From f507560a2a3cf2cdbeca5ba21d130d74a783bc76 Mon Sep 17 00:00:00 2001
From: changwoo-lee <45101999+changwoo-lee@users.noreply.github.com>
Date: Fri, 3 May 2024 20:40:43 +0000
Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20changwoo?=
 =?UTF-8?q?-lee/bspme@ad50543b516976c27bd045ac34caf86afa5cb210=20?=
 =?UTF-8?q?=F0=9F=9A=80?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 index.html  | 2 +-
 pkgdown.yml | 2 +-
 search.json | 2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/index.html b/index.html
index 9e0a944..c0d984c 100644
--- a/index.html
+++ b/index.html
@@ -151,7 +151,7 @@ <h2 id="examples">Examples<a class="anchor" aria-label="anchor" href="#examples"
 <div class="section level2">
 <h2 id="acknowldegements">Acknowldegements<a class="anchor" aria-label="anchor" href="#acknowldegements"></a>
 </h2>
-<p>This work was supported by the National Institute of Environmental Health Sciences (NIEHS) of the National Institute of Health (NIH) under R01ES031990.</p>
+<p>This work was supported by the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH) under R01ES031990.</p>
 </div>
 </div>
   </main><aside class="col-md-3"><div class="links">
diff --git a/pkgdown.yml b/pkgdown.yml
index d580c8d..bee36cd 100644
--- a/pkgdown.yml
+++ b/pkgdown.yml
@@ -3,7 +3,7 @@ pkgdown: 2.0.9
 pkgdown_sha: ~
 articles:
   no2-exposure-and-health-data-analysis: no2-exposure-and-health-data-analysis.html
-last_built: 2024-05-03T20:07Z
+last_built: 2024-05-03T20:40Z
 urls:
   reference: https://changwoo-lee.github.io/bspme/reference
   article: https://changwoo-lee.github.io/bspme/articles
diff --git a/search.json b/search.json
index 92aae9d..0cc2782 100644
--- a/search.json
+++ b/search.json
@@ -1 +1 @@
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Interpretation of Sections 15 and 16","title":"GNU General Public License","text":"disclaimer warranty limitation liability provided given local legal effect according terms, reviewing courts shall apply local law closely approximates absolute waiver civil liability connection Program, unless warranty assumption liability accompanies copy Program return fee. END TERMS CONDITIONS","code":""},{"path":"https://changwoo-lee.github.io/bspme/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. 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If not, see <http://www.gnu.org/licenses/>. <program>  Copyright (C) <year>  <name of author> 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":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"obtain-posterior-predictive-distribution-at-health-subject-locations","dir":"Articles","previous_headings":"","what":"Obtain posterior predictive distribution at health subject locations","title":"NO2 exposure and health data analysis","text":"example, use Gaussian process obtain predictive distribution \\((X_1,\\dots,X_n)\\) ln(NO2) exposure health subject locations. posterior predictive distribution (ppd) n-dimensional multivariate normal \\(N(\\mu_X, Q_X^{-1})\\), used prior incorporate prediction uncertainty spatial correlation NO2 exposures health subject locations estimation health model. Specifically, use exponential covariance kernel fixed range 8 (km) standard deviation 1.  covariance \\((X_1,\\dots,X_n)\\) contains spatial dependency information second order, precision matrix (inverse covariance matrix) \\(Q_X\\) dense matrix. comes fitting Bayesian linear model spatial exposure measurement error, naive implementation n-dimensional multivariate normal prior \\(N(\\mu_X, Q_X^{-1})\\) takes cubic time complexity posterior inference MCMC iteration, becomes infeasible run \\(n\\) large tens thousands.","code":"coords_monitor = cbind(data_jan10$lon, data_jan10$lat) coords_health = cbind(health_sim$lon, health_sim$lat) a = 8; sigma = 1; # assume known # distance calculation distmat_xx <- rdist.earth(coords_monitor, miles = F) distmat_xy <- rdist.earth(coords_monitor, coords_health, miles = F) distmat_yy <- rdist.earth(coords_health, miles = F)  # GP covariance matrices Sigmaxx = fields::Matern(distmat_xx, smoothness = 0.5, range = a, phi = sigma^2) Sigmaxy = fields::Matern(distmat_xy, smoothness = 0.5, range = a, phi = sigma^2) Sigmayy = fields::Matern(distmat_yy, smoothness = 0.5, range = a, phi = sigma^2)  # posterior predictive mean and covariance X_mean <- t(Sigmaxy) %*% solve(Sigmaxx, data_jan10$lnNO2) X_cov <- Sigmayy - t(Sigmaxy) %*% solve(Sigmaxx,Sigmaxy) # n_y by n_y  # visualization of posterior predictive distribution. Black triangle is monitoring station locations. par(mfrow = c(1,2), mar = c(4, 4, 4, 6)) quilt.plot(cbind(health_sim$lon, health_sim$lat),           X_mean, main = \"posterior predictive mean at health subject locations\",           xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5), legend.lab = \"ln(NO2)\") maps::map(\"county\", \"Texas\", add = T) # posterior predictive sd of exposure at health data quilt.plot(cbind(health_sim$lon, health_sim$lat),           sqrt(diag(X_cov)), main = \"posterior predictive stdev at health subject locations\",           xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5), legend.lab = \"ln(NO2)\") maps::map(\"county\", \"Texas\", add = T)"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"run-vecchia-approximation-to-get-sparse-precision-matrix","dir":"Articles","previous_headings":"","what":"Run Vecchia approximation to get sparse precision matrix","title":"NO2 exposure and health data analysis","text":"propose use Vecchia approximation get new multivariate normal prior \\((X_1,\\dots,X_n)\\) sparse precision matrix, crucial scalable implementation health models spatial exposure measurement error preserving spatial dependency information original covariance matrix closely possible.","code":"#Run the Vecchia approximation to get the sparse precision matrix.  # Vecchia approximation run_vecchia = vecchia_cov(X_cov, coords = cbind(health_sim$lon, health_sim$lat),             n.neighbors = 10) Q_sparse = run_vecchia$Q run_vecchia$cputime #> Time difference of 0.5541399 secs"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"fit-bspme-with-sparse-precision-matrix","dir":"Articles","previous_headings":"","what":"Fit bspme with sparse precision matrix","title":"NO2 exposure and health data analysis","text":"can now fit main function bspme package sparse precision matrix. takes less 10 seconds (1) run Vecchia approximation (2) run MCMC algorithm total 1100 iterations (Macbook M1 max CPU 64GB memory).","code":"# fit the model fit_me = blm_me(Y = health_sim$Y,               X_mean = X_mean,               X_prec = Q_sparse, # sparse precision matrix               Z = health_sim$Z,               nburn = 100, # increase as needed               nsave = 1000, # increase as needed               nthin = 1) # increase as needed fit_me$cputime #> Time difference of 6.949046 secs summary(fit_me$posterior) # posterior summary #>  #> Iterations = 1:1000 #> Thinning interval = 1  #> Number of chains = 1  #> Sample size per chain = 1000  #>  #> 1. Empirical mean and standard deviation for each variable, #>    plus standard error of the mean: #>  #>                Mean      SD Naive SE Time-series SE #> (Intercept)  0.7490 0.08557 0.002706       0.012488 #> exposure.1  -1.9544 0.05068 0.001603       0.007311 #> covariate.1  2.9578 0.04151 0.001313       0.003219 #> sigma2_Y     0.9546 0.14537 0.004597       0.029898 #>  #> 2. Quantiles for each variable: #>  #>                2.5%     25%     50%     75%   97.5% #> (Intercept)  0.5882  0.6911  0.7466  0.8024  0.9289 #> exposure.1  -2.0437 -1.9883 -1.9556 -1.9218 -1.8421 #> covariate.1  2.8702  2.9305  2.9584  2.9863  3.0382 #> sigma2_Y     0.6677  0.8498  0.9593  1.0553  1.2473"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"fit-bspme-without-sparse-precision-matrix-slow","dir":"Articles","previous_headings":"","what":"Fit bspme without sparse precision matrix (slow)","title":"NO2 exposure and health data analysis","text":"mentioned , precision matrix dense, posterior computation becomes infeasible \\(n\\) large, tens thousands. See following example \\(n=2000\\). Even 11 MCMC iterations, takes 15 seconds run MCMC algorithm. number iterations (1100), take 1500 seconds (.e. 25 minutes) run MCMC algorithm. gap becomes much larger number health subject locations grows, tens thousands. , naive algorithm uses dense precision matrix becomes infeasible, can carry inference using Vecchia approximation. quality Vecchia approximation depends size nearest neighbors, controlled argument n.neighbors vecchia_cov function. larger n.neighbors, better approximation. simulation study Lee et al. (2024) demonstrates Vecchia approximation relatively small n.neighbors can provide scalable two-stage Bayesian inference method results (e.g. RMSE, empirical coverage) close based dense precision matrix fully Bayesian analysis.","code":"# fit the model, without vecchia approximation # I will only run 11 iteration for illustration purpose Q_dense = solve(X_cov) # inverting 2000 x 2000 matrix, takes some time # run fit_me_dense = blm_me(Y = health_sim$Y,                 X_mean = X_mean,                 X_prec = Q_dense, # dense precision                 Z = health_sim$Z,                 nburn = 1, # Running only 11 iterations for an illustration purpose                 nsave = 10, # Running only 11 iterations for an illustration purpose                 nthin = 1) # Running only 11 iterations for an illustration purpose fit_me_dense$cputime #> Time difference of 16.86512 secs"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"NO2 exposure and health data analysis","text":"Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634.","code":""},{"path":"https://changwoo-lee.github.io/bspme/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Changwoo Lee. Author, maintainer. Eun Sug Park. Author.","code":""},{"path":"https://changwoo-lee.github.io/bspme/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Lee C, Park E (2024). bspme: Bayesian Spatial Measurement Error Models. R package version 1.0.1, https://changwoo-lee.github.io/bspme/.","code":"@Manual{,   title = {bspme: Bayesian Spatial Measurement Error Models},   author = {Changwoo Lee and Eun Sug Park},   year = {2024},   note = {R package version 1.0.1},   url = {https://changwoo-lee.github.io/bspme/}, }"},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"bspme","dir":"","previous_headings":"","what":"Bayesian Spatial Measurement Error Models","title":"Bayesian Spatial Measurement Error Models","text":"Go package website: [link] See vignette NO2 exposure simulated health data: [link] See bspme_1.0.1.pdf pdf file package manual. bspme R package provides fast, scalable inference tools Bayesian spatial exposure measurement error models, namely, Bayesian linear generalized linear models presence spatial exposure measurement error covariate(s). models typically arise two-stage Bayesian analysis environmental exposures health outcomes. first-stage model, predictions covariate interest (“exposure”) uncertainty information (typically contained MCMC samples) obtained used form multivariate normal prior distribution X ∼ N(μ,Σ) exposure second-stage regression model. Naive, non-sparse choices precision matrix Q = Σ−1 multivariate normal (sample precision matrix) lead MCMC posterior inference algorithm infeasible run large number subjects n cubic computational cost associated n-dimensional MVN prior. sparse precision matrix Q obtained Vecchia approximation, bspme package offers fast, scalable algorithms conduct posterior inference large health datasets, number subjects n possibly reaching tens thousands. details, please see following paper: Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634. https://arxiv.org/abs/2401.00634","code":""},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"installation","dir":"","previous_headings":"","what":"Installation","title":"Bayesian Spatial Measurement Error Models","text":"can install R package bspme following code: browse see vignettes, run","code":"# install.packages(\"devtools\") devtools::install_github(\"changwoo-lee/bspme\", build_vignettes = T) browseVignettes(\"bspme\")"},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"functionality","dir":"","previous_headings":"","what":"Functionality","title":"Bayesian Spatial Measurement Error Models","text":"see function description R environment, run following lines:","code":"?blm_me ?bglm_me ?vecchia_cov"},{"path":[]},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"examples","dir":"","previous_headings":"","what":"Examples","title":"Bayesian Spatial Measurement Error Models","text":"Please see vignette “NO2-exposure--health-data-analysis”.","code":""},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"acknowldegements","dir":"","previous_headings":"","what":"Acknowldegements","title":"Bayesian Spatial Measurement Error Models","text":"work supported National Institute Environmental Health Sciences (NIEHS) National Institute Health (NIH) R01ES031990.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/NO2_Jan2012.html","id":null,"dir":"Reference","previous_headings":"","what":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","title":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","text":"dataset contains daily average NO2 (nitrogen dioxide) concentrations obtained 21 monitoring stations around Harris County, Texas, January 2012.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/NO2_Jan2012.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","text":"","code":"data(NO2_Jan2012)"},{"path":"https://changwoo-lee.github.io/bspme/reference/NO2_Jan2012.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","text":"data frame 651 (21 sites x 31 days) rows 5 variables: date date POSIXct format site_name monitoring station name lon monitoring station longitude lat monitoring station latitude lnNO2 natural logarithm daily average NO2 concentrations measured parts per billion volume (ppbv)","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":null,"dir":"Reference","previous_headings":"","what":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"function fits Bayesian generalized linear model presence spatial exposure measurement error covariate(s) \\(X\\). One important features function allows sparse matrix input prior precision matrix \\(X\\) scalable computation. version 1.0.0, Bayesian logistic regression model supported among GLMs, function bglm_me() runs Gibbs sampler carry posterior inference using Polya-Gamma augmentation (Polson et al., 2013). See \"Details\" section model description Lee et al. (2024) application example environmental epidemiology.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"","code":"bglm_me(   Y,   X_mean,   X_prec,   Z,   family = binomial(link = \"logit\"),   nburn = 5000,   nsave = 5000,   nthin = 1,   prior = NULL,   saveX = FALSE )"},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"Y vector<int>, n 1 binary response vector. X_mean vector<num>, n 1 prior mean vector \\(\\mu_X\\). q multiple exposures subject measurement error, can length q list n 1 vectors. X_prec matrix<num>, n n prior precision matrix \\(Q_X\\), allows sparse format Matrix package. q multiple exposures subject measurement error, can length q list n n matrices. Z matrix<num>, n p matrix containing p covariates subject measurement error. family class family, description error distribution link function used model. Currently, supports binomial(link = \"logit\"). nburn integer, number burn-iterations (default=5000). nsave integer, number posterior samples (default=5000). Total number MCMC iteration nburn + nsave * nthin. nthin integer, thin-rate (default=1). prior list, list prior parameters regression model. Default list(var_beta = 100). saveX logical, default FALSE, whether save posterior samples X (exposure).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"List following: posterior nsave (q+p) matrix posterior samples \\(\\beta_x\\) \\(\\beta_z\\) coda::mcmc object. time time taken running MCMC (seconds) X_save (saveX = TRUE) posterior samples X","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"Let \\(Y_i\\) binary response, \\(X_i\\) \\(q\\times 1\\) covariate vector subject spatial exposure measurement error, \\(Z_i\\) \\(p\\times 1\\) covariate vector without measurement error. Consider logistic regression model, independently \\(=1,\\dots,n,\\) $$\\log(Pr(Y_i = 1)/Pr(Y_i = 0)) = \\beta_0 + X_i^\\top \\beta_X +  Z_i^\\top \\beta_Z.$$ Spatial exposure measurement error \\(X_i\\) (\\(=1,\\dots,n\\)) incorporated model using multivariate normal prior. example, \\(q=1\\), \\(n-\\)dimensional multivariate normal prior \\(X = (X_1,\\dots,X_n)^\\top\\), $$(X_1,\\dots,X_n)\\sim N_n(\\mu_X, Q_X^{-1}).$$ importantly, allows sparse matrix input prior precision matrix \\(Q_X\\) scalable computation, can obtained Vecchia approximation. \\(q>1\\), \\(q\\) independent \\(n-\\)dimensional multivariate normal priors assumed. consider normal priors regression coefficients, $$\\beta_0 \\sim N(0, V_\\beta), \\quad \\beta_{X,j} \\stackrel{iid}{\\sim} N(0, V_\\beta), \\quad \\beta_{Z,k} \\stackrel{iid}{\\sim} N(0, V_\\beta)$$ var_beta corresponds \\(V_\\beta\\).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"Polson, N. G., Scott, J. G., & Windle, J. (2013). Bayesian inference logistic models using Pólya–Gamma latent variables. Journal American statistical Association, 108(504), 1339-1349. Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"","code":"if (FALSE) { library(bspme) data(NO2_Jan2012) data(health_sim) library(fields) library(maps) # Obtain the predicted exposure mean and covariance at simulated health subject locations # based on NO2 data obtained on Jan 10, 2012 # using a Gaussian process prior with mean zero and exponential covariance kernel # with a fixed range 8 (in km) and standard deviation 1.  # exposure data data_jan10 = NO2_Jan2012[NO2_Jan2012$date == as.POSIXct(\"2012-01-10\"),] coords_monitor = cbind(data_jan10$lon, data_jan10$lat)  # health data coords_health = cbind(health_sim$lon, health_sim$lat)  distmat_xx <- rdist.earth(coords_monitor, miles = F) distmat_xy <- rdist.earth(coords_monitor, coords_health, miles = F) distmat_yy <- rdist.earth(coords_health, miles = F)  a = 8; sigma = 1; # assume known  Sigmaxx = fields::Matern(distmat_xx, smoothness = 0.5, range = a, phi = sigma^2) Sigmaxy = fields::Matern(distmat_xy, smoothness = 0.5, range = a, phi = sigma^2) Sigmayy = fields::Matern(distmat_yy, smoothness = 0.5, range = a, phi = sigma^2)  # posterior predictive mean and covariance of exposure at health subject locations X_mean <- t(Sigmaxy) %*% solve(Sigmaxx, data_jan10$lnNO2) X_cov <- Sigmayy - t(Sigmaxy) %*% solve(Sigmaxx,Sigmaxy) # n_y by n_y  # visualize # monitoring station exposure data quilt.plot(cbind(data_jan10$lon, data_jan10$lat),            data_jan10$lnNO2, main = \"NO2 exposures (in log) at 21 monitoring stations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive mean of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            X_mean, main = \"posterior predictive mean of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive sd of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            sqrt(diag(X_cov)), main = \"posterior predictive sd of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)   # vecchia approximation run_vecchia = vecchia_cov(X_cov, coords = cbind(health_sim$lon, health_sim$lat),                           n.neighbors = 10) Q_sparse = run_vecchia$Q run_vecchia$cputime  # fit the model, binary response fit = bglm_me(Y = health_sim$Ybinary,                 X_mean = X_mean,                 X_prec = Q_sparse, # sparse precision matrix                 Z = health_sim$Z,                 family = binomial(link = \"logit\"),                 nburn = 5000,                 nsave = 5000,                 nthin = 1) fit$cputime summary(fit$posterior) library(bayesplot) bayesplot::mcmc_trace(fit$posterior) }"},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":null,"dir":"Reference","previous_headings":"","what":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"function fits Bayesian linear regression model presence spatial exposure measurement error covariate(s) \\(X\\). One important features function allows sparse matrix input prior precision matrix \\(X\\) scalable computation. Function blm_me() runs Gibbs sampler carry posterior inference;  see \"Details\" section model description, Lee et al. (2024) application example environmental epidemiology.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"","code":"blm_me(   Y,   X_mean,   X_prec,   Z,   nburn = 5000,   nsave = 5000,   nthin = 1,   prior = NULL,   saveX = FALSE )"},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"Y vector<int>, n 1 continuous response vector. X_mean vector<num>, n 1 prior mean vector \\(\\mu_X\\). q multiple exposures subject measurement error, can length q list n 1 vectors. X_prec matrix<num>, n n prior precision matrix \\(Q_X\\), allows sparse format Matrix package. q multiple exposures subject measurement error, can length q list n n matrices. Z matrix<num>, n p matrix containing p covariates subject measurement error. nburn integer, number burn-iterations (default=5000). nsave integer, number posterior samples (default=5000). Total number MCMC iteration nburn + nsave * nthin. nthin integer, thin-rate (default=1). prior list, list prior parameters regression model. Default list(var_beta = 100, a_Y = 0.01, b_Y = 0.01). saveX logical, default FALSE, whether save posterior samples X (exposure).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"list following: posterior nsave (q + p + 1) matrix posterior samples \\(\\beta_X\\), \\(\\beta_Z\\), \\(\\sigma_Y^2\\) coda::mcmc object. time time taken running MCMC (seconds) X_save (saveX = TRUE) posterior samples X","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"Let \\(Y_i\\) continuous response, \\(X_i\\) \\(q\\times 1\\) covariate vector subject spatial exposure measurement error, \\(Z_i\\) \\(p\\times 1\\) covariate vector without measurement error. Consider normal linear regression model, $$Y_i = \\beta_0 + X_i^\\top \\beta_X +  Z_i^\\top \\beta_Z + \\epsilon_i,\\quad  \\epsilon_i \\stackrel{iid}{\\sim} N(0, \\sigma^2_Y), \\quad =1,\\dots,n.$$ Spatial exposure measurement error \\(X_i\\) \\(=1,\\dots,n\\) incorporated model using multivariate normal prior. example \\(q=1\\), \\(n-\\)dimensional multivariate normal prior \\(X = (X_1,\\dots,X_n)^\\top\\), $$(X_1,\\dots,X_n)\\sim N_n(\\mu_X, Q_X^{-1}).$$ importantly, allows sparse matrix input prior precision matrix \\(Q_X\\) scalable computation, can obtained Vecchia approximation. \\(q>1\\), \\(q\\) independent \\(n-\\)dimensional multivariate normal priors assumed. consider semiconjugate priors regression coefficients error variance, $$\\beta_0 \\sim N(0, V_\\beta), \\quad \\beta_{X,j} \\stackrel{iid}{\\sim} N(0, V_\\beta), \\quad \\beta_{Z,k} \\stackrel{iid}{\\sim} N(0, V_\\beta), \\quad \\sigma_Y^2 \\sim IG(a_Y, b_Y).$$ var_beta corresponds \\(V_\\beta\\), a_Y b_Y correspond hyperparameters inverse gamma prior \\(\\sigma^2_Y\\).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"","code":"if (FALSE) { library(bspme) data(NO2_Jan2012) data(health_sim) library(fields) library(maps) # Obtain the predicted exposure mean and covariance at simulated health subject locations # based on NO2 data obtained on Jan 10, 2012 # using a Gaussian process prior with mean zero and exponential covariance kernel # with a fixed range 8 (in km) and standard deviation 1.  # exposure data data_jan10 = NO2_Jan2012[NO2_Jan2012$date == as.POSIXct(\"2012-01-10\"),] coords_monitor = cbind(data_jan10$lon, data_jan10$lat)  # health data coords_health = cbind(health_sim$lon, health_sim$lat)  distmat_xx <- rdist.earth(coords_monitor, miles = F) distmat_xy <- rdist.earth(coords_monitor, coords_health, miles = F) distmat_yy <- rdist.earth(coords_health, miles = F)  a = 8; sigma = 1; # assume known  Sigmaxx = fields::Matern(distmat_xx, smoothness = 0.5, range = a, phi = sigma^2) Sigmaxy = fields::Matern(distmat_xy, smoothness = 0.5, range = a, phi = sigma^2) Sigmayy = fields::Matern(distmat_yy, smoothness = 0.5, range = a, phi = sigma^2)  # posterior predictive mean and covariance of exposure at health subject locations X_mean <- t(Sigmaxy) %*% solve(Sigmaxx, data_jan10$lnNO2) X_cov <- Sigmayy - t(Sigmaxy) %*% solve(Sigmaxx,Sigmaxy) # n_y by n_y  # visualize # monitoring station exposure data quilt.plot(cbind(data_jan10$lon, data_jan10$lat),            data_jan10$lnNO2, main = \"NO2 exposures (in log) at 21 monitoring stations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive mean of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            X_mean, main = \"posterior predictive mean of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive sd of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            sqrt(diag(X_cov)), main = \"posterior predictive sd of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)   # vecchia approximation run_vecchia = vecchia_cov(X_cov, coords = cbind(health_sim$lon, health_sim$lat),                           n.neighbors = 10) Q_sparse = run_vecchia$Q run_vecchia$cputime  # fit the model, continuous response fit = blm_me(Y = health_sim$Y,                 X_mean = X_mean,                 X_prec = Q_sparse, # sparse precision matrix                 Z = health_sim$Z,                 nburn = 5000,                 nsave = 5000,                 nthin = 1) fit$cputime summary(fit$posterior) library(bayesplot) bayesplot::mcmc_trace(fit$posterior) }"},{"path":"https://changwoo-lee.github.io/bspme/reference/bspme-package.html","id":null,"dir":"Reference","previous_headings":"","what":"bspme: Bayesian Spatial Measurement Error Models — bspme-package","title":"bspme: Bayesian Spatial Measurement Error Models — bspme-package","text":"Scalable methods fitting Bayesian linear generalized linear models presence spatial exposure measurement error. models typically arise two-stage Bayesian analysis environmental exposures health outcomes. first-stage model, predictions covariate interest (''exposure'') uncertainty information (typically contained MCMC samples) used form multivariate normal prior distribution exposure second-stage regression model. package also provides implementation methods used Lee et al. (2024) https://arxiv.org/abs/2401.00634.","code":""},{"path":[]},{"path":"https://changwoo-lee.github.io/bspme/reference/bspme-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"bspme: Bayesian Spatial Measurement Error Models — bspme-package","text":"Maintainer: Changwoo Lee c.lee@stat.tamu.edu Authors: Eun Sug Park","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/health_sim.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulated health data — health_sim","title":"Simulated health data — health_sim","text":"Simulated health data associated ln(NO2) concentration Jan 10, 2012. details, see health_sim.R.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/health_sim.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulated health data — health_sim","text":"","code":"data(health_sim)"},{"path":"https://changwoo-lee.github.io/bspme/reference/health_sim.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Simulated health data — health_sim","text":"data frame n = 2000 rows 6 variables: Y simulated continuous health outcome Ybinary simulated binary health outcome lon simulated health subject longitude lat simulated health subject latitude Z simulated covariate (p=1) subject measurement error X_true true ln(NO2) exposure used simulating health outcome","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":null,"dir":"Reference","previous_headings":"","what":"Run Vecchia approximation given a covariance matrix — vecchia_cov","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"Given multivariate normal (MVN) distribution covariance matrix \\(\\Sigma\\), function finds sparse precision matrix (inverse covariance) \\(Q\\) based Vecchia approximation (Vecchia 1988, Katzfuss Guinness 2021), \\(N(\\mu, Q^{-1})\\) sparse MVN approximates original MVN \\(N(\\mu, \\Sigma)\\). algorithm based pseudocode 2 Finley et al. (2019).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"","code":"vecchia_cov(Sigma, coords, n.neighbors, ord = NULL, KLdiv = FALSE)"},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"Sigma matrix<num>, n n covariance matrix coords matrix<num>, n 2 coordinate matrix nearest neighborhood search n.neighbors integer, number nearest neighbors (k) determine conditioning set Vecchia approximation ord vector<int>, length n vector, ordering data. NULL, ordering based first coordinate used. KLdiv logical, TRUE, return KL divergence \\(D_{KL}(p || \\tilde{p})\\) \\(p\\) multivariate normal original covariance matrix \\(\\tilde{p}\\) approximated multivariate normal sparse precision matrix.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"list following: Q n n sparse precision matrix Matrix format ord ordering used Vecchia approximation cputime time taken run Vecchia approximation KLdiv (KLdiv = TRUE) KL divergence \\(D_{KL}(p || \\tilde{p})\\) \\(p\\) multivariate normal original covariance matrix \\(\\tilde{p}\\) approximated multivariate normal sparse precision matrix.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"Vecchia, . V. (1988). Estimation model identification continuous spatial processes. Journal Royal Statistical Society Series B: Statistical Methodology, 50(2), 297-312. Katzfuss, M., & Guinness, J. (2021). General Framework Vecchia Approximations Gaussian Processes. Statistical Science, 36(1). Finley, . O., Datta, ., Cook, B. D., Morton, D. C., Andersen, H. E., & Banerjee, S. (2019). Efficient algorithms Bayesian nearest neighbor Gaussian processes. Journal Computational Graphical Statistics, 28(2), 401-414. Zhang, L., (2020), public Github repository https://github.com/LuZhangstat/NNGP_STAN.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"","code":"n = 1000 coords = cbind(runif(n), runif(n)) Sigma = fields::Exp.cov(coords, aRange = 1) fit5 = vecchia_cov(Sigma, coords, n.neighbors = 5, KLdiv = TRUE) fit5$KLdiv #> [1] 1097.493 fit10 = vecchia_cov(Sigma, coords, n.neighbors = 10, KLdiv = TRUE) fit10$KLdiv #> [1] 921.3916"}]
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Interpretation of Sections 15 and 16","title":"GNU General Public License","text":"disclaimer warranty limitation liability provided given local legal effect according terms, reviewing courts shall apply local law closely approximates absolute waiver civil liability connection Program, unless warranty assumption liability accompanies copy Program return fee. END TERMS CONDITIONS","code":""},{"path":"https://changwoo-lee.github.io/bspme/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. 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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 <http://www.gnu.org/philosophy/--lgpl.html>.","code":"<one line to give the program's name and a brief idea of what it does.> Copyright (C) <year>  <name of author>  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 <http://www.gnu.org/licenses/>. <program>  Copyright (C) <year>  <name of author> 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":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"obtain-posterior-predictive-distribution-at-health-subject-locations","dir":"Articles","previous_headings":"","what":"Obtain posterior predictive distribution at health subject locations","title":"NO2 exposure and health data analysis","text":"example, use Gaussian process obtain predictive distribution \\((X_1,\\dots,X_n)\\) ln(NO2) exposure health subject locations. posterior predictive distribution (ppd) n-dimensional multivariate normal \\(N(\\mu_X, Q_X^{-1})\\), used prior incorporate prediction uncertainty spatial correlation NO2 exposures health subject locations estimation health model. Specifically, use exponential covariance kernel fixed range 8 (km) standard deviation 1.  covariance \\((X_1,\\dots,X_n)\\) contains spatial dependency information second order, precision matrix (inverse covariance matrix) \\(Q_X\\) dense matrix. comes fitting Bayesian linear model spatial exposure measurement error, naive implementation n-dimensional multivariate normal prior \\(N(\\mu_X, Q_X^{-1})\\) takes cubic time complexity posterior inference MCMC iteration, becomes infeasible run \\(n\\) large tens thousands.","code":"coords_monitor = cbind(data_jan10$lon, data_jan10$lat) coords_health = cbind(health_sim$lon, health_sim$lat) a = 8; sigma = 1; # assume known # distance calculation distmat_xx <- rdist.earth(coords_monitor, miles = F) distmat_xy <- rdist.earth(coords_monitor, coords_health, miles = F) distmat_yy <- rdist.earth(coords_health, miles = F)  # GP covariance matrices Sigmaxx = fields::Matern(distmat_xx, smoothness = 0.5, range = a, phi = sigma^2) Sigmaxy = fields::Matern(distmat_xy, smoothness = 0.5, range = a, phi = sigma^2) Sigmayy = fields::Matern(distmat_yy, smoothness = 0.5, range = a, phi = sigma^2)  # posterior predictive mean and covariance X_mean <- t(Sigmaxy) %*% solve(Sigmaxx, data_jan10$lnNO2) X_cov <- Sigmayy - t(Sigmaxy) %*% solve(Sigmaxx,Sigmaxy) # n_y by n_y  # visualization of posterior predictive distribution. Black triangle is monitoring station locations. par(mfrow = c(1,2), mar = c(4, 4, 4, 6)) quilt.plot(cbind(health_sim$lon, health_sim$lat),           X_mean, main = \"posterior predictive mean at health subject locations\",           xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5), legend.lab = \"ln(NO2)\") maps::map(\"county\", \"Texas\", add = T) # posterior predictive sd of exposure at health data quilt.plot(cbind(health_sim$lon, health_sim$lat),           sqrt(diag(X_cov)), main = \"posterior predictive stdev at health subject locations\",           xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5), legend.lab = \"ln(NO2)\") maps::map(\"county\", \"Texas\", add = T)"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"run-vecchia-approximation-to-get-sparse-precision-matrix","dir":"Articles","previous_headings":"","what":"Run Vecchia approximation to get sparse precision matrix","title":"NO2 exposure and health data analysis","text":"propose use Vecchia approximation get new multivariate normal prior \\((X_1,\\dots,X_n)\\) sparse precision matrix, crucial scalable implementation health models spatial exposure measurement error preserving spatial dependency information original covariance matrix closely possible.","code":"#Run the Vecchia approximation to get the sparse precision matrix.  # Vecchia approximation run_vecchia = vecchia_cov(X_cov, coords = cbind(health_sim$lon, health_sim$lat),             n.neighbors = 10) Q_sparse = run_vecchia$Q run_vecchia$cputime #> Time difference of 0.5541399 secs"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"fit-bspme-with-sparse-precision-matrix","dir":"Articles","previous_headings":"","what":"Fit bspme with sparse precision matrix","title":"NO2 exposure and health data analysis","text":"can now fit main function bspme package sparse precision matrix. takes less 10 seconds (1) run Vecchia approximation (2) run MCMC algorithm total 1100 iterations (Macbook M1 max CPU 64GB memory).","code":"# fit the model fit_me = blm_me(Y = health_sim$Y,               X_mean = X_mean,               X_prec = Q_sparse, # sparse precision matrix               Z = health_sim$Z,               nburn = 100, # increase as needed               nsave = 1000, # increase as needed               nthin = 1) # increase as needed fit_me$cputime #> Time difference of 6.949046 secs summary(fit_me$posterior) # posterior summary #>  #> Iterations = 1:1000 #> Thinning interval = 1  #> Number of chains = 1  #> Sample size per chain = 1000  #>  #> 1. Empirical mean and standard deviation for each variable, #>    plus standard error of the mean: #>  #>                Mean      SD Naive SE Time-series SE #> (Intercept)  0.7490 0.08557 0.002706       0.012488 #> exposure.1  -1.9544 0.05068 0.001603       0.007311 #> covariate.1  2.9578 0.04151 0.001313       0.003219 #> sigma2_Y     0.9546 0.14537 0.004597       0.029898 #>  #> 2. Quantiles for each variable: #>  #>                2.5%     25%     50%     75%   97.5% #> (Intercept)  0.5882  0.6911  0.7466  0.8024  0.9289 #> exposure.1  -2.0437 -1.9883 -1.9556 -1.9218 -1.8421 #> covariate.1  2.8702  2.9305  2.9584  2.9863  3.0382 #> sigma2_Y     0.6677  0.8498  0.9593  1.0553  1.2473"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"fit-bspme-without-sparse-precision-matrix-slow","dir":"Articles","previous_headings":"","what":"Fit bspme without sparse precision matrix (slow)","title":"NO2 exposure and health data analysis","text":"mentioned , precision matrix dense, posterior computation becomes infeasible \\(n\\) large, tens thousands. See following example \\(n=2000\\). Even 11 MCMC iterations, takes 15 seconds run MCMC algorithm. number iterations (1100), take 1500 seconds (.e. 25 minutes) run MCMC algorithm. gap becomes much larger number health subject locations grows, tens thousands. , naive algorithm uses dense precision matrix becomes infeasible, can carry inference using Vecchia approximation. quality Vecchia approximation depends size nearest neighbors, controlled argument n.neighbors vecchia_cov function. larger n.neighbors, better approximation. simulation study Lee et al. (2024) demonstrates Vecchia approximation relatively small n.neighbors can provide scalable two-stage Bayesian inference method results (e.g. RMSE, empirical coverage) close based dense precision matrix fully Bayesian analysis.","code":"# fit the model, without vecchia approximation # I will only run 11 iteration for illustration purpose Q_dense = solve(X_cov) # inverting 2000 x 2000 matrix, takes some time # run fit_me_dense = blm_me(Y = health_sim$Y,                 X_mean = X_mean,                 X_prec = Q_dense, # dense precision                 Z = health_sim$Z,                 nburn = 1, # Running only 11 iterations for an illustration purpose                 nsave = 10, # Running only 11 iterations for an illustration purpose                 nthin = 1) # Running only 11 iterations for an illustration purpose fit_me_dense$cputime #> Time difference of 16.86512 secs"},{"path":"https://changwoo-lee.github.io/bspme/articles/no2-exposure-and-health-data-analysis.html","id":"references","dir":"Articles","previous_headings":"","what":"References","title":"NO2 exposure and health data analysis","text":"Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634.","code":""},{"path":"https://changwoo-lee.github.io/bspme/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Changwoo Lee. Author, maintainer. Eun Sug Park. Author.","code":""},{"path":"https://changwoo-lee.github.io/bspme/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Lee C, Park E (2024). bspme: Bayesian Spatial Measurement Error Models. R package version 1.0.1, https://changwoo-lee.github.io/bspme/.","code":"@Manual{,   title = {bspme: Bayesian Spatial Measurement Error Models},   author = {Changwoo Lee and Eun Sug Park},   year = {2024},   note = {R package version 1.0.1},   url = {https://changwoo-lee.github.io/bspme/}, }"},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"bspme","dir":"","previous_headings":"","what":"Bayesian Spatial Measurement Error Models","title":"Bayesian Spatial Measurement Error Models","text":"Go package website: [link] See vignette NO2 exposure simulated health data: [link] See bspme_1.0.1.pdf pdf file package manual. bspme R package provides fast, scalable inference tools Bayesian spatial exposure measurement error models, namely, Bayesian linear generalized linear models presence spatial exposure measurement error covariate(s). models typically arise two-stage Bayesian analysis environmental exposures health outcomes. first-stage model, predictions covariate interest (“exposure”) uncertainty information (typically contained MCMC samples) obtained used form multivariate normal prior distribution X ∼ N(μ,Σ) exposure second-stage regression model. Naive, non-sparse choices precision matrix Q = Σ−1 multivariate normal (sample precision matrix) lead MCMC posterior inference algorithm infeasible run large number subjects n cubic computational cost associated n-dimensional MVN prior. sparse precision matrix Q obtained Vecchia approximation, bspme package offers fast, scalable algorithms conduct posterior inference large health datasets, number subjects n possibly reaching tens thousands. details, please see following paper: Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634. https://arxiv.org/abs/2401.00634","code":""},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"installation","dir":"","previous_headings":"","what":"Installation","title":"Bayesian Spatial Measurement Error Models","text":"can install R package bspme following code: browse see vignettes, run","code":"# install.packages(\"devtools\") devtools::install_github(\"changwoo-lee/bspme\", build_vignettes = T) browseVignettes(\"bspme\")"},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"functionality","dir":"","previous_headings":"","what":"Functionality","title":"Bayesian Spatial Measurement Error Models","text":"see function description R environment, run following lines:","code":"?blm_me ?bglm_me ?vecchia_cov"},{"path":[]},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"examples","dir":"","previous_headings":"","what":"Examples","title":"Bayesian Spatial Measurement Error Models","text":"Please see vignette “NO2-exposure--health-data-analysis”.","code":""},{"path":"https://changwoo-lee.github.io/bspme/index.html","id":"acknowldegements","dir":"","previous_headings":"","what":"Acknowldegements","title":"Bayesian Spatial Measurement Error Models","text":"work supported National Institute Environmental Health Sciences (NIEHS) National Institutes Health (NIH) R01ES031990.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/NO2_Jan2012.html","id":null,"dir":"Reference","previous_headings":"","what":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","title":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","text":"dataset contains daily average NO2 (nitrogen dioxide) concentrations obtained 21 monitoring stations around Harris County, Texas, January 2012.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/NO2_Jan2012.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","text":"","code":"data(NO2_Jan2012)"},{"path":"https://changwoo-lee.github.io/bspme/reference/NO2_Jan2012.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Daily average NO2 concentrations in and around the Harris County, Texas, in Jan 2012 — NO2_Jan2012","text":"data frame 651 (21 sites x 31 days) rows 5 variables: date date POSIXct format site_name monitoring station name lon monitoring station longitude lat monitoring station latitude lnNO2 natural logarithm daily average NO2 concentrations measured parts per billion volume (ppbv)","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":null,"dir":"Reference","previous_headings":"","what":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"function fits Bayesian generalized linear model presence spatial exposure measurement error covariate(s) \\(X\\). One important features function allows sparse matrix input prior precision matrix \\(X\\) scalable computation. version 1.0.0, Bayesian logistic regression model supported among GLMs, function bglm_me() runs Gibbs sampler carry posterior inference using Polya-Gamma augmentation (Polson et al., 2013). See \"Details\" section model description Lee et al. (2024) application example environmental epidemiology.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"","code":"bglm_me(   Y,   X_mean,   X_prec,   Z,   family = binomial(link = \"logit\"),   nburn = 5000,   nsave = 5000,   nthin = 1,   prior = NULL,   saveX = FALSE )"},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"Y vector<int>, n 1 binary response vector. X_mean vector<num>, n 1 prior mean vector \\(\\mu_X\\). q multiple exposures subject measurement error, can length q list n 1 vectors. X_prec matrix<num>, n n prior precision matrix \\(Q_X\\), allows sparse format Matrix package. q multiple exposures subject measurement error, can length q list n n matrices. Z matrix<num>, n p matrix containing p covariates subject measurement error. family class family, description error distribution link function used model. Currently, supports binomial(link = \"logit\"). nburn integer, number burn-iterations (default=5000). nsave integer, number posterior samples (default=5000). Total number MCMC iteration nburn + nsave * nthin. nthin integer, thin-rate (default=1). prior list, list prior parameters regression model. Default list(var_beta = 100). saveX logical, default FALSE, whether save posterior samples X (exposure).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"List following: posterior nsave (q+p) matrix posterior samples \\(\\beta_x\\) \\(\\beta_z\\) coda::mcmc object. time time taken running MCMC (seconds) X_save (saveX = TRUE) posterior samples X","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"Let \\(Y_i\\) binary response, \\(X_i\\) \\(q\\times 1\\) covariate vector subject spatial exposure measurement error, \\(Z_i\\) \\(p\\times 1\\) covariate vector without measurement error. Consider logistic regression model, independently \\(=1,\\dots,n,\\) $$\\log(Pr(Y_i = 1)/Pr(Y_i = 0)) = \\beta_0 + X_i^\\top \\beta_X +  Z_i^\\top \\beta_Z.$$ Spatial exposure measurement error \\(X_i\\) (\\(=1,\\dots,n\\)) incorporated model using multivariate normal prior. example, \\(q=1\\), \\(n-\\)dimensional multivariate normal prior \\(X = (X_1,\\dots,X_n)^\\top\\), $$(X_1,\\dots,X_n)\\sim N_n(\\mu_X, Q_X^{-1}).$$ importantly, allows sparse matrix input prior precision matrix \\(Q_X\\) scalable computation, can obtained Vecchia approximation. \\(q>1\\), \\(q\\) independent \\(n-\\)dimensional multivariate normal priors assumed. consider normal priors regression coefficients, $$\\beta_0 \\sim N(0, V_\\beta), \\quad \\beta_{X,j} \\stackrel{iid}{\\sim} N(0, V_\\beta), \\quad \\beta_{Z,k} \\stackrel{iid}{\\sim} N(0, V_\\beta)$$ var_beta corresponds \\(V_\\beta\\).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"Polson, N. G., Scott, J. G., & Windle, J. (2013). Bayesian inference logistic models using Pólya–Gamma latent variables. Journal American statistical Association, 108(504), 1339-1349. Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/bglm_me.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Bayesian generalized linear models with spatial exposure measurement error. — bglm_me","text":"","code":"if (FALSE) { library(bspme) data(NO2_Jan2012) data(health_sim) library(fields) library(maps) # Obtain the predicted exposure mean and covariance at simulated health subject locations # based on NO2 data obtained on Jan 10, 2012 # using a Gaussian process prior with mean zero and exponential covariance kernel # with a fixed range 8 (in km) and standard deviation 1.  # exposure data data_jan10 = NO2_Jan2012[NO2_Jan2012$date == as.POSIXct(\"2012-01-10\"),] coords_monitor = cbind(data_jan10$lon, data_jan10$lat)  # health data coords_health = cbind(health_sim$lon, health_sim$lat)  distmat_xx <- rdist.earth(coords_monitor, miles = F) distmat_xy <- rdist.earth(coords_monitor, coords_health, miles = F) distmat_yy <- rdist.earth(coords_health, miles = F)  a = 8; sigma = 1; # assume known  Sigmaxx = fields::Matern(distmat_xx, smoothness = 0.5, range = a, phi = sigma^2) Sigmaxy = fields::Matern(distmat_xy, smoothness = 0.5, range = a, phi = sigma^2) Sigmayy = fields::Matern(distmat_yy, smoothness = 0.5, range = a, phi = sigma^2)  # posterior predictive mean and covariance of exposure at health subject locations X_mean <- t(Sigmaxy) %*% solve(Sigmaxx, data_jan10$lnNO2) X_cov <- Sigmayy - t(Sigmaxy) %*% solve(Sigmaxx,Sigmaxy) # n_y by n_y  # visualize # monitoring station exposure data quilt.plot(cbind(data_jan10$lon, data_jan10$lat),            data_jan10$lnNO2, main = \"NO2 exposures (in log) at 21 monitoring stations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive mean of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            X_mean, main = \"posterior predictive mean of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive sd of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            sqrt(diag(X_cov)), main = \"posterior predictive sd of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)   # vecchia approximation run_vecchia = vecchia_cov(X_cov, coords = cbind(health_sim$lon, health_sim$lat),                           n.neighbors = 10) Q_sparse = run_vecchia$Q run_vecchia$cputime  # fit the model, binary response fit = bglm_me(Y = health_sim$Ybinary,                 X_mean = X_mean,                 X_prec = Q_sparse, # sparse precision matrix                 Z = health_sim$Z,                 family = binomial(link = \"logit\"),                 nburn = 5000,                 nsave = 5000,                 nthin = 1) fit$cputime summary(fit$posterior) library(bayesplot) bayesplot::mcmc_trace(fit$posterior) }"},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":null,"dir":"Reference","previous_headings":"","what":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"function fits Bayesian linear regression model presence spatial exposure measurement error covariate(s) \\(X\\). One important features function allows sparse matrix input prior precision matrix \\(X\\) scalable computation. Function blm_me() runs Gibbs sampler carry posterior inference;  see \"Details\" section model description, Lee et al. (2024) application example environmental epidemiology.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"","code":"blm_me(   Y,   X_mean,   X_prec,   Z,   nburn = 5000,   nsave = 5000,   nthin = 1,   prior = NULL,   saveX = FALSE )"},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"Y vector<int>, n 1 continuous response vector. X_mean vector<num>, n 1 prior mean vector \\(\\mu_X\\). q multiple exposures subject measurement error, can length q list n 1 vectors. X_prec matrix<num>, n n prior precision matrix \\(Q_X\\), allows sparse format Matrix package. q multiple exposures subject measurement error, can length q list n n matrices. Z matrix<num>, n p matrix containing p covariates subject measurement error. nburn integer, number burn-iterations (default=5000). nsave integer, number posterior samples (default=5000). Total number MCMC iteration nburn + nsave * nthin. nthin integer, thin-rate (default=1). prior list, list prior parameters regression model. Default list(var_beta = 100, a_Y = 0.01, b_Y = 0.01). saveX logical, default FALSE, whether save posterior samples X (exposure).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"list following: posterior nsave (q + p + 1) matrix posterior samples \\(\\beta_X\\), \\(\\beta_Z\\), \\(\\sigma_Y^2\\) coda::mcmc object. time time taken running MCMC (seconds) X_save (saveX = TRUE) posterior samples X","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"Let \\(Y_i\\) continuous response, \\(X_i\\) \\(q\\times 1\\) covariate vector subject spatial exposure measurement error, \\(Z_i\\) \\(p\\times 1\\) covariate vector without measurement error. Consider normal linear regression model, $$Y_i = \\beta_0 + X_i^\\top \\beta_X +  Z_i^\\top \\beta_Z + \\epsilon_i,\\quad  \\epsilon_i \\stackrel{iid}{\\sim} N(0, \\sigma^2_Y), \\quad =1,\\dots,n.$$ Spatial exposure measurement error \\(X_i\\) \\(=1,\\dots,n\\) incorporated model using multivariate normal prior. example \\(q=1\\), \\(n-\\)dimensional multivariate normal prior \\(X = (X_1,\\dots,X_n)^\\top\\), $$(X_1,\\dots,X_n)\\sim N_n(\\mu_X, Q_X^{-1}).$$ importantly, allows sparse matrix input prior precision matrix \\(Q_X\\) scalable computation, can obtained Vecchia approximation. \\(q>1\\), \\(q\\) independent \\(n-\\)dimensional multivariate normal priors assumed. consider semiconjugate priors regression coefficients error variance, $$\\beta_0 \\sim N(0, V_\\beta), \\quad \\beta_{X,j} \\stackrel{iid}{\\sim} N(0, V_\\beta), \\quad \\beta_{Z,k} \\stackrel{iid}{\\sim} N(0, V_\\beta), \\quad \\sigma_Y^2 \\sim IG(a_Y, b_Y).$$ var_beta corresponds \\(V_\\beta\\), a_Y b_Y correspond hyperparameters inverse gamma prior \\(\\sigma^2_Y\\).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"Lee, C. J., Symanski, E., Rammah, ., Kang, D. H., Hopke, P. K., & Park, E. S. (2024). scalable two-stage Bayesian approach accounting exposure measurement error environmental epidemiology. arXiv preprint arXiv:2401.00634.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/blm_me.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Bayesian linear regression models with spatial exposure measurement error. — blm_me","text":"","code":"if (FALSE) { library(bspme) data(NO2_Jan2012) data(health_sim) library(fields) library(maps) # Obtain the predicted exposure mean and covariance at simulated health subject locations # based on NO2 data obtained on Jan 10, 2012 # using a Gaussian process prior with mean zero and exponential covariance kernel # with a fixed range 8 (in km) and standard deviation 1.  # exposure data data_jan10 = NO2_Jan2012[NO2_Jan2012$date == as.POSIXct(\"2012-01-10\"),] coords_monitor = cbind(data_jan10$lon, data_jan10$lat)  # health data coords_health = cbind(health_sim$lon, health_sim$lat)  distmat_xx <- rdist.earth(coords_monitor, miles = F) distmat_xy <- rdist.earth(coords_monitor, coords_health, miles = F) distmat_yy <- rdist.earth(coords_health, miles = F)  a = 8; sigma = 1; # assume known  Sigmaxx = fields::Matern(distmat_xx, smoothness = 0.5, range = a, phi = sigma^2) Sigmaxy = fields::Matern(distmat_xy, smoothness = 0.5, range = a, phi = sigma^2) Sigmayy = fields::Matern(distmat_yy, smoothness = 0.5, range = a, phi = sigma^2)  # posterior predictive mean and covariance of exposure at health subject locations X_mean <- t(Sigmaxy) %*% solve(Sigmaxx, data_jan10$lnNO2) X_cov <- Sigmayy - t(Sigmaxy) %*% solve(Sigmaxx,Sigmaxy) # n_y by n_y  # visualize # monitoring station exposure data quilt.plot(cbind(data_jan10$lon, data_jan10$lat),            data_jan10$lnNO2, main = \"NO2 exposures (in log) at 21 monitoring stations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive mean of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            X_mean, main = \"posterior predictive mean of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)  # posterior predictive sd of exposure at health subject locations quilt.plot(cbind(health_sim$lon, health_sim$lat),            sqrt(diag(X_cov)), main = \"posterior predictive sd of exposure at health subject locations\",            xlab = \"longitude\", ylab= \"latitude\", xlim = c(-96.5, -94.5), ylim = c(29, 30.5)) maps::map(\"county\", \"Texas\", add = T)   # vecchia approximation run_vecchia = vecchia_cov(X_cov, coords = cbind(health_sim$lon, health_sim$lat),                           n.neighbors = 10) Q_sparse = run_vecchia$Q run_vecchia$cputime  # fit the model, continuous response fit = blm_me(Y = health_sim$Y,                 X_mean = X_mean,                 X_prec = Q_sparse, # sparse precision matrix                 Z = health_sim$Z,                 nburn = 5000,                 nsave = 5000,                 nthin = 1) fit$cputime summary(fit$posterior) library(bayesplot) bayesplot::mcmc_trace(fit$posterior) }"},{"path":"https://changwoo-lee.github.io/bspme/reference/bspme-package.html","id":null,"dir":"Reference","previous_headings":"","what":"bspme: Bayesian Spatial Measurement Error Models — bspme-package","title":"bspme: Bayesian Spatial Measurement Error Models — bspme-package","text":"Scalable methods fitting Bayesian linear generalized linear models presence spatial exposure measurement error. models typically arise two-stage Bayesian analysis environmental exposures health outcomes. first-stage model, predictions covariate interest (''exposure'') uncertainty information (typically contained MCMC samples) used form multivariate normal prior distribution exposure second-stage regression model. package also provides implementation methods used Lee et al. (2024) https://arxiv.org/abs/2401.00634.","code":""},{"path":[]},{"path":"https://changwoo-lee.github.io/bspme/reference/bspme-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"bspme: Bayesian Spatial Measurement Error Models — bspme-package","text":"Maintainer: Changwoo Lee c.lee@stat.tamu.edu Authors: Eun Sug Park","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/health_sim.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulated health data — health_sim","title":"Simulated health data — health_sim","text":"Simulated health data associated ln(NO2) concentration Jan 10, 2012. details, see health_sim.R.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/health_sim.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulated health data — health_sim","text":"","code":"data(health_sim)"},{"path":"https://changwoo-lee.github.io/bspme/reference/health_sim.html","id":"format","dir":"Reference","previous_headings":"","what":"Format","title":"Simulated health data — health_sim","text":"data frame n = 2000 rows 6 variables: Y simulated continuous health outcome Ybinary simulated binary health outcome lon simulated health subject longitude lat simulated health subject latitude Z simulated covariate (p=1) subject measurement error X_true true ln(NO2) exposure used simulating health outcome","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":null,"dir":"Reference","previous_headings":"","what":"Run Vecchia approximation given a covariance matrix — vecchia_cov","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"Given multivariate normal (MVN) distribution covariance matrix \\(\\Sigma\\), function finds sparse precision matrix (inverse covariance) \\(Q\\) based Vecchia approximation (Vecchia 1988, Katzfuss Guinness 2021), \\(N(\\mu, Q^{-1})\\) sparse MVN approximates original MVN \\(N(\\mu, \\Sigma)\\). algorithm based pseudocode 2 Finley et al. (2019).","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"","code":"vecchia_cov(Sigma, coords, n.neighbors, ord = NULL, KLdiv = FALSE)"},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"Sigma matrix<num>, n n covariance matrix coords matrix<num>, n 2 coordinate matrix nearest neighborhood search n.neighbors integer, number nearest neighbors (k) determine conditioning set Vecchia approximation ord vector<int>, length n vector, ordering data. NULL, ordering based first coordinate used. KLdiv logical, TRUE, return KL divergence \\(D_{KL}(p || \\tilde{p})\\) \\(p\\) multivariate normal original covariance matrix \\(\\tilde{p}\\) approximated multivariate normal sparse precision matrix.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"list following: Q n n sparse precision matrix Matrix format ord ordering used Vecchia approximation cputime time taken run Vecchia approximation KLdiv (KLdiv = TRUE) KL divergence \\(D_{KL}(p || \\tilde{p})\\) \\(p\\) multivariate normal original covariance matrix \\(\\tilde{p}\\) approximated multivariate normal sparse precision matrix.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"Vecchia, . V. (1988). Estimation model identification continuous spatial processes. Journal Royal Statistical Society Series B: Statistical Methodology, 50(2), 297-312. Katzfuss, M., & Guinness, J. (2021). General Framework Vecchia Approximations Gaussian Processes. Statistical Science, 36(1). Finley, . O., Datta, ., Cook, B. D., Morton, D. C., Andersen, H. E., & Banerjee, S. (2019). Efficient algorithms Bayesian nearest neighbor Gaussian processes. Journal Computational Graphical Statistics, 28(2), 401-414. Zhang, L., (2020), public Github repository https://github.com/LuZhangstat/NNGP_STAN.","code":""},{"path":"https://changwoo-lee.github.io/bspme/reference/vecchia_cov.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Run Vecchia approximation given a covariance matrix — vecchia_cov","text":"","code":"n = 1000 coords = cbind(runif(n), runif(n)) Sigma = fields::Exp.cov(coords, aRange = 1) fit5 = vecchia_cov(Sigma, coords, n.neighbors = 5, KLdiv = TRUE) fit5$KLdiv #> [1] 1097.493 fit10 = vecchia_cov(Sigma, coords, n.neighbors = 10, KLdiv = TRUE) fit10$KLdiv #> [1] 921.3916"}]