersphere(d, n_points = 1)"},{"path":"/reference/sample_hypersphere.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"d Number dimensions. n_points Number sampled observations.","code":""},{"path":"/reference/sample_hypersphere.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"Data matrix sampled observations.","code":""},{"path":"/reference/sample_hypersphere.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Generate random sample from the hypersphere — sample_hypersphere","text":"","code":"x_sp <- sample_hypersphere(3,100)"},{"path":"/reference/select_h.html","id":null,"dir":"Reference","previous_headings":"","what":"Select the value of the kernel tuning parameter — select_h","title":"Select the value of the kernel tuning parameter — select_h","text":"function computes kernel bandwidth Gaussian kernel normality, two-sample k-sample kernel-based quadratic distance (KBQD) tests.","code":""},{"path":"/reference/select_h.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Select the value of the kernel tuning parameter — select_h","text":"","code":"select_h( x, y = NULL, alternative = NULL, method = \"subsampling\", b = 0.8, B = 100, delta_dim = 1, delta = NULL, h_values = NULL, Nrep = 50, n_cores = 2, Quantile = 0.95, power.plot = TRUE )"},{"path":"/reference/select_h.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Select the value of the kernel tuning parameter — select_h","text":"x Data set observations X. y Numeric matrix vector data values. Depending input y, selection h performed corresponding test. y = NULL, function performs tests normality x. y data matrix, dimensions x, function performs two-sample test x y. y numeric factor vector, indicating group memberships observation, function performs k-sample test. alternative Family alternative chosen selecting h, \"location\", \"scale\" \"skewness\". method method used critical value estimation (\"subsampling\", \"bootstrap\", \"permutation\"). b size subsamples used subsampling algorithm . B number iterations use critical value estimation, B = 150 default. delta_dim Vector coefficient alternative respect dimension delta Vector parameter values indicating chosen alternatives h_values Values tuning parameter used selection Nrep Number bootstrap/permutation/subsampling replications. n_cores Number cores used parallel h selection algorithm. provided, function detect available cores. Quantile quantile use critical value estimation, 0.95 default value. power.plot Logical. TRUE, displayed plot power values h_values delta.","code":""},{"path":"/reference/select_h.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Select the value of the kernel tuning parameter — select_h","text":"list following attributes: h_sel selected value tuning parameter h; power matrix power values computed considered values delta h_values; power.plot power plots (power.plot TRUE).","code":""},{"path":"/reference/select_h.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Select the value of the kernel tuning parameter — select_h","text":"function performs selection optimal value tuning parameter \\(h\\) normal kernel function, normality test, two-sample k-sample KBQD tests. performs small simulation study, generating samples according family alternative specified, chosen values h_values delta. consider target alternatives \\(F_\\delta(\\hat{\\mathbf{\\mu}}, \\hat{\\mathbf{\\Sigma}}, \\hat{\\mathbf{\\lambda}})\\), \\(\\hat{\\mathbf{\\mu}}, \\hat{\\mathbf{\\Sigma}}\\) \\(\\hat{\\mathbf{\\lambda}}\\) indicate location, covariance skewness parameter estimates pooled sample. Compute estimates mean \\(\\hat{\\mu}\\), covariance matrix \\(\\hat{\\Sigma}\\) skewness \\(\\hat{\\lambda}\\) pooled sample. Choose family alternatives \\(F_\\delta = F_\\delta(\\hat{\\mu} ,\\hat{\\Sigma}, \\hat{\\lambda})\\). value \\(\\delta\\) \\(h\\): Generate \\(\\mathbf{X}_1,\\ldots,\\mathbf{X}_{k-1} \\sim F_0\\), \\(\\delta=0\\); Generate \\(\\mathbf{X}_k \\sim F_\\delta\\); Compute \\(k\\)-sample test statistic \\(\\mathbf{X}_1, \\mathbf{X}_2, \\ldots, \\mathbf{X}_k\\) kernel parameter \\(h\\); Compute power test. greater 0.5, select \\(h\\) optimal value. optimal value selected, choose \\(h\\) corresponds maximum power. available alternative location alternatives, \\(F_\\delta = SN_d(\\hat{\\mu} + \\delta,\\hat{\\Sigma}, \\hat{\\lambda})\\),\\(\\delta = 0.2, 0.3, 0.4\\); scale alternatives, \\(F_\\delta = SN_d(\\hat{\\mu} ,\\hat{\\Sigma}*\\delta, \\hat{\\lambda})\\), \\(\\delta = 0.1, 0.3, 0.5\\); skewness alternatives, \\(F_\\delta = SN_d(\\hat{\\mu} ,\\hat{\\Sigma}, \\hat{\\lambda} + \\delta)\\), \\(\\delta = 0.2, 0.3, 0.6\\). values \\(h = 0.6, 1, 1.4, 1.8, 2.2\\) \\(N=50\\) set default values. function select_h() allows user set values \\(\\delta\\) \\(h\\) extensive grid search. suggest set extensive grid search computational resources permit.","code":""},{"path":"/reference/select_h.html","id":"note","dir":"Reference","previous_headings":"","what":"Note","title":"Select the value of the kernel tuning parameter — select_h","text":"Please aware select_h() function may take significant amount time run, especially larger datasets using larger number parameters h_values delta. Consider applying function large complex data.","code":""},{"path":"/reference/select_h.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Select the value of the kernel tuning parameter — select_h","text":"Markatou, M. Saraceno, G. (2024). “Unified Framework Multivariate Two- k-Sample Kernel-based Quadratic Distance Goodness--Fit Tests.” https://doi.org/10.48550/arXiv.2407.16374 Saraceno, G., Markatou, M., Mukhopadhyay, R. Golzy, M. (2024). Goodness--Fit Clustering Spherical Data: QuadratiK package R Python. https://arxiv.org/abs/2402.02290.","code":""},{"path":[]},{"path":"/reference/select_h.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Select the value of the kernel tuning parameter — select_h","text":"","code":"# Select the value of h using the mid-power algorithm # \\donttest{ x <- matrix(rnorm(100),ncol=2) y <- matrix(rnorm(100),ncol=2) h_sel <- select_h(x,y,\"skewness\") h_sel #> $h_sel #> [1] 1.6 #> #> $power #> h delta power #> 20 1.6 0.6 0.46 #> 23 2.8 0.6 0.40 #> 21 2.0 0.6 0.38 #> 22 2.4 0.6 0.34 #> 24 3.2 0.6 0.34 #> 13 2.0 0.3 0.32 #> 19 1.2 0.6 0.30 #> 18 0.8 0.6 0.26 #> 15 2.8 0.3 0.22 #> 17 0.4 0.6 0.20 #> 12 1.6 0.3 0.20 #> 4 1.6 0.2 0.18 #> 16 3.2 0.3 0.18 #> 6 2.4 0.2 0.16 #> 9 0.4 0.3 0.12 #> 5 2.0 0.2 0.12 #> 14 2.4 0.3 0.12 #> 10 0.8 0.3 0.10 #> 11 1.2 0.3 0.10 #> 7 2.8 0.2 0.10 #> 1 0.4 0.2 0.06 #> 3 1.2 0.2 0.06 #> 8 3.2 0.2 0.06 #> 2 0.8 0.2 0.04 #> #> $power.plot #> # }"},{"path":"/reference/stats_clusters.html","id":null,"dir":"Reference","previous_headings":"","what":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"Method objects class pkbc computes descriptive variable respect detected groups. Method objects class pkbc computes descriptive statistics variable respect detected groups.","code":""},{"path":"/reference/stats_clusters.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"","code":"stats_clusters(object, ...) # S4 method for pkbc stats_clusters(object, k)"},{"path":"/reference/stats_clusters.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"object Object class pkbc. ... possible additional inputs k Number clusters used.","code":""},{"path":"/reference/stats_clusters.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"List computed descriptive statistics dimension.","code":""},{"path":"/reference/stats_clusters.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"function computes mean, standard deviation, median, inter-quantile range, minimum maximum variable data set given final membership assigned clustering algorithm.","code":""},{"path":[]},{"path":"/reference/stats_clusters.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Descriptive statistics for the clusters identified by the Poisson\r\nkernel-based clustering. — stats_clusters","text":"","code":"#We generate three samples of 100 observations from 3-dimensional #Poisson kernel-based densities with rho=0.8 and different mean directions dat<-matrix(rnorm(300),ncol=3) #Perform the clustering algorithm pkbc_res<- pkbc(dat, 3) stats_clusters(pkbc_res, 3) #> [[1]] #> Group 1 Group 2 Group 3 Overall #> mean -0.53360502 0.2185515 0.007405148 0.04344402 #> sd 0.34258917 0.5269849 NA 0.58084656 #> median -0.55364837 0.2151044 0.007405148 0.07318718 #> IQR 0.53206272 0.9127813 0.000000000 0.93759633 #> min -0.99487956 -0.9642025 0.007405148 -0.99487956 #> max 0.07466077 0.9873736 0.007405148 0.98737364 #> #> [[2]] #> Group 1 Group 2 Group 3 Overall #> mean 0.1608868 -0.1893987 0.8593866 -0.0983452 #> sd 0.3330494 0.6198886 NA 0.5890662 #> median 0.1009154 -0.3009186 0.8593866 -0.1183249 #> IQR 0.5036676 0.9810143 0.0000000 1.0017535 #> min -0.4511650 -0.9888086 0.8593866 -0.9888086 #> max 0.7908170 0.9928995 0.8593866 0.9928995 #> #> [[3]] #> Group 1 Group 2 Group 3 Overall #> mean -0.62082799 0.09072061 -0.5112727 -0.0789555 #> sd 0.29913303 0.50819936 NA 0.5547490 #> median -0.65141402 0.08038502 -0.5112727 -0.1405382 #> IQR 0.47845791 0.77660008 0.0000000 0.8655454 #> min -0.95185464 -0.99561644 -0.5112727 -0.9956164 #> max -0.09110684 0.96905979 -0.5112727 0.9690598 #>"},{"path":"/reference/summary-kb.test-method.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"summary method class kb.test","code":""},{"path":"/reference/summary-kb.test-method.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"","code":"# S4 method for kb.test summary(object)"},{"path":"/reference/summary-kb.test-method.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"object Object class kb.test","code":""},{"path":"/reference/summary-kb.test-method.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"List following components: summary_tables Table computed descriptive statistics per variable (per group available). test_results Data frame results performed kernel-based quadratic distance test. qqplots Figure qq-plots variable.","code":""},{"path":[]},{"path":"/reference/summary-kb.test-method.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary,kb.test-method","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) # Normality test my_test <- kb.test(x, h=0.5) summary(my_test) #> #> Kernel-based quadratic distance Normality test #> Statistic Test_Statistic Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> $summary_tables #> V1 V2 #> mean 0.2225601 -0.01276311 #> sd 1.0933236 1.14937193 #> median 0.2061474 0.14471308 #> IQR 1.3443250 1.40042674 #> min -1.5643294 -4.11865455 #> max 3.5340310 2.55373421 #> #> $test_results #> Statistic Test_Statistic Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> #> $qqplots #>"},{"path":"/reference/summary-pk.test-method.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"summary method class pk.test","code":""},{"path":"/reference/summary-pk.test-method.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"","code":"# S4 method for pk.test summary(object)"},{"path":"/reference/summary-pk.test-method.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"object Object class pk.test","code":""},{"path":"/reference/summary-pk.test-method.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"List following components: summary_tables Table computed descriptive statistics per variable. test_results Data frame results performed Poisson kernel-based test. qqplots Figure qq-plots variable uniform distribution.","code":""},{"path":[]},{"path":"/reference/summary-pk.test-method.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary,pk.test-method","text":"","code":"# create a pk.test object x_sp <- sample_hypersphere(3, n_points=100) unif_test <- pk.test(x_sp,rho=0.8) summary(unif_test) #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> $summary_tables #> V1 V2 V3 #> mean -0.09616741 0.08709887 -0.01460631 #> sd 0.57359035 0.57642804 0.57585233 #> median -0.17876519 0.20832080 -0.04974959 #> IQR 0.95429937 0.89814932 1.00133903 #> min -0.99451952 -0.97609712 -0.92478198 #> max 0.94196356 0.98234388 0.93822530 #> #> $test_results #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> #> $qqplots #>"},{"path":"/reference/summary.kb.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary.kb.test","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"summary method class kb.test","code":""},{"path":"/reference/summary.kb.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"","code":"# S4 method for kb.test summary(object)"},{"path":"/reference/summary.kb.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"object Object class kb.test","code":""},{"path":"/reference/summary.kb.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"List following components: summary_tables Table computed descriptive statistics per variable (per group available). test_results Data frame results performed kernel-based quadratic distance test. qqplots Figure qq-plots variable.","code":""},{"path":[]},{"path":"/reference/summary.kb.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary.kb.test","text":"","code":"# create a kb.test object x <- matrix(rnorm(100),ncol=2) # Normality test my_test <- kb.test(x, h=0.5) summary(my_test) #> #> Kernel-based quadratic distance Normality test #> Statistic Value Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> $summary_tables #> V1 V2 #> mean 0.2225601 -0.01276311 #> sd 1.0933236 1.14937193 #> median 0.2061474 0.14471308 #> IQR 1.3443250 1.40042674 #> min -1.5643294 -4.11865455 #> max 3.5340310 2.55373421 #> #> $test_results #> Statistic Value Critical_Value Reject_H0 #> 1 Un -0.07807674 1.734051 FALSE #> 2 Vn 0.55877213 6.071062 FALSE #> #> $qqplots #>"},{"path":"/reference/summary.pk.test.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing kernel-based quadratic distance results — summary.pk.test","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"summary method class pk.test","code":""},{"path":"/reference/summary.pk.test.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"","code":"# S4 method for pk.test summary(object)"},{"path":"/reference/summary.pk.test.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"object Object class pk.test","code":""},{"path":"/reference/summary.pk.test.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"List following components: summary_tables Table computed descriptive statistics per variable. test_results Data frame results performed Poisson kernel-based test. qqplots Figure qq-plots variable uniform distribution.","code":""},{"path":[]},{"path":"/reference/summary.pk.test.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing kernel-based quadratic distance results — summary.pk.test","text":"","code":"# create a pk.test object x_sp <- sample_hypersphere(3, n_points=100) unif_test <- pk.test(x_sp,rho=0.8) summary(unif_test) #> #> Poisson Kernel-based quadratic distance test of #> Uniformity on the Sphere #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> $summary_tables #> V1 V2 V3 #> mean -0.09616741 0.08709887 -0.01460631 #> sd 0.57359035 0.57642804 0.57585233 #> median -0.17876519 0.20832080 -0.04974959 #> IQR 0.95429937 0.89814932 1.00133903 #> min -0.99451952 -0.97609712 -0.92478198 #> max 0.94196356 0.98234388 0.93822530 #> #> $test_results #> Test_Statistics Critical_Value Reject_H0 #> 1 0.2207214 1.722025 FALSE #> 2 45.0602797 52.230766 FALSE #> #> $qqplots #>"},{"path":"/reference/summary.pkbc.html","id":null,"dir":"Reference","previous_headings":"","what":"Summarizing PKBD mixture Fits — summary.pkbc","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"Summary method class \"pkbc\"","code":""},{"path":"/reference/summary.pkbc.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"","code":"# S4 method for pkbc summary(object)"},{"path":"/reference/summary.pkbc.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"object Object class pkbc","code":""},{"path":"/reference/summary.pkbc.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"Display logLikelihood values within cluster sum squares (wcss) values number clusters provided. values estimated mixing proportions showed together table assigned memberships.","code":""},{"path":[]},{"path":"/reference/summary.pkbc.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Summarizing PKBD mixture Fits — summary.pkbc","text":"","code":"dat <- rbind(matrix(rnorm(100),2),matrix(rnorm(100,5),2)) res <- pkbc(dat,2:4) summary(res) #> Poisson Kernel-Based Clustering on the Sphere (pkbc) Results #> ------------------------------------------------------------ #> #> Summary: #> nClust LogLik WCSS #> [1,] 2 744.1161 4.978247 #> [2,] 3 1104.1246 4.049000 #> [3,] 4 1391.2432 4.000000 #> #> Results for 2 clusters: #> Estimated Mixing Proportions (alpha): #> [1] 0.2879668 0.7120332 #> #> Clustering table: #> #> 1 2 #> 1 3 #> #> #> Results for 3 clusters: #> Estimated Mixing Proportions (alpha): #> [1] 0.25 0.25 0.50 #> #> Clustering table: #> #> 1 2 3 #> 1 1 2 #> #> #> Results for 4 clusters: #> Estimated Mixing Proportions (alpha): #> [1] 0.25 0.25 0.25 0.25 #> #> Clustering table: #> #> 1 2 3 4 #> 1 1 1 1 #> #>"},{"path":"/reference/var_k.html","id":null,"dir":"Reference","previous_headings":"","what":"Exact variance of k-sample test — var_k","title":"Exact variance of k-sample test — var_k","text":"Compute exact variance kernel test k-sample problem null hypothesis F1=...=Fk.","code":""},{"path":"/reference/var_k.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Exact variance of k-sample test — var_k","text":"","code":"var_k(Kcen, sizes, cum_size)"},{"path":"/reference/var_k.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Exact variance of k-sample test — var_k","text":"Kcen matrix centered kernel values sizes vector indicating sample's size. cum_size vector indicating sample's cumulative sizes.","code":""},{"path":"/reference/var_k.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Exact variance of k-sample test — var_k","text":"value computed variance.","code":""},{"path":"/reference/var_norm.html","id":null,"dir":"Reference","previous_headings":"","what":"Exact variance of normality test — var_norm","title":"Exact variance of normality test — var_norm","text":"Compute exact variance kernel test normality null hypothesis G=N(0,).","code":""},{"path":"/reference/var_norm.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Exact variance of normality test — var_norm","text":"","code":"var_norm(Sigma_h, V, n)"},{"path":"/reference/var_norm.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Exact variance of normality test — var_norm","text":"Sigma_h covariance matrix gaussian kernel V Covariance matrix tested distribution G n sample size","code":""},{"path":"/reference/var_norm.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Exact variance of normality test — var_norm","text":"value computed variance","code":""},{"path":"/reference/var_two.html","id":null,"dir":"Reference","previous_headings":"","what":"Exact variance of two-sample test — var_two","title":"Exact variance of two-sample test — var_two","text":"Compute exact variance kernel test two-sample problem null hypothesis F=G.","code":""},{"path":"/reference/var_two.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Exact variance of two-sample test — var_two","text":"","code":"var_two(Kcen, nsamples)"},{"path":"/reference/var_two.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Exact variance of two-sample test — var_two","text":"Kcen matrix centered kernel values nsamples vector indicating sample's membership.","code":""},{"path":"/reference/var_two.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Exact variance of two-sample test — var_two","text":"value computed variance.","code":""},{"path":"/reference/wine.html","id":null,"dir":"Reference","previous_headings":"","what":"Wine data set — wine","title":"Wine data set — wine","text":"wine data frame 178 rows 14 columns. first 13 variables report 13 constituents found three types wines. last column indicates class labels (1,2 3).","code":""},{"path":"/reference/wine.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Wine data set — wine","text":"","code":"wine"},{"path":"/reference/wine.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Wine data set — wine","text":"data frame containing following columns: Alcohol Malic acid Ash Alcalinity ash Magnesium Total phenols Flavanoids Nonflavanoid phenols Proanthocyanins Color intensity Hue OD280/OD315 diluted wines Proline y: class membership","code":""},{"path":"/reference/wine.html","id":"source","dir":"Reference","previous_headings":"","what":"Source","title":"Wine data set — wine","text":"Aeberhard, S. Forina, M. (1991). Wine. UCI Machine Learning Repository. https://doi.org/10.24432/C5PC7J.","code":""},{"path":"/reference/wine.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Wine data set — wine","text":"data results chemical analysis wines grown region Italy derived three different cultivars. analysis determined quantities 13 constituents found three types wines.","code":""},{"path":"/reference/wine.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Wine data set — wine","text":"Aeberhard, S., Coomans, D. De Vel, O. (1994). Comparative analysis statistical pattern recognition methods high dimensional settings. Pattern Recognition, 27(8), 1065-1077.","code":""},{"path":"/reference/wine.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Wine data set — wine","text":"","code":"data(wine) summary(wine) #> Alcohol Malicacid Ash Alcalinity_of_ash #> Min. :11.03 Min. :0.740 Min. :1.360 Min. :10.60 #> 1st Qu.:12.36 1st Qu.:1.603 1st Qu.:2.210 1st Qu.:17.20 #> Median :13.05 Median :1.865 Median :2.360 Median :19.50 #> Mean :13.00 Mean :2.336 Mean :2.367 Mean :19.49 #> 3rd Qu.:13.68 3rd Qu.:3.083 3rd Qu.:2.558 3rd Qu.:21.50 #> Max. :14.83 Max. :5.800 Max. :3.230 Max. :30.00 #> Magnesium Total_phenols Flavanoids Nonflavanoid_phenols #> Min. : 70.00 Min. :0.980 Min. :0.340 Min. :0.1300 #> 1st Qu.: 88.00 1st Qu.:1.742 1st Qu.:1.205 1st Qu.:0.2700 #> Median : 98.00 Median :2.355 Median :2.135 Median :0.3400 #> Mean : 99.74 Mean :2.295 Mean :2.029 Mean :0.3619 #> 3rd Qu.:107.00 3rd Qu.:2.800 3rd Qu.:2.875 3rd Qu.:0.4375 #> Max. :162.00 Max. :3.880 Max. :5.080 Max. :0.6600 #> Proanthocyanins Color_intensity Hue #> Min. :0.410 Min. : 1.280 Min. :0.4800 #> 1st Qu.:1.250 1st Qu.: 3.220 1st Qu.:0.7825 #> Median :1.555 Median : 4.690 Median :0.9650 #> Mean :1.591 Mean : 5.058 Mean :0.9574 #> 3rd Qu.:1.950 3rd Qu.: 6.200 3rd Qu.:1.1200 #> Max. :3.580 Max. :13.000 Max. :1.7100 #> X0D280_0D315_of_diluted_wines Proline y #> Min. :1.270 Min. : 278.0 Min. :1.000 #> 1st Qu.:1.938 1st Qu.: 500.5 1st Qu.:1.000 #> Median :2.780 Median : 673.5 Median :2.000 #> Mean :2.612 Mean : 746.9 Mean :1.938 #> 3rd Qu.:3.170 3rd Qu.: 985.0 3rd Qu.:3.000 #> Max. :4.000 Max. :1680.0 Max. :3.000"},{"path":"/reference/wireless.html","id":null,"dir":"Reference","previous_headings":"","what":"Wireless Indoor Localization — wireless","title":"Wireless Indoor Localization — wireless","text":"wireless data frame 2000 rows 8 columns. first 7 variables report measurements Wi-Fi signal strength received 7 Wi-Fi routers office location Pittsburgh (USA). last column indicates class labels.","code":""},{"path":"/reference/wireless.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Wireless Indoor Localization — wireless","text":"","code":"wireless"},{"path":"/reference/wireless.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Wireless Indoor Localization — wireless","text":"data frame containing following columns: V1 Signal strength router 1. V2 Signal strength router 2. V3 Signal strength router 3. V4 Signal strength router 4. V5 Signal strength router 5. V6 Signal strength router 6. V7 Signal strength router 7. V8 Group memberships, 1 4.","code":""},{"path":"/reference/wireless.html","id":"source","dir":"Reference","previous_headings":"","what":"Source","title":"Wireless Indoor Localization — wireless","text":"Bhatt, R. (2017). Wireless Indoor Localization. UCI Machine Learning Repository. https://doi.org/10.24432/C51880.","code":""},{"path":"/reference/wireless.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Wireless Indoor Localization — wireless","text":"Wi-Fi signal strength measured dBm, decibel milliwatts, expressed negative value ranging -100 0. labels correspond 4 different rooms. total, 4 groups 500 observations .","code":""},{"path":"/reference/wireless.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Wireless Indoor Localization — wireless","text":"Rohra, J.G., Perumal, B., Narayanan, S.J., Thakur, P. Bhatt, R.B. (2017). \"User Localization Indoor Environment Using Fuzzy Hybrid Particle Swarm Optimization & Gravitational Search Algorithm Neural Networks\". : Deep, K., et al. Proceedings Sixth International Conference Soft Computing Problem Solving. Advances Intelligent Systems Computing, vol 546. Springer, Singapore. https://doi.org/10.1007/978-981-10-3322-3_27","code":""},{"path":"/reference/wireless.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Wireless Indoor Localization — wireless","text":"","code":"data(wireless) summary(wireless) #> V1 V2 V3 V4 #> Min. :-74.00 Min. :-74.00 Min. :-73.00 Min. :-77.00 #> 1st Qu.:-61.00 1st Qu.:-58.00 1st Qu.:-58.00 1st Qu.:-63.00 #> Median :-55.00 Median :-56.00 Median :-55.00 Median :-56.00 #> Mean :-52.33 Mean :-55.62 Mean :-54.96 Mean :-53.57 #> 3rd Qu.:-46.00 3rd Qu.:-53.00 3rd Qu.:-51.00 3rd Qu.:-46.00 #> Max. :-10.00 Max. :-45.00 Max. :-40.00 Max. :-11.00 #> V5 V6 V7 V8 #> Min. :-89.00 Min. :-97.00 Min. :-98.00 Min. :1.00 #> 1st Qu.:-69.00 1st Qu.:-86.00 1st Qu.:-87.00 1st Qu.:1.75 #> Median :-64.00 Median :-82.00 Median :-83.00 Median :2.50 #> Mean :-62.64 Mean :-80.98 Mean :-81.73 Mean :2.50 #> 3rd Qu.:-56.00 3rd Qu.:-77.00 3rd Qu.:-78.00 3rd Qu.:3.25 #> Max. :-36.00 Max. :-61.00 Max. :-63.00 Max. :4.00"},{"path":[]},{"path":"/news/index.html","id":"new-features-1-1-3","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.3 (Development version)","text":"function rpkb() returns directly generated data set list.","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-3","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.3 (Development version)","text":"packages ggplot2 foreach called full packages using importFrom. nClust argument pkbc function default value. usage standardGeneric (see https://adv-r.hadley.nz/s4.html#s4-generics).","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-3","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.3 (Development version)","text":"goodpractice suggestions. specify rho pk.test() equal 1.","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-3","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.3 (Development version)","text":"Uniform notation: d dimension Eucldean space spherical data lying (d-1)-dimensional sphere Specify dimension input matrix x clustering functions. Improve explanation Average Silhouette Width. Better explanation motivation L2 normalization wireless data set application.","code":""},{"path":"/news/index.html","id":"quadratik-112-2024-10-29","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.1.2 (2024-10-29)","title":"QuadratiK 1.1.2 (2024-10-29)","text":"CRAN release: 2024-10-29","code":""},{"path":"/news/index.html","id":"new-features-1-1-2","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.2 (2024-10-29)","text":"Added NEWS.md file. detailed information README file documentation (methods usable clearer broader audience).","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-2","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.2 (2024-10-29)","text":"Added badges rOpenSci review status, test coverage, R CMD CHECK, CRAN downloads. Addition test codes testthat package. particular, testing random sampling PKBD wrapped Cauchy distribution equivalent.","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-2","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.2 (2024-10-29)","text":"scatter-plots related clustering algorithm results. computation evaluation measures (pkdc_validation function) true labels provided.","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-2","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.2 (2024-10-29)","text":"help documentation methods defined classes package can accessed directly (addition roxygen tag name aliases).","code":""},{"path":"/news/index.html","id":"quadratik-111-2024-06-05","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.1.1 (2024-06-05)","title":"QuadratiK 1.1.1 (2024-06-05)","text":"CRAN release: 2024-06-05","code":""},{"path":"/news/index.html","id":"new-features-1-1-1","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.1 (2024-06-05)","text":"Improve computation variance k-sample test Add breast_cancer wine data sets","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-1","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.1 (2024-06-05)","text":"print() summary() methods kb.test object two k-sample tests","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-1","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.1 (2024-06-05)","text":"Computation test statistics normality test","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-1","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.1 (2024-06-05)","text":"Correction typos","code":""},{"path":"/news/index.html","id":"quadratik-110-2024-05-14","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.1.0 (2024-05-14)","title":"QuadratiK 1.1.0 (2024-05-14)","text":"CRAN release: 2024-05-14","code":""},{"path":"/news/index.html","id":"new-features-1-1-0","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.1.0 (2024-05-14)","text":"Add GitHub links DESCRIPTION file Add unit tests Add CONTRIBUTING.md Add srr standards categories: General, Clustering Probability Distributions Add methods clustering object pkbc: show(), summary(), estract_stats(), plot() predict() Add vignette generate_rpkb.Rmd Add CODE_OF_CONDUCT.md Create GitHub page using pkgdown Add V-statistic Normality test, corresponding Critical Value Degrees Freedom function DOF_norm() Update kb.test S4 class slots, kb.test() function, unit-test codes, summary() print() method including V-statistic Add Trace statistic two k-sample tests Add computation Variance two k-sample tests","code":""},{"path":"/news/index.html","id":"minor-improvements-1-1-0","dir":"Changelog","previous_headings":"","what":"MINOR IMPROVEMENTS","title":"QuadratiK 1.1.0 (2024-05-14)","text":"Addition GitHub workflows R-CMD-check, test-coverage pkgcheck Update codes check arguments function pk.test() Update codes check arguments function kb.test() select_h() Modify indentation assignments (using ‘<-’) following goodpractice standards Update references README.md file","code":""},{"path":"/news/index.html","id":"bug-fixes-1-1-0","dir":"Changelog","previous_headings":"","what":"BUG FIXES","title":"QuadratiK 1.1.0 (2024-05-14)","text":"Add codemeta.json .Rbuildignore Add CONTRIBUTING.md .Rbuildignore Fix warning internal function compare_qq() Add doc folder .gitignore Modify returned test statistics multiplying sample size Remove “Nonparam” centering option Normality test Critical value V-statistic divided variance function pk.test()","code":""},{"path":"/news/index.html","id":"deprecated-and-defunct-1-1-0","dir":"Changelog","previous_headings":"","what":"DEPRECATED AND DEFUNCT","title":"QuadratiK 1.1.0 (2024-05-14)","text":"summary_stat() validation() substituted estract_stats() pkbc_validation()","code":""},{"path":"/news/index.html","id":"documentation-fixes-1-1-0","dir":"Changelog","previous_headings":"","what":"DOCUMENTATION FIXES","title":"QuadratiK 1.1.0 (2024-05-14)","text":"C++ functions set internal functions Correction typos Remove Date DESCRIPTION Update References DESCRIPTION file","code":""},{"path":"/news/index.html","id":"quadratik-100-2024-02-23","dir":"Changelog","previous_headings":"","what":"QuadratiK 1.0.0 (2024-02-23)","title":"QuadratiK 1.0.0 (2024-02-23)","text":"CRAN release: 2024-02-23","code":""},{"path":"/news/index.html","id":"new-features-1-0-0","dir":"Changelog","previous_headings":"","what":"NEW FEATURES","title":"QuadratiK 1.0.0 (2024-02-23)","text":"released CRAN","code":""}] diff --git a/src/QuadratiK.dll b/src/QuadratiK.dll index f906f691e0117018b9b8f13e751a0513e4508efe..ef2ea572b88b9359fb1e6f552af13106938a544f 100644 GIT binary patch delta 84 zcmV~$%MCV%vjqiwB@tBP!J_c delta 84 zcmZpeGoodness-of-Fit Tests

## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ -## Test Statistic: -0.9873721 0.6813104 -## Critical Value: 1.759572 8.901682 +## Test Statistic: -0.6798404 0.7516341 +## Critical Value: 1.751246 8.901682 ## H0 is rejected: FALSE FALSE ## Selected tuning parameter h: 0.4

If needed, we can specify \(\mu\) @@ -446,8 +446,8 @@

Goodness-of-Fit Tests

## Kernel-based quadratic distance Normality test ## U-statistic V-statistic ## ------------------------------------------------ -## Test Statistic: -0.6104855 0.7908929 -## Critical Value: 1.637207 8.901682 +## Test Statistic: -0.3672468 0.8436093 +## Critical Value: 2.305779 8.901682 ## H0 is rejected: FALSE FALSE ## Selected tuning parameter h: 0.4

Two-sample test

@@ -463,8 +463,8 @@

Goodness-of-Fit Tests

## Kernel-based quadratic distance two-sample test ## U-statistic Dn Trace ## ------------------------------------------------ -## Test Statistic: 5.825482 11.02655 -## Critical Value: 0.4229694 0.8015014 +## Test Statistic: 5.74045 11.20924 +## Critical Value: 0.6623309 1.29477 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4 @@ -482,8 +482,8 @@

Goodness-of-Fit Tests

## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ -## Test Statistic: 7.70439 11.29268 -## Critical Value: 0.7086637 1.039496 +## Test Statistic: 7.420552 11.26751 +## Critical Value: 0.8736407 1.327543 ## H0 is rejected: TRUE TRUE ## CV method: subsampling ## Selected tuning parameter h: 0.4 @@ -508,13 +508,13 @@

Test for uniformity on the sphere

## U-statistic: ## ## H0 is rejected: FALSE -## Statistic Un: 0.2882334 -## Critical value: 1.735172 +## Statistic Un: -1.4196 +## Critical value: 1.866933 ## ## V-statistic: ## ## H0 is rejected: FALSE -## Statistic Vn: 18.77083 +## Statistic Vn: 13.54515 ## Critical value: 23.22949 @@ -534,21 +534,21 @@

Poisson kernel-based distribution (PKBD)

x <- rpkb(n = 100, mu = mu, rho = rho) head(x) 
##           [,1]        [,2]        [,3]
-## [1,] 0.9775137  0.19476960  0.08081988
-## [2,] 0.6859924 -0.66157455  0.30287544
-## [3,] 0.9970575 -0.06396612 -0.04224593
-## [4,] 0.9990864 -0.01407321 -0.04035286
-## [5,] 0.9790443 -0.16982909  0.11238456
-## [6,] 0.9882565  0.12532919  0.08741655
+## [1,] 0.9567587 0.03814016 0.28837147 +## [2,] 0.9889017 0.14159121 -0.04500343 +## [3,] 0.9891182 -0.12795485 0.07261316 +## [4,] 0.8536550 -0.10580009 0.50997996 +## [5,] 0.9735937 -0.10170156 -0.20438216 +## [6,] 0.7637669 -0.41480680 -0.49456597 
dens_x <- dpkb(x, mu = mu, rho = rho)
 head(dens_x)
-
##             [,1]
-## [1,]  1.33328770
-## [2,]  0.03465772
-## [3,]  7.99197356
-## [4,] 12.03265310
-## [5,]  1.45040792
-## [6,]  2.75170555
+
##            [,1]
+## [1,] 0.58082769
+## [2,] 2.91315586
+## [3,] 2.97090321
+## [4,] 0.10575374
+## [5,] 1.09568914
+## [6,] 0.05266002

Clustering Algorithm for Spherical Data

@@ -575,38 +575,38 @@

Clustering Algorithm for Spherical Data

## ## Summary: ## nClust LogLik WCSS -## [1,] 2 -622.3166 398.7752 -## [2,] 3 -310.4306 320.8642 -## [3,] 4 -301.8025 319.5156 +## [1,] 2 -612.1145 402.2211 +## [2,] 3 -319.1574 319.2195 +## [3,] 4 -310.2422 319.0612 ## ## Results for 2 clusters: ## Estimated Mixing Proportions (alpha): -## [1] 0.7104349 0.2895651 +## [1] 0.7007738 0.2992262 ## ## Clustering table: ## ## 1 2 -## 211 89 +## 209 91 ## ## ## Results for 3 clusters: ## Estimated Mixing Proportions (alpha): -## [1] 0.3318428 0.3453057 0.3228515 +## [1] 0.3365681 0.3319747 0.3314572 ## ## Clustering table: ## ## 1 2 3 -## 100 102 98 +## 102 99 99 ## ## ## Results for 4 clusters: ## Estimated Mixing Proportions (alpha): -## [1] 0.32391578 0.31729722 0.02786929 0.33091772 +## [1] 0.003546364 0.331499072 0.336611237 0.328343327 ## ## Clustering table: ## ## 1 2 3 4 -## 98 92 8 102 +## 1 99 102 98

The software includes additional graphical functions, aiding users in validating and representing the cluster results as well as enhancing the interpretability and usability of the analysis.

@@ -620,25 +620,25 @@

Clustering Algorithm for Spherical Data

val_res
## $metrics
 ##             2         3         4
-## ASW 0.5105145 0.6937623 0.6060139
+## ASW 0.5177927 0.7065232 0.5851571
 ## 
 ## $IGP
 ## $IGP[[1]]
 ## NULL
 ## 
 ## $IGP[[2]]
-## [1] 0.9949495 1.0000000
+## [1] 1.000000 0.990566
 ## 
 ## $IGP[[3]]
-## [1] 0.9897959 0.9900990 1.0000000
+## [1] 1.00 0.99 1.00
 ## 
 ## $IGP[[4]]
-## [1] 1.0000000 1.0000000 1.0000000 0.9897959
+## [1] 1 1 1 1 
# Plot method for the pkbc object:
 # - scatter plot of data points on the sphere
 # - elbow plot for helping the choice of the number of clusters
 plot(cluster_res)
-

+

diff --git a/vignettes/TwoSample_test.html b/vignettes/TwoSample_test.html index 5d98ac9..e0744f0 100644 --- a/vignettes/TwoSample_test.html +++ b/vignettes/TwoSample_test.html @@ -422,7 +422,7 @@

Non-parametric two-sample test

two_test@h$h_sel
## [1] 1.6
two_test@h$power.plot
-

+

For more details visit the help documentation of the select_h() function.

help(select_h)
diff --git a/vignettes/generate_rpkb.html b/vignettes/generate_rpkb.html index 318d9ab..ba2af61 100644 --- a/vignettes/generate_rpkb.html +++ b/vignettes/generate_rpkb.html @@ -9231,9 +9231,9 @@

Random sampling from the Poisson # replication file. rglwidget()

-
- -
+ + 
close3d()
diff --git a/vignettes/kSample_test.html b/vignettes/kSample_test.html index b30b466..31ac7a5 100644 --- a/vignettes/kSample_test.html +++ b/vignettes/kSample_test.html @@ -423,9 +423,9 @@

k-sample test

alternatives. Here we consider the location alternatives.

set.seed(2468)
 h_k <- select_h(x=x, y=y, alternative="location")
-

+

h_k$h_sel
-
## [1] 1.6
+
## [1] 2

The select_h function has also generated a figure displaying the obtained power versus the considered \(h\), for each value of alternative \(\delta\) considered.

We can now perform the \(k\)-sample @@ -437,11 +437,11 @@

k-sample test

## Kernel-based quadratic distance k-sample test ## U-statistic Dn Trace ## ------------------------------------------------ -## Test Statistic: 4.238404 6.834896 -## Critical Value: 1.113956 1.797719 +## Test Statistic: 4.504812 7.668806 +## Critical Value: 1.079711 1.839428 ## H0 is rejected: TRUE TRUE ## CV method: subsampling -## Selected tuning parameter h: 1.6 +## Selected tuning parameter h: 2

The function kb.test() returns an object of class kb.test. The show method for the kb.test object shows the computed statistics with @@ -456,8 +456,8 @@

k-sample test

## 
 ##  Kernel-based quadratic distance k-sample test 
 ##   Statistic    Value Critical_Value Reject_H0
-## 1        Dn 4.238404       1.113956      TRUE
-## 2     Trace 6.834896       1.797719      TRUE
+## 1 Dn 4.504812 1.079711 TRUE +## 2 Trace 7.668806 1.839428 TRUE 
summary_ktest$summary_tables
## [[1]]
 ##            Group 1    Group 2    Group 3      Overall
diff --git a/vignettes/wireless_clustering.html b/vignettes/wireless_clustering.html
index fc06b8e..c6357b5 100644
--- a/vignettes/wireless_clustering.html
+++ b/vignettes/wireless_clustering.html
@@ -366,14 +366,20 @@ 
The Wireless Indoor Localization Data

 Notice that, the Wi-Fi signal strength is measured in dBm, decibel
 milliwatts, which is expressed as a negative value ranging from -100 to
 0. In total, we have 500 observations for each room.

-
Given that Wi-Fi signal strength values are inherently bounded within
-a certain range, it is possible to consider the spherically transformed
-data points using \(L_2\)
-normalization. This transformation maps the data onto the surface of a
-7-dimensional sphere, ensuring that each observation has a uniform
-length. By projecting the data onto this high-dimensional sphere, we can
-take advantage of spherical geometry, and consequently perform the
-proposed clustering algorithm.

+
In many wireless applications, the relative signal strengths across
+routers are more relevant to the underlying spatial patterns and device
+positioning than the absolute magnitudes. Additionally, absolute signal
+strength can be affected by noise, device orientation or environmental
+factors. In this case, it is reasonable to consider the spherically
+transformed data points using L2 normalization. This transformation maps
+the data onto the surface of a 6-dimensional sphere, ensuring that each
+observation has a uniform length. Given that absolute signal strength is
+not critical to the research question, the spherical representation
+provides a meaningful and interpretable framework for studying the data
+set. In general, it is appropriate to consider spherically transformed
+data points when: (i) the absolute length of the measurements is
+irrelevant or too noisy; (ii) if we are more interested in the relative
+distributions (or angular relationships) between data points.

 
We perform the clustering algorithm on the wireless data
 set. We consider the \(K= 3, 4, 5\) as
 possible values for the number of clusters.