From d0bb70411a79a865e66c2474ac1e5f1831547cd8 Mon Sep 17 00:00:00 2001 From: GitHub Actions Date: Fri, 15 Mar 2024 03:05:29 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20Xiangyun?= =?UTF-8?q?Huang/data-analysis-in-action@1508d0a52fed26c8728627d632947e49c?= =?UTF-8?q?81a12ab=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- analyze-text-data.html | 4 +- analyze-time-series-data.html | 4 +- .../figure-html/fig-two-samples-means-1.pdf | Bin 12052 -> 12054 bytes .../figure-html/fig-two-samples-means-1.tex | 2 +- index.html | 31 +++-- interactive-applications.html | 4 +- interactive-graphics.html | 72 ++++++------ interactive-tables.html | 8 +- intro.html | 106 +++++++++--------- numerical-optimization.html | 12 +- .../fig-one-dimensional-optimization-1.pdf | Bin 7015 -> 7015 bytes .../fig-one-dimensional-optimization-1.tex | 2 +- optimization-problems.html | 36 +++--- .../figure-html/fig-power-t-test-1.pdf | Bin 17130 -> 17128 bytes .../figure-html/fig-power-t-test-1.tex | 2 +- .../figure-html/fig-state-x77-bubble-1.png | Bin 180327 -> 179900 bytes search.json | 12 +- wrangling-objects.html | 2 +- 18 files changed, 146 insertions(+), 151 deletions(-) diff --git a/analyze-text-data.html b/analyze-text-data.html index 0644f856..3b9414d7 100644 --- a/analyze-text-data.html +++ b/analyze-text-data.html @@ -596,8 +596,8 @@

n_check_convergence = 25, progressbar = FALSE )
-
#> INFO  [15:28:26.129] early stopping at 175 iteration
-#> INFO  [15:28:26.608] early stopping at 50 iteration
+
#> INFO  [03:04:22.578] early stopping at 175 iteration
+#> INFO  [03:04:23.061] early stopping at 50 iteration

下图展示主题的分布，各个主题及其所占比例。

diff --git a/analyze-time-series-data.html b/analyze-time-series-data.html index 37dd57cd..b950dbb6 100644 --- a/analyze-time-series-data.html +++ b/analyze-time-series-data.html @@ -714,8 +714,8 @@

dyUnzoom()
-
- +
+
图 18.6: 美团股价变化趋势 diff --git a/common-statistical-tests_files/figure-html/fig-two-samples-means-1.pdf b/common-statistical-tests_files/figure-html/fig-two-samples-means-1.pdf index d3914043a82411cc22bba23ce8110a0acb804c7f..d058ce1601af16940143779a3eb71c1134523a1f 100644 GIT binary patch delta 3815 zcmai0Wmpr88YQK>8I8n;#D>HOVRT9;2#g%yP+^FU78s1~7#)IAf^;J#NQZO?NFyT! zWD-NT-{Y@)f8F2j`E$;5&U;R(UAkS(2qj8ZK}CfU>G}F83_z!C@rwny zT=HCVY>_8!NOG<}^suCgw*UxDi6OzQt+GuVyd%q$|B^E>LS>nip)X0lJlgAOQ;?z0 zo?ANW5k^Z^rpq3Z6riAp`c*r7*TMXw>JkJ0MN7v{#*Zv>bB?KuA9i6B(fzv4R z$;k}Ajxnl);yn@F$3*l2)_r~;K(3_Y-to{_XBrg6=(?X zVjuYVY?oHlob4n1_wPbE97u!RCRJf1NU()}#KVnB>{GmF!vxXxFs?vCDt$wn%Ger; z1?bvWeID*NI4pr7K{>I~_GqT_0mC?uX!32`qTS}E$oRJq*hP~| zU9a5R#jSkfm#=+TQ*@SUZMZU6yp;mr8W_{UC%k55vt(lIn0Q&4U4>6YFuIPeih~K2 zt6-?Db_T$!;+@*GwuLLWCb^!Fu?Y;=J>0UFU7^Iv0B9_P_{Ys(%ItM zz=pohySw)l>k|%Vp6_lN0(82{KzrCqIy37+G$>mU%xz$CAIiOyj?2Y!{U-V?HWy`8 z?=9HEbgx&eR?#p$bUl-~Eq=O1C33L!UJG*%ucIqb-d6Y&IN{Ues=mEICLu2fM(3R@ z=@4;++o+1y(UMf6?%_jCGSwL;QIP9ASXqZ`8Qm#^Ka|gVWf`(BFY-8qiWd}HLol=z zS(_X4Vt(oU{grtOPKN9>$onINdIf+3@os7dYHe68;<3t?TOZed*eH70oG(1e-xcNtnFDvG^ z>vFCYjaj~Ko!$nlFxRe{_+A8{*~u zgsr6v%5WtVU@$p)Ag^bJ<@rW=zb-k-pd(itr@X9& zm)ox!t`2U$u=fZ5O}zxY2!@OWfRamFB(;j+vZ_sOFwdpt0OfKQankpP-ge%x=1?T? 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R 语言数据分析实战

发布日期
-

2024年3月14日

+

2024年3月15日

@@ -446,17 +446,15 @@

R 语言数据分析实战

本书初稿是在 RStudio IDE 内使用 Quarto 编辑的，Quarto 是继R Markdown之后，一个新的开源的科学和技术发布系统，它基于 Pandoc支持输出多种格式的书稿，比如 HTML 网页、EPUB 电子书、DOCX 文档和 PDF 便携式文档等。Quarto 吸收了过去 10 年 R Markdown 取得的经验和教训，在书籍写作、创建博客、制作简历和幻灯片等系列场景中支持更加统一的使用语法，一份源文档输出多种格式，使文档内容在不同场景中的迁移成本更低。了解更多 Quarto 特性，请访问 https://quarto.org/。

书中的代码字体采用美观的 Source Code Pro 字体， 为方便跨操作系统编译书籍电子版，正文的中文字体采用开源的 fandol 字体。此外，考虑到美观性，本书图形使用了 Noto 系列中英文字体，它们来自 Google Fonts 字体库，分别是 Noto Sans 无衬线英文字体和 Noto Serif SC 宋体中文字体。

书中 R 包名以粗体表示，如 knitr 包，函数名以等宽体表示，如 plot()，函数的参数名同理。代码块内注释用 # 表示，运行结果每一行开头以 #> 标记。本书写作过程中，依赖 knitr (Xie 2015)、ggplot2 (Wickham 2016) 和 lattice (Sarkar 2008) 等众多 R 包。考虑到要同时支持 DOCX、EPUB、PDF 和 HTML 四种书籍格式，书中使用 knitr 包和 gt 包制作静态的表格。

-

为方便测试贡献者提供的 PR，本书托管在 Github 上，同时启用 Github Action 服务，为书籍自定义了一个可复现全书内容的运行环境，包括 R 软件、扩展包和系统软件依赖，详见仓库中的 DESCRIPTION 文件。你现在看到的是在线编译版本，使用 Quarto 1.4.550，最新一次编译时间是 2024-03-14 23:25:46。

+

为方便测试贡献者提供的 PR，本书托管在 Github 上，同时启用 Github Action 服务，为书籍自定义了一个可复现全书内容的运行环境，包括 R 软件、扩展包和系统软件依赖，详见仓库中的 DESCRIPTION 文件。你现在看到的是在线编译版本，使用 Quarto 1.4.550，最新一次编译时间是 2024-03-15 11:01:54。

xfun::session_info(packages = c(
-  "ggplot2", "gganimate", "ggrepel", "ggdensity",
-  "ggridges", "ggsignif", "ggforce", "ggbeeswarm",
-  "ggeffects", "ggnewscale", "patchwork", "shiny",
-  "plotly", "lattice", "igraph", "tidygraph", "ggraph",
-  "dplyr", "purrr", "tidyr", "httr", "data.table",
-  "rsconnect", "knitr", "rmarkdown", "gt", "DT",
-  "showtext", "gifski", "tinytex", "magick"
-), dependencies = FALSE)
+ "ggplot2", "gganimate", "ggrepel", "ggridges", "ggbeeswarm", + "patchwork", "shiny", "plotly", "lattice", "igraph", + "tidygraph", "ggraph", "dplyr", "purrr", "tidyr", "httr", + "data.table", "rsconnect", "knitr", "rmarkdown", "gt", "DT", + "showtext", "gifski", "tinytex", "magick" +), dependencies = FALSE)
#> R version 4.3.3 (2024-02-29)
 #> Platform: x86_64-pc-linux-gnu (64-bit)
@@ -470,13 +468,12 @@ 
R 语言数据分析实战

 #> 
 #> Package version:
 #>   data.table_1.15.2 dplyr_1.1.4       DT_0.32           gganimate_1.0.9  
-#>   ggbeeswarm_0.7.2  ggdensity_1.0.0   ggeffects_1.5.0   ggforce_0.4.2    
-#>   ggnewscale_0.4.10 ggplot2_3.5.0     ggraph_2.2.1      ggrepel_0.9.5    
-#>   ggridges_0.5.6    ggsignif_0.6.4    gifski_1.12.0.2   gt_0.10.1        
-#>   httr_1.4.7        igraph_2.0.3      knitr_1.45        lattice_0.22.5   
-#>   magick_2.8.3      patchwork_1.2.0   plotly_4.10.4     purrr_1.0.2      
-#>   rmarkdown_2.26    rsconnect_1.2.1   shiny_1.8.0       showtext_0.9.7   
-#>   tidygraph_1.3.1   tidyr_1.3.1       tinytex_0.49     
+#>   ggbeeswarm_0.7.2  ggplot2_3.5.0     ggraph_2.2.1      ggrepel_0.9.5    
+#>   ggridges_0.5.6    gifski_1.12.0.2   gt_0.10.1         httr_1.4.7       
+#>   igraph_2.0.3      knitr_1.45        lattice_0.22.5    magick_2.8.3     
+#>   patchwork_1.2.0   plotly_4.10.4     purrr_1.0.2       rmarkdown_2.26   
+#>   rsconnect_1.2.1   shiny_1.8.0       showtext_0.9.7    tidygraph_1.3.1  
+#>   tidyr_1.3.1       tinytex_0.49     
 #> 
 #> Pandoc version: 3.1.11
 #> 
diff --git a/interactive-applications.html b/interactive-applications.html
index 811a5fae..b0ab1643 100644
--- a/interactive-applications.html
+++ b/interactive-applications.html
@@ -959,8 +959,8 @@ 

 

 

-

-
+

+
 

 

 图 8.1: Shiny 生态系统
diff --git a/interactive-graphics.html b/interactive-graphics.html
index 580800f6..a3c04c9e 100644
--- a/interactive-graphics.html
+++ b/interactive-graphics.html
@@ -581,8 +581,8 @@ 
  plotly::add_markers(color = ~mag)
-
- +
+
图 6.2: 给散点图配色 @@ -601,8 +601,8 @@

)

-
- +
+
图 6.3: 设置刻度及标签 @@ -629,8 +629,8 @@

)

-
- +
+
图 6.4: 添加各处标题 @@ -657,8 +657,8 @@

)

-
- +
+
图 6.5: 设置主题风格 @@ -761,8 +761,8 @@

)

-
- +
+
图 6.7: 柱形图 @@ -782,8 +782,8 @@

)

-
- +
+
图 6.8: 曲线图 @@ -879,8 +879,8 @@

)

-
- +
+
图 6.12: 不同深度下地震震级的分布 @@ -900,8 +900,8 @@

)

-
- +
+
图 6.13: 不同深度下地震震级的分布 @@ -1017,8 +1017,8 @@

)

-
- +
+
图 6.16: 1999-2022 年 Martin Maechler 和 Brian Ripley 的代码提交量变化 @@ -1107,8 +1107,8 @@ 

plotly::ggplotly(p)
-
- +
+

当使用配置函数 config() 设置参数选项 staticPlot = TRUE，可将原本交互式的动态图形转为非交互式的静态图形。

@@ -1116,8 +1116,8 @@ 

plotly::ggplotly(p) |> 
   plotly::config(staticPlot = TRUE)

-
- +
+
@@ -1136,8 +1136,8 @@

plotly::style(hoveron = "points", hoverinfo = "x+y+text", hoverlabel = list(bgcolor = "white"))

-
- +
+
@@ -1223,8 +1223,8 @@

)
-
- +
+
图 6.20: 添加水印图片 @@ -1255,9 +1255,9 @@

htmltools::tagList(p1, p2)

-
-
- +
+
+
图 6.21: 上下布局 @@ -1271,8 +1271,8 @@

nrows = 2, margin = 0.05, shareX = TRUE, titleY = TRUE)

-
- +
+
图 6.22: 上下布局 @@ -1292,8 +1292,8 @@

)

-
- +
+
图 6.23: 灵活布局 @@ -1320,9 +1320,9 @@

-
-
- +
+
+

diff --git a/interactive-tables.html b/interactive-tables.html index bba4d742..fc83a074 100644 --- a/interactive-tables.html +++ b/interactive-tables.html @@ -499,8 +499,8 @@

-
- +
+
@@ -539,8 +539,8 @@

-
- +
+
diff --git a/intro.html b/intro.html index f75ff246..ec3cbb13 100644 --- a/intro.html +++ b/intro.html @@ -630,23 +630,23 @@

介绍<
-
- diff --git a/numerical-optimization.html b/numerical-optimization.html index be3acbdc..df05ed56 100644 --- a/numerical-optimization.html +++ b/numerical-optimization.html @@ -1829,11 +1829,11 @@

nlp <- ROI_solve(op, solver = "nloptr.directL") nlp$solution

-
#> [1]  0.00000 22.22222
+
#> [1] -21.99115   0.00000
nlp$objval
-
#> [1] -0.9734211
+
#> [1] -1

结果还是陷入局部最优解。运筹优化方面的商业软件，著名的有 Lingo 和 Matlab，下面采用 Lingo 20 求解，Lingo 代码如下：

@@ -2176,14 +2176,14 @@

# 目标函数值 nlp$objval

-
#> [1] 368.1061
+
#> [1] 368.1059
# 最优解
 nlp$solution
#>  [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
 #>  [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
-#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109093
+#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109096
 #> [25] 4.000000
@@ -2537,11 +2537,11 @@ 

nlp <- ROI_solve(op, solver = "nloptr.isres", start = c(1, 5, 5, 1))
 nlp$solution
-
#> [1] 1.276795 4.607639 3.992841 1.093756
+
#> [1] 1.131671 4.721246 3.868353 1.216995
nlp$objval
-
#> [1] 17.78648
+
#> [1] 17.25686

可以看出，nloptr 提供的优化能力可以覆盖 Ipopt 求解器，从以上求解的情况来看，推荐使用 nloptr.slsqp 求解器，这也是 Octave 的选择。

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nlp <- ROI_solve(op, solver = "nloptr.directL")
 nlp$solution
-
#> [1] 63.934432  6.121382
+
#> [1] 63.934193  6.121331
nlp$objval
@@ -1099,33 +1099,31 @@

control = list(maximize = TRUE) )

-
#> iter:  0  f-value:  -2968.501  pgrad:  7.572203 
-#> iter:  10  f-value:  -1992.18  pgrad:  3.503607 
-#> iter:  20  f-value:  -1991.277  pgrad:  2.365923 
-#> iter:  30  f-value:  -1990.325  pgrad:  1.405494 
-#> iter:  40  f-value:  -1989.979  pgrad:  1.709452 
-#> iter:  50  f-value:  -1989.946  pgrad:  0.02604793 
+
#> iter:  0  f-value:  -2301.554  pgrad:  4.703557 
+#> iter:  10  f-value:  -1990.644  pgrad:  2.361774 
+#> iter:  20  f-value:  -1989.947  pgrad:  0.2183447 
+#> iter:  30  f-value:  -1989.946  pgrad:  0.005798029 
 #>   Successful convergence.

 
ans
#> $par
-#> [1] 0.3598808 1.2560870 2.6633987
+#> [1] 0.6401137 2.6634054 1.2560965
 #> 
 #> $value
 #> [1] -1989.946
 #> 
 #> $gradient
-#> [1] 1.818989e-05
+#> [1] 0.0002660272
 #> 
 #> $fn.reduction
-#> [1] -978.5554
+#> [1] -311.6079
 #> 
 #> $iter
-#> [1] 59
+#> [1] 39
 #> 
 #> $feval
-#> [1] 61
+#> [1] 41
 #> 
 #> $convergence
 #> [1] 0
@@ -1145,15 +1143,15 @@ 
hess
#>           [,1]       [,2]       [,3]
-#> [1,] -907.1131  270.22803  341.25563
-#> [2,]  270.2280 -113.47865  -61.68173
-#> [3,]  341.2556  -61.68173 -192.78329
+#> [1,] -907.1064 -341.25220 -270.22960 +#> [2,] -341.2522 -192.78034 -61.68217 +#> [3,] -270.2296 -61.68217 -113.48058 
# 标准差
 se <- sqrt(diag(solve(-hess)))
 se
-
#> [1] 0.1946826 0.3500306 0.2504754
+
#> [1] 0.1946849 0.2504791 0.3500301

multiStart 从不同初始值出发寻找全局最大值，先找一系列局部极大值，通过比较获得全局最大值。

@@ -1174,9 +1172,9 @@

pmat[!duplicated(pmat), ] 
#>         fvalue parameter 1 parameter 2 parameter 3
-#> [1,] -1989.946      0.6401      2.6634      1.2561
-#> [2,] -1989.946      0.3599      1.2561      2.6634
-#> [3,] -1999.873      0.3187      1.9075      2.2792
+#> [1,] -1989.946 0.3599 1.2561 2.6634 +#> [2,] -1995.912 0.3990 2.5910 1.8588 +#> [3,] -1989.946 0.6401 2.6634 1.2561

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z_6d`cu?eLX`q}5l1268DR71GdN|Q76jOi6iLVo>v6%jJsw5LY`xpMmHdGc`-SboyF zR!c)du3R}ZE@Wp9Nq?zkTQhS%QYW-~!XcI)2GTQ_D~5EuN%sNLQ5`z;khJcoJ*i1a z)@Fv9`=>3U1`*vxS8re6>ZixfSdE@@c`cQHFSiLBU%8$>oe#&f{Qyoq!5{^K!fhns zd^^tCXx7hfTDH}-wZyqM=jKw^PTC)JF4Tr2vHBp6AtWg>9tp@^5X(-uU$}-#Ne|zp zsiAQ(ti3N*jM*k~_dO%Xf#BzmKfn7_&NBH;2}<4#O@h$b=;D{%mxzjC5ftd1{q~^X z5vhUHyao+amP(&xcN)u%47_|GgQa`SuzlCA?@qnxvvi@Qco9`5{uTRf4aTjtrK}*&Ox&j9x8J@hFbsUY(9#4OE*JbSYOwEj8)J!>*$HAIC+HHj8JD{3rQnwO8+Ul zPWK&KkeSJ7F-{^2i$3Ri!-wbE+h`uX6&d+DU9qNh0viXoJnig+NKny|k`mWrj5R{J z@FG5bDpSsagDXCK=%)KVt6W0{0S#Ffw2k%H#(-w*c;ek4M-~Jy%TuPm9yqvaH&$K1 zxxLk_B?I)#%)ax!Nf5wHK}WCBM72<7kkWdqRncIm58ADRetX)PjQrA_tp}Qio~8 zDy~w0?4`oExqvikTF-cvh<9+Bzji6zTG&^v$6!H@?)3WOM*}IK9&F(mq2nq&efpEE zCT#f#fj82#Ybw}#@5^73taplMtz(z>3W`Xl_}L6Qr_u8HP!mLoqC?s!k_33%wn?RV z0Bxu_gUZUvxRTL6vsJ8ahJ+*)6r8c}z7`k==XcyVGZ2!gQ!iK<*xA{=|M20)h|KS8 zDK9u~0WH&yEK%{oW%PJr;+9@*JuGATcJH3EW*~>3>oLlKW|$48hj1O(?WhZ5;pHjn z2^8~`h*#LNJO@^np%wMc%0Ii2=yZz5z$Lr1gw1|RJAv~6$__OenfWxUASsTBnig@Dx1#e1>h?&H4MR+E`+4QQ#I;7hu5im z$-b0ItSLN%FvqGQ;xC9!!2YQ~t(37e>VZcLPgQcV^G0E>k=C)vI6xMZSiFY>%o{ek zZ@+nS4J-Jj75E}WNxxpbW?CK5;ElFz7K-9|nBy#E%2Hmlc5V6GdanW~buf&ByWC=> z&7CvHR-2!tcD3xqf+auZcSPA2U>mfyQhht@ob~G?QuZZXi13kj!whC4#&?dFCnHma zE=geI^^v5LPcdV_zhG>fm51  数据对象" @@ -1124,7 +1124,7 @@ "href": "analyze-text-data.html#sec-topic-models", "title": "17  文本数据分析", "section": "\n17.4 主题的探索", - "text": "17.4 主题的探索\n益辉的日志是没有分类和标签的，所以，先聚类，接着逐个分析每个类代表的实际含义。然后，将聚类的结果作为结果标签，再应用多分类回归模型，最后联合聚类、分类模型，从无监督转化到有监督模型。\ntopicmodels (Grün 和 Hornik 2011) 基于 tm (Feinerer, Hornik, 和 Meyer 2008) 支持潜在狄利克雷分配（Latent Dirichlet Allocation，简称 LDA） 和 Correlated Topics Models (CTM) 文本主题建模，这一套工具比较适合英文文本分词、向量化和建模。text2vec 包支持多个统计模型，如LDA 、LSA 、GloVe 等，文本向量化后，结合统计学习模型，可用于分类、回归、聚类等任务，更多详情见 https://text2vec.org。\n接下来使用 David M. Blei 等提出 LDA 算法做主题建模，详情见 LDA 算法原始论文。\n\nlibrary(text2vec)\n\n首先将所有日志分词、向量化，构建文档-词矩阵 document-term matrix (DTM)\n\n# 移除链接\nremove_links <- function(x) {\n gsub(pattern = \"(<http.*?>)|(\\\\(http.*?\\\\))|(<www.*?>)|(\\\\(www.*?>\\\\))\", replacement = \"\", x)\n}\n# 清理、分词、清理\nfile_list1 <- lapply(file_list, remove_yaml)\nfile_list1 <- lapply(file_list1, remove_links)\nfile_list1 <- lapply(file_list1, segment, jiebar = jieba_seg)\nfile_list1 <- lapply(file_list1, remove_number_english)\n\n去掉没啥实际意义的词（比如单个字），极高频词和极低频词。\n\n# Token 化\nit <- itoken(file_list1, ids = 1:length(file_list1), progressbar = FALSE)\nv <- create_vocabulary(it)\n# 去掉单个字 减少 3K\nv <- v[nchar(v$term) > 1,]\n# 去掉极高频词和极低频词 减少 1.4W\nv <- prune_vocabulary(v, term_count_min = 10, doc_proportion_max = 0.2)\n\n采用 LDA（Latent Dirichlet Allocation）算法建模\n\n# 词向量化\nvectorizer <- vocab_vectorizer(v)\n# 文档-词矩阵 DTM\ndtm <- create_dtm(it, vectorizer, type = \"dgTMatrix\")\n# 10 个主题\nlda_model <- LDA$new(n_topics = 9, doc_topic_prior = 0.1, topic_word_prior = 0.01)\n# 训练模型\ndoc_topic_distr <- lda_model$fit_transform(\n x = dtm, n_iter = 1000, convergence_tol = 0.001, \n n_check_convergence = 25, progressbar = FALSE\n )\n\n#> INFO [15:28:26.129] early stopping at 175 iteration\n#> INFO [15:28:26.608] early stopping at 50 iteration\n\n\n下图展示主题的分布，各个主题及其所占比例。\n\nbarplot(\n doc_topic_distr[1, ], xlab = \"主题\", ylab = \"比例\", \n ylim = c(0, 1), names.arg = 1:ncol(doc_topic_distr)\n)\n\n\n\n\n\n\n图 17.3: 主题分布\n\n\n\n\n将 9 个主题的 Top 12 词分别打印出来。\n\nlda_model$get_top_words(n = 12, topic_number = 1L:9L, lambda = 0.3)\n\n#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] \n#> [1,] \"例子\" \"一首\" \"社会\" \"统计\" \"代码\" \"记得\" \"吱吱\" \"时代\" \"网站\" \n#> [2,] \"翻译\" \"歌词\" \"观点\" \"会议\" \"函数\" \"不知\" \"照片\" \"意义\" \"数据\" \n#> [3,] \"字符\" \"手机\" \"痛苦\" \"模型\" \"文档\" \"同学\" \"好吃\" \"媒体\" \"图形\" \n#> [4,] \"特征\" \"这首\" \"教育\" \"论文\" \"文件\" \"阿姨\" \"我家\" \"文化\" \"域名\" \n#> [5,] \"作品\" \"首歌\" \"人类\" \"老师\" \"变量\" \"居然\" \"家里\" \"现实\" \"软件\" \n#> [6,] \"中文\" \"遗憾\" \"追求\" \"分布\" \"字体\" \"看见\" \"味道\" \"社交\" \"服务器\"\n#> [7,] \"排版\" \"艺术\" \"思考\" \"小子\" \"元素\" \"学校\" \"厨房\" \"社区\" \"邮件\" \n#> [8,] \"意思\" \"小说\" \"强烈\" \"统计学\" \"语法\" \"路上\" \"在家\" \"眼中\" \"提供\" \n#> [9,] \"风格\" \"生活\" \"成功\" \"参加\" \"编译\" \"听说\" \"黄瓜\" \"避免\" \"编辑\" \n#> [10,] \"主题\" \"诗词\" \"接受\" \"报告\" \"图片\" \"印象\" \"包子\" \"造成\" \"系统\" \n#> [11,] \"伟大\" \"鸡蛋\" \"工作\" \"检验\" \"参数\" \"当时\" \"叶子\" \"事实\" \"浏览器\"\n#> [12,] \"表示\" \"人间\" \"努力\" \"学生\" \"生成\" \"名字\" \"辣椒\" \"政治\" \"注册\"\n\n\n结果有点意思，说明益辉喜欢读书写作（主题 1、3、8）、诗词歌赋（主题 2）、统计图形（主题 4）、代码编程（主题 5）、回忆青春（主题 6）、做菜吃饭（7）、倒腾网站（主题 9）。\n\n\n\n\n\n\n注释\n\n\n\n提示：参考论文 (Zhang, Li, 和 Zhang 2023) 根据 perplexities 做交叉验证选择最合适的主题数量。", + "text": "17.4 主题的探索\n益辉的日志是没有分类和标签的，所以，先聚类，接着逐个分析每个类代表的实际含义。然后，将聚类的结果作为结果标签，再应用多分类回归模型，最后联合聚类、分类模型，从无监督转化到有监督模型。\ntopicmodels (Grün 和 Hornik 2011) 基于 tm (Feinerer, Hornik, 和 Meyer 2008) 支持潜在狄利克雷分配（Latent Dirichlet Allocation，简称 LDA） 和 Correlated Topics Models (CTM) 文本主题建模，这一套工具比较适合英文文本分词、向量化和建模。text2vec 包支持多个统计模型，如LDA 、LSA 、GloVe 等，文本向量化后，结合统计学习模型，可用于分类、回归、聚类等任务，更多详情见 https://text2vec.org。\n接下来使用 David M. Blei 等提出 LDA 算法做主题建模，详情见 LDA 算法原始论文。\n\nlibrary(text2vec)\n\n首先将所有日志分词、向量化，构建文档-词矩阵 document-term matrix (DTM)\n\n# 移除链接\nremove_links <- function(x) {\n gsub(pattern = \"(<http.*?>)|(\\\\(http.*?\\\\))|(<www.*?>)|(\\\\(www.*?>\\\\))\", replacement = \"\", x)\n}\n# 清理、分词、清理\nfile_list1 <- lapply(file_list, remove_yaml)\nfile_list1 <- lapply(file_list1, remove_links)\nfile_list1 <- lapply(file_list1, segment, jiebar = jieba_seg)\nfile_list1 <- lapply(file_list1, remove_number_english)\n\n去掉没啥实际意义的词（比如单个字），极高频词和极低频词。\n\n# Token 化\nit <- itoken(file_list1, ids = 1:length(file_list1), progressbar = FALSE)\nv <- create_vocabulary(it)\n# 去掉单个字 减少 3K\nv <- v[nchar(v$term) > 1,]\n# 去掉极高频词和极低频词 减少 1.4W\nv <- prune_vocabulary(v, term_count_min = 10, doc_proportion_max = 0.2)\n\n采用 LDA（Latent Dirichlet Allocation）算法建模\n\n# 词向量化\nvectorizer <- vocab_vectorizer(v)\n# 文档-词矩阵 DTM\ndtm <- create_dtm(it, vectorizer, type = \"dgTMatrix\")\n# 10 个主题\nlda_model <- LDA$new(n_topics = 9, doc_topic_prior = 0.1, topic_word_prior = 0.01)\n# 训练模型\ndoc_topic_distr <- lda_model$fit_transform(\n x = dtm, n_iter = 1000, convergence_tol = 0.001, \n n_check_convergence = 25, progressbar = FALSE\n )\n\n#> INFO [03:04:22.578] early stopping at 175 iteration\n#> INFO [03:04:23.061] early stopping at 50 iteration\n\n\n下图展示主题的分布，各个主题及其所占比例。\n\nbarplot(\n doc_topic_distr[1, ], xlab = \"主题\", ylab = \"比例\", \n ylim = c(0, 1), names.arg = 1:ncol(doc_topic_distr)\n)\n\n\n\n\n\n\n图 17.3: 主题分布\n\n\n\n\n将 9 个主题的 Top 12 词分别打印出来。\n\nlda_model$get_top_words(n = 12, topic_number = 1L:9L, lambda = 0.3)\n\n#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] \n#> [1,] \"例子\" \"一首\" \"社会\" \"统计\" \"代码\" \"记得\" \"吱吱\" \"时代\" \"网站\" \n#> [2,] \"翻译\" \"歌词\" \"观点\" \"会议\" \"函数\" \"不知\" \"照片\" \"意义\" \"数据\" \n#> [3,] \"字符\" \"手机\" \"痛苦\" \"模型\" \"文档\" \"同学\" \"好吃\" \"媒体\" \"图形\" \n#> [4,] \"特征\" \"这首\" \"教育\" \"论文\" \"文件\" \"阿姨\" \"我家\" \"文化\" \"域名\" \n#> [5,] \"作品\" \"首歌\" \"人类\" \"老师\" \"变量\" \"居然\" \"家里\" \"现实\" \"软件\" \n#> [6,] \"中文\" \"遗憾\" \"追求\" \"分布\" \"字体\" \"看见\" \"味道\" \"社交\" \"服务器\"\n#> [7,] \"排版\" \"艺术\" \"思考\" \"小子\" \"元素\" \"学校\" \"厨房\" \"社区\" \"邮件\" \n#> [8,] \"意思\" \"小说\" \"强烈\" \"统计学\" \"语法\" \"路上\" \"在家\" \"眼中\" \"提供\" \n#> [9,] \"风格\" \"生活\" \"成功\" \"参加\" \"编译\" \"听说\" \"黄瓜\" \"避免\" \"编辑\" \n#> [10,] \"主题\" \"诗词\" \"接受\" \"报告\" \"图片\" \"印象\" \"包子\" \"造成\" \"系统\" \n#> [11,] \"伟大\" \"鸡蛋\" \"工作\" \"检验\" \"参数\" \"当时\" \"叶子\" \"事实\" \"浏览器\"\n#> [12,] \"表示\" \"人间\" \"努力\" \"学生\" \"生成\" \"名字\" \"辣椒\" \"政治\" \"注册\"\n\n\n结果有点意思，说明益辉喜欢读书写作（主题 1、3、8）、诗词歌赋（主题 2）、统计图形（主题 4）、代码编程（主题 5）、回忆青春（主题 6）、做菜吃饭（7）、倒腾网站（主题 9）。\n\n\n\n\n\n\n注释\n\n\n\n提示：参考论文 (Zhang, Li, 和 Zhang 2023) 根据 perplexities 做交叉验证选择最合适的主题数量。", "crumbs": [ "数据建模", "17  文本数据分析" @@ -1333,7 +1333,7 @@ "href": "numerical-optimization.html#sec-nonlinear-optimization", "title": "20  数值优化", "section": "\n20.4 非线性优化", - "text": "20.4 非线性优化\n非线性优化按是否带有约束，以及约束是线性还是非线性，分为无约束优化、箱式约束优化、线性约束优化和非线性约束优化。箱式约束可看作是线性约束的特殊情况。\n\nR 软件内置的非线性优化函数\n\n\nnlm()\nnlminb()\nconstrOptim()\noptim()\n\n\n\n无约束\n支持\n支持\n不支持\n支持\n\n\n箱式约束\n不支持\n支持\n支持\n支持\n\n\n线性约束\n不支持\n不支持\n支持\n不支持\n\n\n\nR 软件内置的 stats 包有 4 个数值优化方面的函数，函数 nlm() 可求解无约束优化问题，函数 nlminb() 可求解无约束、箱式约束优化问题，函数 constrOptim() 可求解箱式和线性约束优化。函数 optim() 是通用型求解器，包含多个优化算法，可求解无约束、箱式约束优化问题。尽管这些函数在 R 语言中长期存在，在统计中有广泛的使用，如非线性最小二乘 stats::nls()，极大似然估计 stats4::mle() 和广义最小二乘估计 nlme::gls() 等。但是，这些优化函数的求解能力有重合，使用语法不尽相同，对于非线性约束无能为力，下面仍然主要使用 ROI 包来求解多维非线性优化问题。\n\n20.4.1 一元非线性优化\n求如下一维分段非线性函数的最小值，其函数图像见 图 20.5 ，这个函数是不连续的，更不光滑。\n\\[\nf(x) =\n\\begin{cases}\n10 & x \\in (-\\infty,-1] \\\\\n\\exp(-\\frac{1}{|x-1|}) & x \\in (-1,4) \\\\\n10 & x \\in [4, +\\infty)\n\\end{cases}\n\\]\n\nfn <- function(x) ifelse(x > -1, ifelse(x < 4, exp(-1 / abs(x - 1)), 10), 10)\n\n\n代码op <- par(mar = c(4, 4, 0.5, 0.5))\ncurve(\n expr = fn, from = -2, to = 5, lwd = 2,\n panel.first = grid(),\n xlab = \"$x$\", ylab = \"$f(x)$\"\n)\non.exit(par(op), add = TRUE)\n\n\n\n\n\n\n图 20.5: 一维函数图像\n\n\n\n\n函数 optimize() 可以求解一元函数的极值问题，默认求极小值，参数 f 表示目标函数，参数 interval 表示搜索在此区间内最小值。函数返回一个列表，元素 minimum 表示极小值点，objective 表示极值点对应的目标函数值。\n\noptimize(f = fn, interval = c(-4, 20), maximum = FALSE)\n\n#> $minimum\n#> [1] 19.99995\n#> \n#> $objective\n#> [1] 10\n\noptimize(f = fn, interval = c(-7, 20), maximum = FALSE)\n\n#> $minimum\n#> [1] 0.9992797\n#> \n#> $objective\n#> [1] 0\n\n\n值得注意，对于不连续的分段函数，在不同的区间内搜索极值，可能获得不同的结果，可以绘制函数图像帮助选择最小值。\n\n20.4.2 多元隐函数优化\n这个优化问题来自 1stOpt 软件的帮助文档，下面利用 R 语言来求该多元隐函数的极值。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} y = & ~\\sin\\Big((yx_1 -0.5)^2 + 2x_1 x_2^2 - \\frac{y}{10} \\Big)\\cdot \\\\\n&~\\exp\\Big(-\\Big( \\big(x_1 - 0.5 -\\exp(-x_2 + y)\\big)^2 + x_2^2 - \\frac{y}{5} + 3 \\Big)\\Big)\n\\end{aligned}\n\\]\n其中， \\(x_1 \\in [-1,7],x_2 \\in [-2,2]\\) 。\n对于隐函数 \\(f(x_1,x_2,y)=0\\) ，常规的做法是先计算隐函数的偏导数，并令偏导数为 0，再求解非线性方程组，得到各个驻点，最后，将驻点代入原方程，比较驻点处函数值，根据优化目标选择最大或最小值。\n\\[\n\\begin{aligned}\n\\frac{\\partial f(x_1,x_2,y)}{\\partial x_1} = 0 \\\\\n\\frac{\\partial f(x_1,x_2,y)}{\\partial x_2} = 0\n\\end{aligned}\n\\]\n如果目标函数很复杂，隐函数偏导数难以计算，可以考虑暴力网格搜索。先估计隐函数值 \\(z\\) 的大致范围，给定 \\(x,y\\) 时，计算一元非线性方程的根。\n\nfn <- function(m) {\n subfun <- function(x) {\n f1 <- (m[1] * x - 0.5)^2 + 2 * m[1] * m[2]^2 - x / 10\n f2 <- -((m[1] - 0.5 - exp(-m[2] + x))^2 + m[2]^2 - x / 5 + 3)\n x - sin(f1) * exp(f2)\n }\n uniroot(f = subfun, interval = c(-1, 1))$root\n}\n\n在位置 \\((1,2)\\) 处函数值为 0.0007368468。\n\n# 测试函数 fn\nfn(m = c(1, 2))\n\n#> [1] 0.0007368468\n\n\n将目标区域网格化，通过一元非线性方程求根的方式获得每个格点处的函数值。\n\ndf <- expand.grid(\n x1 = seq(from = -1, to = 7, length.out = 81),\n x2 = seq(from = -2, to = 2, length.out = 41)\n)\n# 计算格点处的函数值\ndf$fn <- apply(df, 1, FUN = fn)\n\n在此基础上，绘制隐函数图像，如 图 20.6 所示，可以获得关于隐函数的大致情况。\n\n代码# 绘图\nwireframe(\n data = df, fn ~ x1 * x2,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]), ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 20.6: 隐函数图像\n\n\n\n\n最后，获得暴力网格搜索的结果，目标函数在 \\((2.8,-0.9)\\) 处取得最小值 \\(-0.02159723\\)。总的来说，这是一个近似结果，如果进一步缩小搜索区域，将网格划分得越细，搜索的结果将越接近全局最小值。\n\ndf[df$fn == min(df$fn), ]\n\n#> x1 x2 fn\n#> 930 2.8 -0.9 -0.02159723\n\n\n将求隐函数极值的问题转为含非线性等式约束的非线性优化问题。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} \\quad & y \\\\\n\\text{s.t.} \\quad & f(x_1,x_2,y) = 0\n\\end{aligned}\n\\]\n由于等式约束非常复杂，手动计算等式约束的雅可比矩阵不可行，可以用 numDeriv 包的函数 jacobian() 计算等式约束的雅可比矩阵。考虑到本例中仅含有一个等式约束，雅可比矩阵退化为梯度向量，这可以用 numDeriv 包的另一个函数 grad() 计算。\n\n# 等式约束\nheq <- function(x) {\n f1 <- (x[1] * x[3] - 0.5)^2 + 2 * x[1] * x[2]^2 - x[3] / 10\n f2 <- (x[1] - 0.5 - exp(-x[2] + x[3]))^2 + x[2]^2 - x[3] / 5 + 3\n x[3] - sin(f1) * exp(-f2)\n}\n# 等式约束的梯度\nheq.jac <- function(x) {\n numDeriv::grad(func = heq, x = x)\n}\n\n函数 L_objective() 表示含 1 个决策变量的线性目标函数，函数 F_constraint() 表示非线性等式约束。\n\n# 定义优化问题\nop <- OP(\n objective = L_objective(L = c(0, 0, 1)),\n constraints = F_constraint(\n # 等式约束\n F = list(heq = heq),\n dir = \"==\",\n rhs = 0,\n # 等式约束的雅可比\n J = list(heq.jac = heq.jac)\n ),\n bounds = V_bound(\n ld = -Inf, ud = Inf,\n li = c(1, 2), ui = c(1, 2),\n lb = c(-1, -2), ub = c(7, 2),\n nobj = 3L\n ),\n maximum = FALSE # 求最小\n)\nop\n\n#> ROI Optimization Problem:\n#> \n#> Minimize a linear objective function of length 3 with\n#> - 3 continuous objective variables,\n#> \n#> subject to\n#> - 1 constraint of type nonlinear.\n#> - 3 lower and 2 upper non-standard variable bounds.\n\n\n将网格搜索的结果作为初值，继续寻找更优的目标函数值。\n\nnlp <- ROI_solve(op,\n solver = \"nloptr.slsqp\", start = c(2.8, -0.9, -0.02159723)\n)\n# 最优解\nnlp$solution\n\n#> [1] 2.89826224 -0.85731584 -0.02335409\n\n# 目标函数值\nnlp$objval\n\n#> [1] -0.02335409\n\n\n可以发现，更优的目标函数值 \\(-0.02335\\) 在 \\((2.898,-0.8573)\\) 取得。\n\n20.4.3 多元无约束优化\n\n20.4.3.1 示例 1\nRastrigin 函数是一个 \\(n\\) 维优化问题测试函数。\n\\[\n\\min_{\\boldsymbol{x}} \\sum_{i=1}^{n}\\big(x_i^2 - 10 \\cos(2\\pi x_i) + 10\\big)\n\\]\n计算函数值的 R 代码如下：\n\nfn <- function(x) {\n sum(x^2 - 10 * cos(2 * pi * x) + 10)\n}\n\n绘制二维情形下的 Rastrigin 函数图像，如 图 20.7 所示，这是一个多模态的函数，有许多局部极小值。如果采用 BFGS 算法寻优容易陷入局部极值点。\n\n代码df <- expand.grid(\n x = seq(-4, 4, length.out = 151),\n y = seq(-4, 4, length.out = 151)\n)\n\ndf$fnxy <- apply(df, 1, fn)\nwireframe(\n data = df, fnxy ~ x * y,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]),\n ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 20.7: 二维 Rastrigin 函数图像\n\n\n\n\n不失一般性，考虑函数维数 \\(n=20\\) ，决策变量 \\(x_i \\in [-50,50], i = 1,2,\\ldots,n\\) 的情况。\n\nop <- OP(\n objective = F_objective(fn, n = 20L),\n bounds = V_bound(ld = -50, ud = 50, nobj = 20L)\n)\n\n调全局优化器求解优化问题。\n\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\n# 最优解\nnlp$solution\n\n#> [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n\n# 目标函数值\nnlp$objval\n\n#> [1] 0\n\n\n\n代码# R 语言内置的非线性优化函数\n# 无约束\nnlm(f = fn, p = rep(1, 20))\noptim(par = rep(1, 20), fn = fn, method = \"BFGS\")\noptim(par = rep(1, 20), fn = fn, method = \"Nelder-Mead\")\n\n# 箱式约束\noptim(par = rep(1, 20), fn = fn, \n lower = -50, upper = 50, method = \"L-BFGS-B\")\nnlminb(start = rep(1, 20), objective = fn, lower = -50, upper = 50)\nconstrOptim(\n theta = rep(1, 20), f = fn, grad = NULL,\n ui = rbind(diag(rep(1, 20)), diag(rep(-1, 20))),\n ci = c(rep(-50, 20), rep(-50, 20))\n)\n\n\n\n20.4.3.2 示例 2\n下面这个优化问题来自 1stOpt 软件帮助手册，是一个无约束非线性优化问题，它的目标函数非常复杂，一般的求解器都无法求解。最优解在 \\((7.999982, 7.999982)\\) 取得，目标函数值为 -7.978832。\n\\[\n\\begin{aligned}\n & \\min_{\\boldsymbol{x}} ~ \\cos(x_1)\\cos(x_2) - \\sum_{i=1}^{5}\\Big( (-1)^i \\cdot i \\cdot 2 \\cdot \\exp\\big(-500 \\cdot ( (x_1 - i \\cdot 2)^2 + (x_2 - i\\cdot 2)^2 ) \\big) \\Big)\n\\end{aligned}\n\\]\n目标函数分两步计算，先计算累加部分的通项，然后代入计算目标函数。\n\nsubfun <- function(i, m) {\n (-1)^i * i * 2 * exp(-500 * ((m[1] - i * 2)^2 + (m[2] - i * 2)^2))\n}\nfn <- function(x) {\n cos(x[1]) * cos(x[2]) -\n sum(mapply(FUN = subfun, i = 1:5, MoreArgs = list(m = x)))\n}\n\n直观起见，绘制目标函数在区域 \\([-50, 50] \\times [-50, 50]\\) 内的图像，如 图 20.8 (a) 所示，可以看到几乎没有变化的梯度，给寻优过程带来很大困难。再将区域 \\([0, 12] \\times [0, 12]\\) 上的三维图像绘制出来，如 图 20.8 (b) 所示，可见，有不少局部陷阱，且分布在 \\(x_2 = x_1\\) 的直线上。\n代码df <- expand.grid(\n x = seq(-50, 50, length.out = 101),\n y = seq(-50, 50, length.out = 101)\n)\ndf$fnxy <- apply(df, 1, fn)\nwireframe(\n data = df, fnxy ~ x * y,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]),\n ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\ndf <- expand.grid(\n x = seq(0, 12, length.out = 151),\n y = seq(0, 12, length.out = 151)\n)\ndf$fnxy <- apply(df, 1, fn)\nwireframe(\n data = df, fnxy ~ x * y,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]),\n ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90), alpha = 0.75, \n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n\n\n\n(a) 区域 \\([-50,50]\\times[-50,50]\\) 内的函数图像\n\n\n\n\n\n\n\n\n\n\n\n(b) 区域 \\([0,12]\\times[0,12]\\) 内的函数图像\n\n\n\n\n\n\n图 20.8: 局部放大前后的函数图像\n\n\n不失一般性，下面考虑 \\(x_1,x_2 \\in [-50,50]\\) ，面对如此复杂的函数，调用全局优化器 nloptr.directL 寻优。\n\nop <- OP(\n objective = F_objective(fn, n = 2L),\n bounds = V_bound(ld = -50, ud = 50, nobj = 2L)\n)\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\nnlp$solution\n\n#> [1] 0.00000 22.22222\n\nnlp$objval\n\n#> [1] -0.9734211\n\n\n结果还是陷入局部最优解。运筹优化方面的商业软件，著名的有 Lingo 和 Matlab，下面采用 Lingo 20 求解，Lingo 代码如下：\nSETS:\nP/1..5/;\nEndsets\nMin=@cos(x1) * @cos(x2) - @Sum(P(j): (-1)^j * j * 2 * @exp(-500 * ((x1 - j * 2)^2 + (x2 - j * 2)^2)));\n@Bnd(-50, x1, 50);\n@Bnd(-50, x2, 50);\n启用全局优化求解器后，在 \\((x_1 = 7.999982, x_2 = 7.999982)\\) 取得最小值 -7.978832。而默认未启用全局优化求解器的情况下，在 \\((x_1 = 18.84956, x_2 = -40.84070)\\) 取得局部极小值 -1.000000。\n在这种情况下，数值优化算法遇到瓶颈，可以采用一些全局随机优化算法，比如 GA 包 (Scrucca 2013) 实现的遗传算法。经过对参数的一些调优，可以获得与商业软件几乎一样的结果。\n\nnlp <- GA::ga(\n type = \"real-valued\",\n fitness = function(x) -fn(x),\n lower = c(0, 0), upper = c(12, 12),\n popSize = 500, maxiter = 100, \n monitor = FALSE, seed = 20232023\n)\n# 最优解\nnlp@solution\n\n#> x1 x2\n#> [1,] 7.999982 7.999981\n\n# 目标函数值\nnlp@fitnessValue\n\n#> [1] 7.978832\n\n\n其中，参数 type 指定决策变量的类型，type = \"real-valued\" 表示目标函数中的决策变量是实值连续的，参数 fitness 是目标函数，函数 ga() 对目标函数求极大，所以，对当前优化问题，添加了一个负号。 参数 popSize 控制种群大小，值越大，运行时间越长，搜索范围越广，获得的全局优化解越好。对于复杂的优化问题，可以不断增加种群大小来寻优，直至增加种群大小也不能获得更好的解。参数 maxiter 控制种群进化的次数，值越大，搜索次数可以越多，获得的解越好。参数 popSize 的影响大于参数 maxiter ，减少陷入局部最优解（陷阱）的可能。根据已知条件尽可能缩小可行域，以减少种群数量，进而缩短算法迭代时间。\n\n20.4.4 多元箱式约束优化\n有如下带箱式约束的多元非线性优化问题，该示例来自函数 nlminb() 的帮助文档，如果没有箱式约束，全局极小值点在 \\((1,1,\\cdots,1)\\) 处取得。\n\\[\n\\begin{aligned}\n \\min_{\\boldsymbol{x}} \\quad & (x_1 - 1)^2 + 4\\sum_{i =1}^{n -1}(x_{i+1} -x_i^2)^2 \\\\\n \\text{s.t.} \\quad & 2 \\leq x_1,x_2,\\cdots,x_n \\leq 4\n\\end{aligned}\n\\]\nR 语言编码的函数代码如下：\n\nfn <- function(x) {\n n <- length(x)\n sum(c(1, rep(4, n - 1)) * (x - c(1, x[-n])^2)^2)\n}\n\n在二维的情形下，可以绘制目标函数的三维图像，见 图 20.9 ，函数曲面和香蕉函数有些相似。\n\n代码dat <- expand.grid(\n x1 = seq(from = 0, to = 4, length.out = 41),\n x2 = seq(from = 0, to = 4, length.out = 41)\n)\ndat$fn <- apply(dat, 1, fn)\n\nwireframe(\n data = dat, fn ~ x1 * x2,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]), ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 20.9: 类香蕉函数的曲面图\n\n\n\n\nBase R 有 3 个函数可以求解这个优化问题，分别是 nlminb() 、constrOptim()和optim() ，因此，不妨在这个示例上，用这 3 个函数分别求解该优化问题，介绍它们的用法，最后，介绍 ROI 包实现的方法。这个优化问题的目标函数是 \\(n\\) 维非线性的，不失一般性，又不让问题变得太过简单，下面考虑 25 维的情况，\n\n20.4.4.1 nlminb()\n\n函数 nlminb() 参数 start 指定迭代初始值，参数 objective 指定目标函数，参数 lower 和 upper 分别指定箱式约束中的下界和上界。给定初值 \\((3, 3, \\cdots, 3)\\)，下界 \\((2,2,\\cdots,2)\\) 和上界 \\((4,4,\\cdots,4)\\) 。nlminb() 帮助文档说该函数出于历史兼容性的原因尚且存在，一般来说，这个函数会一直维护下去的。\n\nnlminb(\n start = rep(3, 25), objective = fn,\n lower = rep(2, 25), upper = rep(4, 25)\n)\n\n#> $par\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109093\n#> [25] 4.000000\n#> \n#> $objective\n#> [1] 368.1059\n#> \n#> $convergence\n#> [1] 0\n#> \n#> $iterations\n#> [1] 6\n#> \n#> $evaluations\n#> function gradient \n#> 10 177 \n#> \n#> $message\n#> [1] \"relative convergence (4)\"\n\n\n从返回结果来看，求解过程成功收敛，最优解的前 23 个决策变量取值为 2，在箱式约束的边界上，第 24 个分量没有边界上，而在内部，第 25 个决策变量取值为 4，也在边界上。目标函数值为 368.1059。\n\n20.4.4.2 constrOptim()\n\n使用 constrOptim() 函数求解，默认求极小，需将箱式或线性不等式约束写成矩阵形式，即 \\(Ax \\geq b\\) 的形式，参数 ui 是 \\(k \\times n\\) 的约束矩阵 \\(A\\)，ci 是右侧 \\(k\\) 维约束向量 \\(b\\)。以上面的优化问题为例，将箱式约束 \\(2 \\leq x_1,x_2 \\leq 4\\) 转化为矩阵形式，约束矩阵和向量分别为：\n\\[\nA = \\begin{bmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n-1 & 0 \\\\\n0 & -1\n\\end{bmatrix}, \\quad\nb = \\begin{bmatrix}\n2 \\\\\n2 \\\\\n-4 \\\\\n-4\n\\end{bmatrix}\n\\]\n\nconstrOptim(\n theta = rep(3, 25), # 初始值\n f = fn, # 目标函数\n method = \"Nelder-Mead\", # 没有提供梯度，则必须用 Nelder-Mead 方法\n ui = rbind(diag(rep(1, 25)), diag(rep(-1, 25))),\n ci = c(rep(2, 25), rep(-4, 25))\n)\n\n#> $par\n#> [1] 2.006142 2.002260 2.003971 2.003967 2.004143 2.004255 2.001178 2.002990\n#> [9] 2.003883 2.006029 2.017345 2.009236 2.000949 2.007793 2.025831 2.007896\n#> [17] 2.004514 2.004381 2.008771 2.015695 2.005803 2.009127 2.017988 2.257782\n#> [25] 3.999846\n#> \n#> $value\n#> [1] 378.4208\n#> \n#> $counts\n#> function gradient \n#> 12048 NA \n#> \n#> $convergence\n#> [1] 1\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 25\n#> \n#> $barrier.value\n#> [1] -0.003278963\n\n\n返回结果中 convergence = 1 表示迭代次数到达默认的极限 maxit = 500 。参考函数 nlminb() 的求解结果，可知还没有收敛。如果没有提供梯度，则必须用 Nelder-Mead 方法，下面增加迭代次数到 1000。\n\nconstrOptim(\n theta = rep(3, 25), # 初始值\n f = fn, # 目标函数\n method = \"Nelder-Mead\", \n control = list(maxit = 1000),\n ui = rbind(diag(rep(1, 25)), diag(rep(-1, 25))),\n ci = c(rep(2, 25), rep(-4, 25))\n)\n\n#> $par\n#> [1] 2.000081 2.000142 2.001919 2.000584 2.000007 2.000003 2.001097 2.001600\n#> [9] 2.000207 2.000042 2.000250 2.000295 2.000580 2.002165 2.000453 2.000932\n#> [17] 2.000456 2.000363 2.000418 2.000474 2.009483 2.001156 2.003173 2.241046\n#> [25] 3.990754\n#> \n#> $value\n#> [1] 370.8601\n#> \n#> $counts\n#> function gradient \n#> 18036 NA \n#> \n#> $convergence\n#> [1] 1\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 19\n#> \n#> $barrier.value\n#> [1] -0.003366467\n\n\n结果有改善，目标函数值从 378.4208 减小到 370.8601，但还是没有收敛，可见 Nelder-Mead 方法在这个优化问题上收敛速度比较慢。下面考虑调用基于梯度的 BFGS 优化算法，这得先计算出来目标函数的梯度。\n\n# 输入 n 维向量，输出 n 维向量\ngr <- function(x) {\n n <- length(x)\n c(2 * (x[1] - 2), rep(0, n - 1))\n +8 * c(0, x[-1] - x[-n]^2)\n -16 * c(x[-n], 0) * c(x[-1] - x[-n]^2, 0)\n}\nconstrOptim(\n theta = rep(3, 25), # 初始值\n f = fn, # 目标函数\n grad = gr,\n method = \"BFGS\", \n control = list(maxit = 1000),\n ui = rbind(diag(rep(1, 25)), diag(rep(-1, 25))),\n ci = c(rep(2, 25), rep(-4, 25))\n)\n\n#> $par\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000001\n#> [25] 3.000000\n#> \n#> $value\n#> [1] 373\n#> \n#> $counts\n#> function gradient \n#> 3721 464 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 3\n#> \n#> $barrier.value\n#> [1] -0.003327104\n\n\n从结果来看，虽然已经收敛，但相比于 Nelder-Mead 方法，目标函数值变大了，可见已陷入局部最优解。\n\n20.4.4.3 optim()\n\n下面再使用函数 optim() 提供的 L-BFGS-B 算法求解优化问题。\n\noptim(\n par = rep(3, 25), fn = fn, gr = NULL, method = \"L-BFGS-B\",\n lower = rep(2, 25), upper = rep(4, 25)\n)\n\n#> $par\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109093\n#> [25] 4.000000\n#> \n#> $value\n#> [1] 368.1059\n#> \n#> $counts\n#> function gradient \n#> 6 6 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> [1] \"CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH\"\n\n\n发现结果和函数 nlminb() 的结果差不多了。\n\noptim(\n par = rep(3, 25), fn = fn, gr = gr, method = \"L-BFGS-B\",\n lower = rep(2, 25), upper = rep(4, 25)\n)\n\n#> $par\n#> [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3\n#> \n#> $value\n#> [1] 373\n#> \n#> $counts\n#> function gradient \n#> 2 2 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> [1] \"CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL\"\n\n\n然而，当在函数 optim() 里提供梯度信息的时候，虽然目标函数及梯度的计算次数变少了，求解速度提升了，但是最优解反而变差了，最优解和在函数 constrOptim() 中设置 method = \"BFGS\" 算法基本一致。\n\n20.4.4.4 ROI 包\n下面通过 ROI 包，分别调用求解器 nloptr.lbfgs 和 nloptr.directL ，发现前者同样陷入局部最优解，而后者可以获得与 nlminb() 函数一致的结果。\n\nop <- OP(\n objective = F_objective(fn, n = 25L, G = gr),\n bounds = V_bound(ld = 2, ud = 4, nobj = 25L)\n)\nnlp <- ROI_solve(op, solver = \"nloptr.lbfgs\", start = rep(3, 25))\n# 目标函数值\nnlp$objval\n\n#> [1] 373\n\n# 最优解\nnlp$solution\n\n#> [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3\n\n\n调全局优化算法。\n\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\n# 目标函数值\nnlp$objval\n\n#> [1] 368.1061\n\n# 最优解\nnlp$solution\n\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109093\n#> [25] 4.000000\n\n\n\n20.4.5 多元线性约束优化\n对于带线性约束的多元非线性优化问题，Base R 提供函数 constrOptim() 来求解，下面的示例来自其帮助文档，这是一个带线性约束的二次规划问题。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}}\n\\quad & - \\begin{bmatrix}\n0 \\\\\n5 \\\\\n0\n\\end{bmatrix}^{\\top} \\boldsymbol{x} +\\frac{1}{2} \\boldsymbol{x}^{\\top}\\boldsymbol{x} \\\\\n\\text{s.t.} \\quad & \\begin{bmatrix}\n-4 & 2 & 0 \\\\\n-3 & 1 & -2 \\\\\n0 & 0 & 1\n\\end{bmatrix}^{\\top}\\boldsymbol{x} \\geq \\begin{bmatrix}\n-8 \\\\\n2 \\\\\n0\n\\end{bmatrix}\n\\end{aligned}\n\\]\n\nfQP <- function(x) {\n -sum(c(0, 5, 0) * x) + 0.5 * sum(x * x)\n}\nAmat <- matrix(c(-4, -3, 0, 2, 1, 0, 0, -2, 1),\n ncol = 3, nrow = 3, byrow = FALSE\n)\nbvec <- c(-8, 2, 0)\n# 目标函数的梯度\ngQP <- function(x) {\n -c(0, 5, 0) + x\n}\nconstrOptim(\n theta = c(2, -1, -1), \n f = fQP, g = gQP, \n ui = t(Amat), ci = bvec\n)\n\n#> $par\n#> [1] 0.4761908 1.0476188 2.0952376\n#> \n#> $value\n#> [1] -2.380952\n#> \n#> $counts\n#> function gradient \n#> 406 81 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 3\n#> \n#> $barrier.value\n#> [1] -0.0006243894\n\n\n在上一节，箱式约束可以看作线性约束的一种特殊情况，ROI 包是支持箱式、线性、二次、锥和非线性约束的。因此，下面给出调用 ROI 包求解上述优化问题的代码。\n\nDmat <- diag(rep(1,3))\ndvec <- c(0, 5, 0)\nop <- OP(\n objective = Q_objective(Q = Dmat, L = -dvec),\n constraints = L_constraint(L = t(Amat), dir = rep(\">=\", 3), rhs = bvec),\n maximum = FALSE\n)\nnlp <- ROI_solve(op, solver = \"nloptr.slsqp\", start = c(0, 1, 2))\n# 最优解\nnlp$solution\n\n#> [1] 0.4761905 1.0476190 2.0952381\n\n# 目标函数值\nnlp$objval\n\n#> [1] -2.380952\n\n\n可见输出结果与函数 constrOptim() 是一致的。\n\n代码# quadprog\nlibrary(quadprog)\nsol <- solve.QP(\n Dmat = Dmat, dvec = dvec, Amat = Amat, bvec = bvec\n)\nsol\n\n\n\n20.4.6 多元非线性约束优化\nnloptr 包的非线性优化能力覆盖开源优化软件 Octave 和 Ipopt 。通过插件包 ROI.plugin.nloptr，ROI 包可以调用 nloptr 包内置的所有求解器，常用的求解器见下表。表中从优化器类型（局部还是全局优化器），支持的约束条件类型（箱式还是非线性），是否需要提供目标函数的梯度、黑塞和约束条件的雅可比矩阵信息等方面归纳各个求解器的能力。\n\n常用的非线性优化求解器\n\n求解器\n类型\n约束\n梯度\n黑塞\n雅可比\n\n\n\nnloptr.lbfgs\n局部\n箱式\n需要\n不需要\n不需要\n\n\nnloptr.slsqp\n局部\n非线性\n需要\n不需要\n需要\n\n\nnloptr.auglag\n局部\n非线性\n需要\n不需要\n需要\n\n\nnloptr.directL\n全局\n箱式\n不需要\n不需要\n不需要\n\n\nnloptr.isres\n全局\n非线性\n不需要\n不需要\n不需要\n\n\n\n\n20.4.6.1 非线性等式约束\n下面这个示例来自 Octave 软件的非线性优化帮助文档，Octave 中的函数 sqp() 使用序列二次优化求解器（successive quadratic programming solver）求解非线性优化问题，示例中该优化问题包含多个非线性等式约束。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} \\quad & \\exp\\big(\\prod_{i=1}^{5} x_i\\big) - \\frac{1}{2}(x_1^3 + x_2^3 + 1)^2 \\\\\n\\text{s.t.} \\quad & \\left\\{\n \\begin{array}{l}\n \\sum_{i=1}^{5}x_i^2 - 10 = 0 \\\\\n x_2 x_3 - 5x_4 x_5 = 0 \\\\\n x_1^3 + x_2^3 + 1 = 0\n \\end{array} \\right.\n\\end{aligned}\n\\]\n目标函数是非线性的，有 5 个变量，约束条件也是非线性的，有 3 个等式约束。先手动计算目标函数的梯度，等式约束的雅可比矩阵。\n\n# 目标函数\nfn <- function(x) {\n exp(prod(x)) - 0.5 * (x[1]^3 + x[2]^3 + 1)^2\n}\n# 目标函数的梯度\ngr <- function(x) {\n c(\n exp(prod(x)) * prod(x[-1]) - 3 * (x[1]^3 + x[2]^3 + 1) * x[1]^2,\n exp(prod(x)) * prod(x[-2]) - 3 * (x[1]^3 + x[2]^3 + 1) * x[2]^2,\n exp(prod(x)) * prod(x[-3]),\n exp(prod(x)) * prod(x[-4]),\n exp(prod(x)) * prod(x[-5])\n )\n}\n# 等式约束\nheq <- function(x) {\n c(\n sum(x^2) - 10,\n x[2] * x[3] - 5 * x[4] * x[5],\n x[1]^3 + x[2]^3 + 1\n )\n}\n# 等式约束的雅可比矩阵\nheq.jac <- function(x) {\n matrix(c(2 * x[1], 2 * x[2], 2 * x[3], 2 * x[4], 2 * x[5],\n 0, x[3], x[2], -5 * x[5], -5 * x[4],\n 3 * x[1]^2, 3 * x[2]^2, 0, 0, 0),\n ncol = 5, byrow = TRUE\n )\n}\n\n在 OP() 函数里定义目标优化的各个成分。\n\n# 定义目标优化\nop <- OP(\n # 5 个决策变量\n objective = F_objective(F = fn, n = 5L, G = gr), \n constraints = F_constraint(\n F = list(heq = heq),\n dir = \"==\",\n rhs = 0,\n # 等式约束的雅可比矩阵\n J = list(heq.jac = heq.jac)\n ),\n bounds = V_bound(ld = -Inf, ud = Inf, nobj = 5L),\n maximum = FALSE # 求最小\n)\nop\n\n#> ROI Optimization Problem:\n#> \n#> Minimize a nonlinear objective function of length 5 with\n#> - 5 continuous objective variables,\n#> \n#> subject to\n#> - 1 constraint of type nonlinear.\n#> - 5 lower and 0 upper non-standard variable bounds.\n\n\n调用 SQP（序列二次优化） 求解器 nloptr.slsqp 。\n\nnlp <- ROI_solve(op,\n solver = \"nloptr.slsqp\",\n start = c(-1.8, 1.7, 1.9, -0.8, -0.8)\n)\n# 最优解\nnlp$solution\n\n#> [1] -1.7171435 1.5957096 1.8272458 -0.7636431 -0.7636431\n\n# 目标函数值\nnlp$objval\n\n#> [1] 0.05394985\n\n\n计算结果和 Octave 的示例一致。\n\n20.4.6.2 多种非线性约束\n\n非线性等式约束\n非线性不等式约束，不等式约束包含等号\n箱式约束\n\n此优化问题来源于 Ipopt 官网的帮助文档，约束条件比较复杂。提供的初始值为 \\(x_0 = (1,5,5,1)\\)，最优解为 \\(x_{\\star} = (1.00000000,4.74299963,3.82114998,1.37940829)\\)。优化问题的具体内容如下：\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} & \\quad x_1 x_4 (x_1 + x_2 + x_3) + x_3 \\\\\n\\text{s.t.} & \\quad \\left\\{\n \\begin{array}{l}\n x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40 \\\\\n x_1 x_2 x_3 x_4 \\geq 25 \\\\\n 1 \\leq x_1, x_2, x_3, x_4 \\leq 5\n \\end{array} \\right.\n\\end{aligned}\n\\]\n下面用 ROI 调 nloptr 包求解，看结果是否和例子一致，nloptr 支持箱式约束且支持不等式约束包含等号。\n\n# 一个 4 维的目标函数\nfn <- function(x) {\n x[1] * x[4] * (x[1] + x[2] + x[3]) + x[3]\n}\n# 目标函数的梯度\ngr <- function(x) {\n c(\n x[4] * (2 * x[1] + x[2] + x[3]), x[1] * x[4],\n x[1] * x[4] + 1, x[1] * (x[1] + x[2] + x[3])\n )\n}\n# 等式约束\nheq <- function(x) {\n sum(x^2)\n}\n# 等式约束的雅可比\nheq.jac <- function(x) {\n 2 * c(x[1], x[2], x[3], x[4])\n}\n# 不等式约束\nhin <- function(x) {\n prod(x)\n}\n# 不等式约束的雅可比\nhin.jac <- function(x) {\n c(prod(x[-1]), prod(x[-2]), prod(x[-3]), prod(x[-4]))\n}\n# 定义目标优化\nop <- OP(\n objective = F_objective(F = fn, n = 4L, G = gr), # 4 个决策变量\n constraints = F_constraint(\n F = list(heq = heq, hin = hin),\n dir = c(\"==\", \">=\"),\n rhs = c(40, 25),\n # 等式和不等式约束的雅可比\n J = list(heq.jac = heq.jac, hin.jac = hin.jac)\n ),\n bounds = V_bound(ld = 1, ud = 5, nobj = 4L),\n maximum = FALSE # 求最小\n)\n\n作为对比参考，先计算目标函数的初始值和最优值。\n\n# 目标函数初始值\nfn(c(1, 5, 5, 1))\n\n#> [1] 16\n\n# 目标函数最优值\nfn(c(1.00000000, 4.74299963, 3.82114998, 1.37940829))\n\n#> [1] 17.01402\n\n\n求解一般的非线性约束问题。\n\n求解器 nloptr.mma / nloptr.cobyla 仅支持非线性不等式约束，不支持等式约束。\n函数 nlminb() 只支持等式约束。\n\n因此，下面分别调用 nloptr.auglag、nloptr.slsqp 和 nloptr.isres 来求解上述优化问题。\n\nnlp <- ROI_solve(op, solver = \"nloptr.auglag\", start = c(1, 5, 5, 1))\nnlp$solution\n\n#> [1] 1.000000 4.743174 3.820922 1.379440\n\nnlp$objval\n\n#> [1] 17.01402\n\n\n\nnlp <- ROI_solve(op, solver = \"nloptr.slsqp\", start = c(1, 5, 5, 1))\nnlp$solution\n\n#> [1] 1.000000 4.742996 3.821155 1.379408\n\nnlp$objval\n\n#> [1] 17.01402\n\n\n\nnlp <- ROI_solve(op, solver = \"nloptr.isres\", start = c(1, 5, 5, 1))\nnlp$solution\n\n#> [1] 1.276795 4.607639 3.992841 1.093756\n\nnlp$objval\n\n#> [1] 17.78648\n\n\n可以看出，nloptr 提供的优化能力可以覆盖 Ipopt 求解器，从以上求解的情况来看，推荐使用 nloptr.slsqp 求解器，这也是 Octave 的选择。", + "text": "20.4 非线性优化\n非线性优化按是否带有约束，以及约束是线性还是非线性，分为无约束优化、箱式约束优化、线性约束优化和非线性约束优化。箱式约束可看作是线性约束的特殊情况。\n\nR 软件内置的非线性优化函数\n\n\nnlm()\nnlminb()\nconstrOptim()\noptim()\n\n\n\n无约束\n支持\n支持\n不支持\n支持\n\n\n箱式约束\n不支持\n支持\n支持\n支持\n\n\n线性约束\n不支持\n不支持\n支持\n不支持\n\n\n\nR 软件内置的 stats 包有 4 个数值优化方面的函数，函数 nlm() 可求解无约束优化问题，函数 nlminb() 可求解无约束、箱式约束优化问题，函数 constrOptim() 可求解箱式和线性约束优化。函数 optim() 是通用型求解器，包含多个优化算法，可求解无约束、箱式约束优化问题。尽管这些函数在 R 语言中长期存在，在统计中有广泛的使用，如非线性最小二乘 stats::nls()，极大似然估计 stats4::mle() 和广义最小二乘估计 nlme::gls() 等。但是，这些优化函数的求解能力有重合，使用语法不尽相同，对于非线性约束无能为力，下面仍然主要使用 ROI 包来求解多维非线性优化问题。\n\n20.4.1 一元非线性优化\n求如下一维分段非线性函数的最小值，其函数图像见 图 20.5 ，这个函数是不连续的，更不光滑。\n\\[\nf(x) =\n\\begin{cases}\n10 & x \\in (-\\infty,-1] \\\\\n\\exp(-\\frac{1}{|x-1|}) & x \\in (-1,4) \\\\\n10 & x \\in [4, +\\infty)\n\\end{cases}\n\\]\n\nfn <- function(x) ifelse(x > -1, ifelse(x < 4, exp(-1 / abs(x - 1)), 10), 10)\n\n\n代码op <- par(mar = c(4, 4, 0.5, 0.5))\ncurve(\n expr = fn, from = -2, to = 5, lwd = 2,\n panel.first = grid(),\n xlab = \"$x$\", ylab = \"$f(x)$\"\n)\non.exit(par(op), add = TRUE)\n\n\n\n\n\n\n图 20.5: 一维函数图像\n\n\n\n\n函数 optimize() 可以求解一元函数的极值问题，默认求极小值，参数 f 表示目标函数，参数 interval 表示搜索在此区间内最小值。函数返回一个列表，元素 minimum 表示极小值点，objective 表示极值点对应的目标函数值。\n\noptimize(f = fn, interval = c(-4, 20), maximum = FALSE)\n\n#> $minimum\n#> [1] 19.99995\n#> \n#> $objective\n#> [1] 10\n\noptimize(f = fn, interval = c(-7, 20), maximum = FALSE)\n\n#> $minimum\n#> [1] 0.9992797\n#> \n#> $objective\n#> [1] 0\n\n\n值得注意，对于不连续的分段函数，在不同的区间内搜索极值，可能获得不同的结果，可以绘制函数图像帮助选择最小值。\n\n20.4.2 多元隐函数优化\n这个优化问题来自 1stOpt 软件的帮助文档，下面利用 R 语言来求该多元隐函数的极值。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} y = & ~\\sin\\Big((yx_1 -0.5)^2 + 2x_1 x_2^2 - \\frac{y}{10} \\Big)\\cdot \\\\\n&~\\exp\\Big(-\\Big( \\big(x_1 - 0.5 -\\exp(-x_2 + y)\\big)^2 + x_2^2 - \\frac{y}{5} + 3 \\Big)\\Big)\n\\end{aligned}\n\\]\n其中， \\(x_1 \\in [-1,7],x_2 \\in [-2,2]\\) 。\n对于隐函数 \\(f(x_1,x_2,y)=0\\) ，常规的做法是先计算隐函数的偏导数，并令偏导数为 0，再求解非线性方程组，得到各个驻点，最后，将驻点代入原方程，比较驻点处函数值，根据优化目标选择最大或最小值。\n\\[\n\\begin{aligned}\n\\frac{\\partial f(x_1,x_2,y)}{\\partial x_1} = 0 \\\\\n\\frac{\\partial f(x_1,x_2,y)}{\\partial x_2} = 0\n\\end{aligned}\n\\]\n如果目标函数很复杂，隐函数偏导数难以计算，可以考虑暴力网格搜索。先估计隐函数值 \\(z\\) 的大致范围，给定 \\(x,y\\) 时，计算一元非线性方程的根。\n\nfn <- function(m) {\n subfun <- function(x) {\n f1 <- (m[1] * x - 0.5)^2 + 2 * m[1] * m[2]^2 - x / 10\n f2 <- -((m[1] - 0.5 - exp(-m[2] + x))^2 + m[2]^2 - x / 5 + 3)\n x - sin(f1) * exp(f2)\n }\n uniroot(f = subfun, interval = c(-1, 1))$root\n}\n\n在位置 \\((1,2)\\) 处函数值为 0.0007368468。\n\n# 测试函数 fn\nfn(m = c(1, 2))\n\n#> [1] 0.0007368468\n\n\n将目标区域网格化，通过一元非线性方程求根的方式获得每个格点处的函数值。\n\ndf <- expand.grid(\n x1 = seq(from = -1, to = 7, length.out = 81),\n x2 = seq(from = -2, to = 2, length.out = 41)\n)\n# 计算格点处的函数值\ndf$fn <- apply(df, 1, FUN = fn)\n\n在此基础上，绘制隐函数图像，如 图 20.6 所示，可以获得关于隐函数的大致情况。\n\n代码# 绘图\nwireframe(\n data = df, fn ~ x1 * x2,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]), ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 20.6: 隐函数图像\n\n\n\n\n最后，获得暴力网格搜索的结果，目标函数在 \\((2.8,-0.9)\\) 处取得最小值 \\(-0.02159723\\)。总的来说，这是一个近似结果，如果进一步缩小搜索区域，将网格划分得越细，搜索的结果将越接近全局最小值。\n\ndf[df$fn == min(df$fn), ]\n\n#> x1 x2 fn\n#> 930 2.8 -0.9 -0.02159723\n\n\n将求隐函数极值的问题转为含非线性等式约束的非线性优化问题。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} \\quad & y \\\\\n\\text{s.t.} \\quad & f(x_1,x_2,y) = 0\n\\end{aligned}\n\\]\n由于等式约束非常复杂，手动计算等式约束的雅可比矩阵不可行，可以用 numDeriv 包的函数 jacobian() 计算等式约束的雅可比矩阵。考虑到本例中仅含有一个等式约束，雅可比矩阵退化为梯度向量，这可以用 numDeriv 包的另一个函数 grad() 计算。\n\n# 等式约束\nheq <- function(x) {\n f1 <- (x[1] * x[3] - 0.5)^2 + 2 * x[1] * x[2]^2 - x[3] / 10\n f2 <- (x[1] - 0.5 - exp(-x[2] + x[3]))^2 + x[2]^2 - x[3] / 5 + 3\n x[3] - sin(f1) * exp(-f2)\n}\n# 等式约束的梯度\nheq.jac <- function(x) {\n numDeriv::grad(func = heq, x = x)\n}\n\n函数 L_objective() 表示含 1 个决策变量的线性目标函数，函数 F_constraint() 表示非线性等式约束。\n\n# 定义优化问题\nop <- OP(\n objective = L_objective(L = c(0, 0, 1)),\n constraints = F_constraint(\n # 等式约束\n F = list(heq = heq),\n dir = \"==\",\n rhs = 0,\n # 等式约束的雅可比\n J = list(heq.jac = heq.jac)\n ),\n bounds = V_bound(\n ld = -Inf, ud = Inf,\n li = c(1, 2), ui = c(1, 2),\n lb = c(-1, -2), ub = c(7, 2),\n nobj = 3L\n ),\n maximum = FALSE # 求最小\n)\nop\n\n#> ROI Optimization Problem:\n#> \n#> Minimize a linear objective function of length 3 with\n#> - 3 continuous objective variables,\n#> \n#> subject to\n#> - 1 constraint of type nonlinear.\n#> - 3 lower and 2 upper non-standard variable bounds.\n\n\n将网格搜索的结果作为初值，继续寻找更优的目标函数值。\n\nnlp <- ROI_solve(op,\n solver = \"nloptr.slsqp\", start = c(2.8, -0.9, -0.02159723)\n)\n# 最优解\nnlp$solution\n\n#> [1] 2.89826224 -0.85731584 -0.02335409\n\n# 目标函数值\nnlp$objval\n\n#> [1] -0.02335409\n\n\n可以发现，更优的目标函数值 \\(-0.02335\\) 在 \\((2.898,-0.8573)\\) 取得。\n\n20.4.3 多元无约束优化\n\n20.4.3.1 示例 1\nRastrigin 函数是一个 \\(n\\) 维优化问题测试函数。\n\\[\n\\min_{\\boldsymbol{x}} \\sum_{i=1}^{n}\\big(x_i^2 - 10 \\cos(2\\pi x_i) + 10\\big)\n\\]\n计算函数值的 R 代码如下：\n\nfn <- function(x) {\n sum(x^2 - 10 * cos(2 * pi * x) + 10)\n}\n\n绘制二维情形下的 Rastrigin 函数图像，如 图 20.7 所示，这是一个多模态的函数，有许多局部极小值。如果采用 BFGS 算法寻优容易陷入局部极值点。\n\n代码df <- expand.grid(\n x = seq(-4, 4, length.out = 151),\n y = seq(-4, 4, length.out = 151)\n)\n\ndf$fnxy <- apply(df, 1, fn)\nwireframe(\n data = df, fnxy ~ x * y,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]),\n ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 20.7: 二维 Rastrigin 函数图像\n\n\n\n\n不失一般性，考虑函数维数 \\(n=20\\) ，决策变量 \\(x_i \\in [-50,50], i = 1,2,\\ldots,n\\) 的情况。\n\nop <- OP(\n objective = F_objective(fn, n = 20L),\n bounds = V_bound(ld = -50, ud = 50, nobj = 20L)\n)\n\n调全局优化器求解优化问题。\n\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\n# 最优解\nnlp$solution\n\n#> [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n\n# 目标函数值\nnlp$objval\n\n#> [1] 0\n\n\n\n代码# R 语言内置的非线性优化函数\n# 无约束\nnlm(f = fn, p = rep(1, 20))\noptim(par = rep(1, 20), fn = fn, method = \"BFGS\")\noptim(par = rep(1, 20), fn = fn, method = \"Nelder-Mead\")\n\n# 箱式约束\noptim(par = rep(1, 20), fn = fn, \n lower = -50, upper = 50, method = \"L-BFGS-B\")\nnlminb(start = rep(1, 20), objective = fn, lower = -50, upper = 50)\nconstrOptim(\n theta = rep(1, 20), f = fn, grad = NULL,\n ui = rbind(diag(rep(1, 20)), diag(rep(-1, 20))),\n ci = c(rep(-50, 20), rep(-50, 20))\n)\n\n\n\n20.4.3.2 示例 2\n下面这个优化问题来自 1stOpt 软件帮助手册，是一个无约束非线性优化问题，它的目标函数非常复杂，一般的求解器都无法求解。最优解在 \\((7.999982, 7.999982)\\) 取得，目标函数值为 -7.978832。\n\\[\n\\begin{aligned}\n & \\min_{\\boldsymbol{x}} ~ \\cos(x_1)\\cos(x_2) - \\sum_{i=1}^{5}\\Big( (-1)^i \\cdot i \\cdot 2 \\cdot \\exp\\big(-500 \\cdot ( (x_1 - i \\cdot 2)^2 + (x_2 - i\\cdot 2)^2 ) \\big) \\Big)\n\\end{aligned}\n\\]\n目标函数分两步计算，先计算累加部分的通项，然后代入计算目标函数。\n\nsubfun <- function(i, m) {\n (-1)^i * i * 2 * exp(-500 * ((m[1] - i * 2)^2 + (m[2] - i * 2)^2))\n}\nfn <- function(x) {\n cos(x[1]) * cos(x[2]) -\n sum(mapply(FUN = subfun, i = 1:5, MoreArgs = list(m = x)))\n}\n\n直观起见，绘制目标函数在区域 \\([-50, 50] \\times [-50, 50]\\) 内的图像，如 图 20.8 (a) 所示，可以看到几乎没有变化的梯度，给寻优过程带来很大困难。再将区域 \\([0, 12] \\times [0, 12]\\) 上的三维图像绘制出来，如 图 20.8 (b) 所示，可见，有不少局部陷阱，且分布在 \\(x_2 = x_1\\) 的直线上。\n代码df <- expand.grid(\n x = seq(-50, 50, length.out = 101),\n y = seq(-50, 50, length.out = 101)\n)\ndf$fnxy <- apply(df, 1, fn)\nwireframe(\n data = df, fnxy ~ x * y,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]),\n ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\ndf <- expand.grid(\n x = seq(0, 12, length.out = 151),\n y = seq(0, 12, length.out = 151)\n)\ndf$fnxy <- apply(df, 1, fn)\nwireframe(\n data = df, fnxy ~ x * y,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]),\n ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90), alpha = 0.75, \n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n\n\n\n(a) 区域 \\([-50,50]\\times[-50,50]\\) 内的函数图像\n\n\n\n\n\n\n\n\n\n\n\n(b) 区域 \\([0,12]\\times[0,12]\\) 内的函数图像\n\n\n\n\n\n\n图 20.8: 局部放大前后的函数图像\n\n\n不失一般性，下面考虑 \\(x_1,x_2 \\in [-50,50]\\) ，面对如此复杂的函数，调用全局优化器 nloptr.directL 寻优。\n\nop <- OP(\n objective = F_objective(fn, n = 2L),\n bounds = V_bound(ld = -50, ud = 50, nobj = 2L)\n)\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\nnlp$solution\n\n#> [1] -21.99115 0.00000\n\nnlp$objval\n\n#> [1] -1\n\n\n结果还是陷入局部最优解。运筹优化方面的商业软件，著名的有 Lingo 和 Matlab，下面采用 Lingo 20 求解，Lingo 代码如下：\nSETS:\nP/1..5/;\nEndsets\nMin=@cos(x1) * @cos(x2) - @Sum(P(j): (-1)^j * j * 2 * @exp(-500 * ((x1 - j * 2)^2 + (x2 - j * 2)^2)));\n@Bnd(-50, x1, 50);\n@Bnd(-50, x2, 50);\n启用全局优化求解器后，在 \\((x_1 = 7.999982, x_2 = 7.999982)\\) 取得最小值 -7.978832。而默认未启用全局优化求解器的情况下，在 \\((x_1 = 18.84956, x_2 = -40.84070)\\) 取得局部极小值 -1.000000。\n在这种情况下，数值优化算法遇到瓶颈，可以采用一些全局随机优化算法，比如 GA 包 (Scrucca 2013) 实现的遗传算法。经过对参数的一些调优，可以获得与商业软件几乎一样的结果。\n\nnlp <- GA::ga(\n type = \"real-valued\",\n fitness = function(x) -fn(x),\n lower = c(0, 0), upper = c(12, 12),\n popSize = 500, maxiter = 100, \n monitor = FALSE, seed = 20232023\n)\n# 最优解\nnlp@solution\n\n#> x1 x2\n#> [1,] 7.999982 7.999981\n\n# 目标函数值\nnlp@fitnessValue\n\n#> [1] 7.978832\n\n\n其中，参数 type 指定决策变量的类型，type = \"real-valued\" 表示目标函数中的决策变量是实值连续的，参数 fitness 是目标函数，函数 ga() 对目标函数求极大，所以，对当前优化问题，添加了一个负号。 参数 popSize 控制种群大小，值越大，运行时间越长，搜索范围越广，获得的全局优化解越好。对于复杂的优化问题，可以不断增加种群大小来寻优，直至增加种群大小也不能获得更好的解。参数 maxiter 控制种群进化的次数，值越大，搜索次数可以越多，获得的解越好。参数 popSize 的影响大于参数 maxiter ，减少陷入局部最优解（陷阱）的可能。根据已知条件尽可能缩小可行域，以减少种群数量，进而缩短算法迭代时间。\n\n20.4.4 多元箱式约束优化\n有如下带箱式约束的多元非线性优化问题，该示例来自函数 nlminb() 的帮助文档，如果没有箱式约束，全局极小值点在 \\((1,1,\\cdots,1)\\) 处取得。\n\\[\n\\begin{aligned}\n \\min_{\\boldsymbol{x}} \\quad & (x_1 - 1)^2 + 4\\sum_{i =1}^{n -1}(x_{i+1} -x_i^2)^2 \\\\\n \\text{s.t.} \\quad & 2 \\leq x_1,x_2,\\cdots,x_n \\leq 4\n\\end{aligned}\n\\]\nR 语言编码的函数代码如下：\n\nfn <- function(x) {\n n <- length(x)\n sum(c(1, rep(4, n - 1)) * (x - c(1, x[-n])^2)^2)\n}\n\n在二维的情形下，可以绘制目标函数的三维图像，见 图 20.9 ，函数曲面和香蕉函数有些相似。\n\n代码dat <- expand.grid(\n x1 = seq(from = 0, to = 4, length.out = 41),\n x2 = seq(from = 0, to = 4, length.out = 41)\n)\ndat$fn <- apply(dat, 1, fn)\n\nwireframe(\n data = dat, fn ~ x1 * x2,\n shade = TRUE, drape = FALSE,\n xlab = expression(x[1]), ylab = expression(x[2]),\n zlab = list(expression(\n italic(f) ~ group(\"(\", list(x[1], x[2]), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 20.9: 类香蕉函数的曲面图\n\n\n\n\nBase R 有 3 个函数可以求解这个优化问题，分别是 nlminb() 、constrOptim()和optim() ，因此，不妨在这个示例上，用这 3 个函数分别求解该优化问题，介绍它们的用法，最后，介绍 ROI 包实现的方法。这个优化问题的目标函数是 \\(n\\) 维非线性的，不失一般性，又不让问题变得太过简单，下面考虑 25 维的情况，\n\n20.4.4.1 nlminb()\n\n函数 nlminb() 参数 start 指定迭代初始值，参数 objective 指定目标函数，参数 lower 和 upper 分别指定箱式约束中的下界和上界。给定初值 \\((3, 3, \\cdots, 3)\\)，下界 \\((2,2,\\cdots,2)\\) 和上界 \\((4,4,\\cdots,4)\\) 。nlminb() 帮助文档说该函数出于历史兼容性的原因尚且存在，一般来说，这个函数会一直维护下去的。\n\nnlminb(\n start = rep(3, 25), objective = fn,\n lower = rep(2, 25), upper = rep(4, 25)\n)\n\n#> $par\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109093\n#> [25] 4.000000\n#> \n#> $objective\n#> [1] 368.1059\n#> \n#> $convergence\n#> [1] 0\n#> \n#> $iterations\n#> [1] 6\n#> \n#> $evaluations\n#> function gradient \n#> 10 177 \n#> \n#> $message\n#> [1] \"relative convergence (4)\"\n\n\n从返回结果来看，求解过程成功收敛，最优解的前 23 个决策变量取值为 2，在箱式约束的边界上，第 24 个分量没有边界上，而在内部，第 25 个决策变量取值为 4，也在边界上。目标函数值为 368.1059。\n\n20.4.4.2 constrOptim()\n\n使用 constrOptim() 函数求解，默认求极小，需将箱式或线性不等式约束写成矩阵形式，即 \\(Ax \\geq b\\) 的形式，参数 ui 是 \\(k \\times n\\) 的约束矩阵 \\(A\\)，ci 是右侧 \\(k\\) 维约束向量 \\(b\\)。以上面的优化问题为例，将箱式约束 \\(2 \\leq x_1,x_2 \\leq 4\\) 转化为矩阵形式，约束矩阵和向量分别为：\n\\[\nA = \\begin{bmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n-1 & 0 \\\\\n0 & -1\n\\end{bmatrix}, \\quad\nb = \\begin{bmatrix}\n2 \\\\\n2 \\\\\n-4 \\\\\n-4\n\\end{bmatrix}\n\\]\n\nconstrOptim(\n theta = rep(3, 25), # 初始值\n f = fn, # 目标函数\n method = \"Nelder-Mead\", # 没有提供梯度，则必须用 Nelder-Mead 方法\n ui = rbind(diag(rep(1, 25)), diag(rep(-1, 25))),\n ci = c(rep(2, 25), rep(-4, 25))\n)\n\n#> $par\n#> [1] 2.006142 2.002260 2.003971 2.003967 2.004143 2.004255 2.001178 2.002990\n#> [9] 2.003883 2.006029 2.017345 2.009236 2.000949 2.007793 2.025831 2.007896\n#> [17] 2.004514 2.004381 2.008771 2.015695 2.005803 2.009127 2.017988 2.257782\n#> [25] 3.999846\n#> \n#> $value\n#> [1] 378.4208\n#> \n#> $counts\n#> function gradient \n#> 12048 NA \n#> \n#> $convergence\n#> [1] 1\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 25\n#> \n#> $barrier.value\n#> [1] -0.003278963\n\n\n返回结果中 convergence = 1 表示迭代次数到达默认的极限 maxit = 500 。参考函数 nlminb() 的求解结果，可知还没有收敛。如果没有提供梯度，则必须用 Nelder-Mead 方法，下面增加迭代次数到 1000。\n\nconstrOptim(\n theta = rep(3, 25), # 初始值\n f = fn, # 目标函数\n method = \"Nelder-Mead\", \n control = list(maxit = 1000),\n ui = rbind(diag(rep(1, 25)), diag(rep(-1, 25))),\n ci = c(rep(2, 25), rep(-4, 25))\n)\n\n#> $par\n#> [1] 2.000081 2.000142 2.001919 2.000584 2.000007 2.000003 2.001097 2.001600\n#> [9] 2.000207 2.000042 2.000250 2.000295 2.000580 2.002165 2.000453 2.000932\n#> [17] 2.000456 2.000363 2.000418 2.000474 2.009483 2.001156 2.003173 2.241046\n#> [25] 3.990754\n#> \n#> $value\n#> [1] 370.8601\n#> \n#> $counts\n#> function gradient \n#> 18036 NA \n#> \n#> $convergence\n#> [1] 1\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 19\n#> \n#> $barrier.value\n#> [1] -0.003366467\n\n\n结果有改善，目标函数值从 378.4208 减小到 370.8601，但还是没有收敛，可见 Nelder-Mead 方法在这个优化问题上收敛速度比较慢。下面考虑调用基于梯度的 BFGS 优化算法，这得先计算出来目标函数的梯度。\n\n# 输入 n 维向量，输出 n 维向量\ngr <- function(x) {\n n <- length(x)\n c(2 * (x[1] - 2), rep(0, n - 1))\n +8 * c(0, x[-1] - x[-n]^2)\n -16 * c(x[-n], 0) * c(x[-1] - x[-n]^2, 0)\n}\nconstrOptim(\n theta = rep(3, 25), # 初始值\n f = fn, # 目标函数\n grad = gr,\n method = \"BFGS\", \n control = list(maxit = 1000),\n ui = rbind(diag(rep(1, 25)), diag(rep(-1, 25))),\n ci = c(rep(2, 25), rep(-4, 25))\n)\n\n#> $par\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000001\n#> [25] 3.000000\n#> \n#> $value\n#> [1] 373\n#> \n#> $counts\n#> function gradient \n#> 3721 464 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 3\n#> \n#> $barrier.value\n#> [1] -0.003327104\n\n\n从结果来看，虽然已经收敛，但相比于 Nelder-Mead 方法，目标函数值变大了，可见已陷入局部最优解。\n\n20.4.4.3 optim()\n\n下面再使用函数 optim() 提供的 L-BFGS-B 算法求解优化问题。\n\noptim(\n par = rep(3, 25), fn = fn, gr = NULL, method = \"L-BFGS-B\",\n lower = rep(2, 25), upper = rep(4, 25)\n)\n\n#> $par\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109093\n#> [25] 4.000000\n#> \n#> $value\n#> [1] 368.1059\n#> \n#> $counts\n#> function gradient \n#> 6 6 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> [1] \"CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH\"\n\n\n发现结果和函数 nlminb() 的结果差不多了。\n\noptim(\n par = rep(3, 25), fn = fn, gr = gr, method = \"L-BFGS-B\",\n lower = rep(2, 25), upper = rep(4, 25)\n)\n\n#> $par\n#> [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3\n#> \n#> $value\n#> [1] 373\n#> \n#> $counts\n#> function gradient \n#> 2 2 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> [1] \"CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL\"\n\n\n然而，当在函数 optim() 里提供梯度信息的时候，虽然目标函数及梯度的计算次数变少了，求解速度提升了，但是最优解反而变差了，最优解和在函数 constrOptim() 中设置 method = \"BFGS\" 算法基本一致。\n\n20.4.4.4 ROI 包\n下面通过 ROI 包，分别调用求解器 nloptr.lbfgs 和 nloptr.directL ，发现前者同样陷入局部最优解，而后者可以获得与 nlminb() 函数一致的结果。\n\nop <- OP(\n objective = F_objective(fn, n = 25L, G = gr),\n bounds = V_bound(ld = 2, ud = 4, nobj = 25L)\n)\nnlp <- ROI_solve(op, solver = \"nloptr.lbfgs\", start = rep(3, 25))\n# 目标函数值\nnlp$objval\n\n#> [1] 373\n\n# 最优解\nnlp$solution\n\n#> [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3\n\n\n调全局优化算法。\n\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\n# 目标函数值\nnlp$objval\n\n#> [1] 368.1059\n\n# 最优解\nnlp$solution\n\n#> [1] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [9] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000\n#> [17] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.109096\n#> [25] 4.000000\n\n\n\n20.4.5 多元线性约束优化\n对于带线性约束的多元非线性优化问题，Base R 提供函数 constrOptim() 来求解，下面的示例来自其帮助文档，这是一个带线性约束的二次规划问题。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}}\n\\quad & - \\begin{bmatrix}\n0 \\\\\n5 \\\\\n0\n\\end{bmatrix}^{\\top} \\boldsymbol{x} +\\frac{1}{2} \\boldsymbol{x}^{\\top}\\boldsymbol{x} \\\\\n\\text{s.t.} \\quad & \\begin{bmatrix}\n-4 & 2 & 0 \\\\\n-3 & 1 & -2 \\\\\n0 & 0 & 1\n\\end{bmatrix}^{\\top}\\boldsymbol{x} \\geq \\begin{bmatrix}\n-8 \\\\\n2 \\\\\n0\n\\end{bmatrix}\n\\end{aligned}\n\\]\n\nfQP <- function(x) {\n -sum(c(0, 5, 0) * x) + 0.5 * sum(x * x)\n}\nAmat <- matrix(c(-4, -3, 0, 2, 1, 0, 0, -2, 1),\n ncol = 3, nrow = 3, byrow = FALSE\n)\nbvec <- c(-8, 2, 0)\n# 目标函数的梯度\ngQP <- function(x) {\n -c(0, 5, 0) + x\n}\nconstrOptim(\n theta = c(2, -1, -1), \n f = fQP, g = gQP, \n ui = t(Amat), ci = bvec\n)\n\n#> $par\n#> [1] 0.4761908 1.0476188 2.0952376\n#> \n#> $value\n#> [1] -2.380952\n#> \n#> $counts\n#> function gradient \n#> 406 81 \n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> NULL\n#> \n#> $outer.iterations\n#> [1] 3\n#> \n#> $barrier.value\n#> [1] -0.0006243894\n\n\n在上一节，箱式约束可以看作线性约束的一种特殊情况，ROI 包是支持箱式、线性、二次、锥和非线性约束的。因此，下面给出调用 ROI 包求解上述优化问题的代码。\n\nDmat <- diag(rep(1,3))\ndvec <- c(0, 5, 0)\nop <- OP(\n objective = Q_objective(Q = Dmat, L = -dvec),\n constraints = L_constraint(L = t(Amat), dir = rep(\">=\", 3), rhs = bvec),\n maximum = FALSE\n)\nnlp <- ROI_solve(op, solver = \"nloptr.slsqp\", start = c(0, 1, 2))\n# 最优解\nnlp$solution\n\n#> [1] 0.4761905 1.0476190 2.0952381\n\n# 目标函数值\nnlp$objval\n\n#> [1] -2.380952\n\n\n可见输出结果与函数 constrOptim() 是一致的。\n\n代码# quadprog\nlibrary(quadprog)\nsol <- solve.QP(\n Dmat = Dmat, dvec = dvec, Amat = Amat, bvec = bvec\n)\nsol\n\n\n\n20.4.6 多元非线性约束优化\nnloptr 包的非线性优化能力覆盖开源优化软件 Octave 和 Ipopt 。通过插件包 ROI.plugin.nloptr，ROI 包可以调用 nloptr 包内置的所有求解器，常用的求解器见下表。表中从优化器类型（局部还是全局优化器），支持的约束条件类型（箱式还是非线性），是否需要提供目标函数的梯度、黑塞和约束条件的雅可比矩阵信息等方面归纳各个求解器的能力。\n\n常用的非线性优化求解器\n\n求解器\n类型\n约束\n梯度\n黑塞\n雅可比\n\n\n\nnloptr.lbfgs\n局部\n箱式\n需要\n不需要\n不需要\n\n\nnloptr.slsqp\n局部\n非线性\n需要\n不需要\n需要\n\n\nnloptr.auglag\n局部\n非线性\n需要\n不需要\n需要\n\n\nnloptr.directL\n全局\n箱式\n不需要\n不需要\n不需要\n\n\nnloptr.isres\n全局\n非线性\n不需要\n不需要\n不需要\n\n\n\n\n20.4.6.1 非线性等式约束\n下面这个示例来自 Octave 软件的非线性优化帮助文档，Octave 中的函数 sqp() 使用序列二次优化求解器（successive quadratic programming solver）求解非线性优化问题，示例中该优化问题包含多个非线性等式约束。\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} \\quad & \\exp\\big(\\prod_{i=1}^{5} x_i\\big) - \\frac{1}{2}(x_1^3 + x_2^3 + 1)^2 \\\\\n\\text{s.t.} \\quad & \\left\\{\n \\begin{array}{l}\n \\sum_{i=1}^{5}x_i^2 - 10 = 0 \\\\\n x_2 x_3 - 5x_4 x_5 = 0 \\\\\n x_1^3 + x_2^3 + 1 = 0\n \\end{array} \\right.\n\\end{aligned}\n\\]\n目标函数是非线性的，有 5 个变量，约束条件也是非线性的，有 3 个等式约束。先手动计算目标函数的梯度，等式约束的雅可比矩阵。\n\n# 目标函数\nfn <- function(x) {\n exp(prod(x)) - 0.5 * (x[1]^3 + x[2]^3 + 1)^2\n}\n# 目标函数的梯度\ngr <- function(x) {\n c(\n exp(prod(x)) * prod(x[-1]) - 3 * (x[1]^3 + x[2]^3 + 1) * x[1]^2,\n exp(prod(x)) * prod(x[-2]) - 3 * (x[1]^3 + x[2]^3 + 1) * x[2]^2,\n exp(prod(x)) * prod(x[-3]),\n exp(prod(x)) * prod(x[-4]),\n exp(prod(x)) * prod(x[-5])\n )\n}\n# 等式约束\nheq <- function(x) {\n c(\n sum(x^2) - 10,\n x[2] * x[3] - 5 * x[4] * x[5],\n x[1]^3 + x[2]^3 + 1\n )\n}\n# 等式约束的雅可比矩阵\nheq.jac <- function(x) {\n matrix(c(2 * x[1], 2 * x[2], 2 * x[3], 2 * x[4], 2 * x[5],\n 0, x[3], x[2], -5 * x[5], -5 * x[4],\n 3 * x[1]^2, 3 * x[2]^2, 0, 0, 0),\n ncol = 5, byrow = TRUE\n )\n}\n\n在 OP() 函数里定义目标优化的各个成分。\n\n# 定义目标优化\nop <- OP(\n # 5 个决策变量\n objective = F_objective(F = fn, n = 5L, G = gr), \n constraints = F_constraint(\n F = list(heq = heq),\n dir = \"==\",\n rhs = 0,\n # 等式约束的雅可比矩阵\n J = list(heq.jac = heq.jac)\n ),\n bounds = V_bound(ld = -Inf, ud = Inf, nobj = 5L),\n maximum = FALSE # 求最小\n)\nop\n\n#> ROI Optimization Problem:\n#> \n#> Minimize a nonlinear objective function of length 5 with\n#> - 5 continuous objective variables,\n#> \n#> subject to\n#> - 1 constraint of type nonlinear.\n#> - 5 lower and 0 upper non-standard variable bounds.\n\n\n调用 SQP（序列二次优化） 求解器 nloptr.slsqp 。\n\nnlp <- ROI_solve(op,\n solver = \"nloptr.slsqp\",\n start = c(-1.8, 1.7, 1.9, -0.8, -0.8)\n)\n# 最优解\nnlp$solution\n\n#> [1] -1.7171435 1.5957096 1.8272458 -0.7636431 -0.7636431\n\n# 目标函数值\nnlp$objval\n\n#> [1] 0.05394985\n\n\n计算结果和 Octave 的示例一致。\n\n20.4.6.2 多种非线性约束\n\n非线性等式约束\n非线性不等式约束，不等式约束包含等号\n箱式约束\n\n此优化问题来源于 Ipopt 官网的帮助文档，约束条件比较复杂。提供的初始值为 \\(x_0 = (1,5,5,1)\\)，最优解为 \\(x_{\\star} = (1.00000000,4.74299963,3.82114998,1.37940829)\\)。优化问题的具体内容如下：\n\\[\n\\begin{aligned}\n\\min_{\\boldsymbol{x}} & \\quad x_1 x_4 (x_1 + x_2 + x_3) + x_3 \\\\\n\\text{s.t.} & \\quad \\left\\{\n \\begin{array}{l}\n x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40 \\\\\n x_1 x_2 x_3 x_4 \\geq 25 \\\\\n 1 \\leq x_1, x_2, x_3, x_4 \\leq 5\n \\end{array} \\right.\n\\end{aligned}\n\\]\n下面用 ROI 调 nloptr 包求解，看结果是否和例子一致，nloptr 支持箱式约束且支持不等式约束包含等号。\n\n# 一个 4 维的目标函数\nfn <- function(x) {\n x[1] * x[4] * (x[1] + x[2] + x[3]) + x[3]\n}\n# 目标函数的梯度\ngr <- function(x) {\n c(\n x[4] * (2 * x[1] + x[2] + x[3]), x[1] * x[4],\n x[1] * x[4] + 1, x[1] * (x[1] + x[2] + x[3])\n )\n}\n# 等式约束\nheq <- function(x) {\n sum(x^2)\n}\n# 等式约束的雅可比\nheq.jac <- function(x) {\n 2 * c(x[1], x[2], x[3], x[4])\n}\n# 不等式约束\nhin <- function(x) {\n prod(x)\n}\n# 不等式约束的雅可比\nhin.jac <- function(x) {\n c(prod(x[-1]), prod(x[-2]), prod(x[-3]), prod(x[-4]))\n}\n# 定义目标优化\nop <- OP(\n objective = F_objective(F = fn, n = 4L, G = gr), # 4 个决策变量\n constraints = F_constraint(\n F = list(heq = heq, hin = hin),\n dir = c(\"==\", \">=\"),\n rhs = c(40, 25),\n # 等式和不等式约束的雅可比\n J = list(heq.jac = heq.jac, hin.jac = hin.jac)\n ),\n bounds = V_bound(ld = 1, ud = 5, nobj = 4L),\n maximum = FALSE # 求最小\n)\n\n作为对比参考，先计算目标函数的初始值和最优值。\n\n# 目标函数初始值\nfn(c(1, 5, 5, 1))\n\n#> [1] 16\n\n# 目标函数最优值\nfn(c(1.00000000, 4.74299963, 3.82114998, 1.37940829))\n\n#> [1] 17.01402\n\n\n求解一般的非线性约束问题。\n\n求解器 nloptr.mma / nloptr.cobyla 仅支持非线性不等式约束，不支持等式约束。\n函数 nlminb() 只支持等式约束。\n\n因此，下面分别调用 nloptr.auglag、nloptr.slsqp 和 nloptr.isres 来求解上述优化问题。\n\nnlp <- ROI_solve(op, solver = \"nloptr.auglag\", start = c(1, 5, 5, 1))\nnlp$solution\n\n#> [1] 1.000000 4.743174 3.820922 1.379440\n\nnlp$objval\n\n#> [1] 17.01402\n\n\n\nnlp <- ROI_solve(op, solver = \"nloptr.slsqp\", start = c(1, 5, 5, 1))\nnlp$solution\n\n#> [1] 1.000000 4.742996 3.821155 1.379408\n\nnlp$objval\n\n#> [1] 17.01402\n\n\n\nnlp <- ROI_solve(op, solver = \"nloptr.isres\", start = c(1, 5, 5, 1))\nnlp$solution\n\n#> [1] 1.131671 4.721246 3.868353 1.216995\n\nnlp$objval\n\n#> [1] 17.25686\n\n\n可以看出，nloptr 提供的优化能力可以覆盖 Ipopt 求解器，从以上求解的情况来看，推荐使用 nloptr.slsqp 求解器，这也是 Octave 的选择。", "crumbs": [ "优化建模", "20  数值优化" @@ -1399,7 +1399,7 @@ "href": "optimization-problems.html#sec-gaussian-process-regression", "title": "21  优化问题", "section": "\n21.3 高斯过程回归", - "text": "21.3 高斯过程回归\n高斯过程回归模型如下：\n\\[\n\\boldsymbol{y}(x) = D\\boldsymbol{\\beta} + S(x)\n\\]\n其中，\\(\\boldsymbol{\\beta}\\) 是一个 \\(p\\times 1\\) 维列向量，随机过程 \\(S(x)\\) 是均值为零，协方差为 \\(V_{\\boldsymbol{\\theta}}\\) 的平稳高斯过程，协方差矩阵 \\(V_{\\boldsymbol{\\theta}}\\) 的元素如下：\n\\[\n\\mathsf{Cov}\\{S(x_i), S(x_j)\\} = \\sigma^2 \\exp(-\\|x_i - x_j\\| / \\phi)\n\\]\n其中， \\(\\boldsymbol{\\theta} = (\\sigma^2,\\phi)\\) 表示与协方差矩阵相关的参数，随机过程 \\(S(x)\\) 的一个实现服从多元正态分布 \\(\\mathrm{MVN}(\\boldsymbol{0},V_{\\boldsymbol{\\theta}})\\) ，则 \\(\\boldsymbol{y}(x)\\) 也服从多元正态分布 \\(\\mathrm{MVN}(D\\boldsymbol{\\beta},V_{\\boldsymbol{\\theta}})\\) 。参数 \\(\\boldsymbol{\\beta}\\) 的广义最小二乘估计为 \\(\\hat{\\boldsymbol{\\beta}}(\\boldsymbol{\\theta}) = (D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}D)^{-1} D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}\\boldsymbol{y}\\) ，关于参数 \\(\\boldsymbol{\\theta}\\) 的剖面对数似然函数如下：\n\\[\n\\log \\mathcal{L}(\\boldsymbol{\\theta}) = -\\frac{n}{2}\\log (2\\pi) - \\frac{1}{2}\\log (\\det V_{\\boldsymbol{\\theta}}) -\\frac{1}{2}\\boldsymbol{y}^{\\top}V_{\\boldsymbol{\\theta}}^{-1}\\big(I - D(D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}D)^{-1}D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}\\big)\\boldsymbol{y}\n\\]\n下面考虑一个来自 MASS 包真实数据 topo。topo 数据集最初来自 John C. Davis （1973年）所著的书《Statistics and Data Analysis in Geology》。后来， J. J. Warnes 和 B. D. Ripley （1987年）以该数据集为例指出空间高斯过程的协方差函数的似然估计中存在的问题(Warnes 和 Ripley 1987)，并将其作为数据集 topo 放在 MASS 包里。Paulo J. Ribeiro Jr 和 Peter J. Diggle （2001年）将该数据集打包成自定义的 geodata 数据类型，放在 geoR 包里，并在他俩合著的书《Model-based Geostatistics》中多次出现。topo 是空间地形数据集，包含有 52 行 3 列，数据点是 310 平方英尺范围内的海拔高度数据，x 坐标每单位 50 英尺，y 坐标单位同 x 坐标，海拔高度 z 单位是英尺。\n\nlibrary(MASS)\ndata(topo)\nstr(topo)\n\n#> 'data.frame': 52 obs. of 3 variables:\n#> $ x: num 0.3 1.4 2.4 3.6 5.7 1.6 2.9 3.4 3.4 4.8 ...\n#> $ y: num 6.1 6.2 6.1 6.2 6.2 5.2 5.1 5.3 5.7 5.6 ...\n#> $ z: int 870 793 755 690 800 800 730 728 710 780 ...\n\n\n根据 topo 数据集， \\(D = \\boldsymbol{1}\\) 是一个 \\(52 \\times 1\\) 的列向量，\\(\\boldsymbol{\\beta} = \\beta\\) 是一个截距项。设置参数初值 \\((\\sigma,\\phi) = (65,2)\\) 。为了与 Ripley 的论文中的图比较，下面扔掉了对数似然函数中常数项，用 R 语言编码的似然函数如下：\n\nlog_lik <- function(x) {\n n <- nrow(topo)\n D <- t(t(rep(1, n)))\n Sigma <- x[1]^2 * exp(-as.matrix(dist(topo[, c(\"x\", \"y\")])) / x[2])\n inv_Sigma <- solve(Sigma)\n P <- diag(1, n) - D %*% solve(t(D) %*% solve(Sigma, D), t(D)) %*% inv_Sigma\n as.vector(-1 / 2 * log(det(Sigma)) - 1 / 2 * t(topo[, \"z\"]) %*% inv_Sigma %*% P %*% topo[, \"z\"])\n}\nlog_lik(x = c(65, 2))\n\n#> [1] -207.1364\n\n\n关于参数的偏导计算复杂，就不计算梯度了，下面调用 R 软件内置的 nlminb 优化器。发现，对不同的初始值，收敛到不同的位置，目标函数值非常接近。\n\nop <- OP(\n objective = F_objective(log_lik, n = 2L),\n bounds = V_bound(lb = c(55, 5), ub = c(75, 8)),\n maximum = TRUE\n)\nnlp <- ROI_solve(op, solver = \"nlminb\", start = c(65, 2))\nnlp$solution\n\n#> [1] 65 5\n\nnlp$objval\n\n#> [1] -197.4197\n\n\n如果初始值靠近局部极值点，则就近收敛到该极值点，比如初值 \\((65, 7)\\) ， \\((70, 7.5)\\) 。\n\nnlp <- ROI_solve(op, solver = \"nlminb\", start = c(65, 7))\nnlp$solution\n\n#> [1] 65 7\n\nnlp$objval\n\n#> [1] -196.9407\n\nnlp <- ROI_solve(op, solver = \"nlminb\", start = c(70, 7.5))\nnlp$solution\n\n#> [1] 70.0 7.5\n\nnlp$objval\n\n#> [1] -196.8441\n\n\n尝试调用来自 nloptr 包的全局优化求解器 nloptr.directL ，大大小小的坑都跳过去了，结果还是比较满意的。\n\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\nnlp$solution\n\n#> [1] 63.934432 6.121382\n\nnlp$objval\n\n#> [1] -196.8158\n\n\n目标区域网格化，计算格点处的似然函数值，然后绘制似然函数图像。\n\ndat <- expand.grid(\n sigma = seq(from = 55, to = 75, length.out = 41),\n phi = seq(from = 5, to = 8, length.out = 31)\n)\ndat$fn <- apply(dat, 1, log_lik)\n\n似然函数关于参数 \\((\\sigma,\\phi)\\) 的三维曲面见 图 21.3 。\n\n代码wireframe(\n data = dat, fn ~ sigma * phi,\n shade = TRUE, drape = FALSE,\n xlab = expression(sigma), ylab = expression(phi),\n zlab = list(expression(\n italic(log-lik) ~ group(\"(\", list(sigma, phi), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 21.3: 对数似然函数的曲面图\n\n\n\n\n等高线图呈现一道非常长且平滑的山岭 long flat ridge，山岭上布满许多局部极大值，普通的数值优化求解器常常陷入其中，只有全局优化求解器才可能找到全局极大值点。高斯过程回归模型的对数似然函数是非凸的，多模态的。\n\n代码levelplot(fn ~ sigma * phi,\n data = dat, aspect = 1,\n xlim = c(54.5, 75.5), ylim = c(4.9, 8.1),\n xlab = expression(sigma), ylab = expression(phi),\n col.regions = cm.colors, contour = TRUE,\n scales = list(\n x = list(alternating = 1, tck = c(1, 0)),\n y = list(alternating = 1, tck = c(1, 0))\n ),\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = 0, units = \"inches\"),\n right.padding = list(x = 0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -.5, units = \"inches\"),\n top.padding = list(x = -.5, units = \"inches\")\n )\n )\n)\n\n\n\n\n\n\n图 21.4: 对数似然函数的等高线图\n\n\n\n\n上图中没有看到许多局部极小值，与作者论文中的图 1 似乎不符。原因是什么？似然函数中涉及到的矩阵运算不精确，应该设计精度更高的运算方式？lattice 包绘图引擎无法展示更加细微的差异？还有一种解释，上图是对的，算法迭代时，对不同的初值，常常收敛到不同的结果，而这些不同的结果都位于岭上不同位置，对应的对数似然值却又几乎一样。\n作为验证，下面调用 nlme 包的 gls() 函数拟合数据，参数的极大似然估计结果与全局优化求解器的结果比较一致。参数估计结果 \\((\\sigma, \\phi)= (63.93429, 6.121352)\\) ，对数似然函数值为 -244.6006 ，自编的似然函数 log_lik() 在最优解处的值为 -196.8158，再加上之前扔掉的常数项 -52 / 2 * log(2 * pi) ，就是 -244.6006 ，丝毫不差。\n\nlibrary(nlme)\nfit_topo_ml <- gls(z ~ 1,\n data = topo, method = \"ML\",\n correlation = corExp(value = 65, form = ~ x + y)\n)\nsummary(fit_topo_ml)\n\n#> Generalized least squares fit by maximum likelihood\n#> Model: z ~ 1 \n#> Data: topo \n#> AIC BIC logLik\n#> 495.2012 501.055 -244.6006\n#> \n#> Correlation Structure: Exponential spatial correlation\n#> Formula: ~x + y \n#> Parameter estimate(s):\n#> range \n#> 6.121352 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 863.708 45.49859 18.98318 0\n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -2.7169766 -1.1919732 -0.5272282 0.1453374 1.5061096 \n#> \n#> Residual standard error: 63.93429 \n#> Degrees of freedom: 52 total; 51 residual\n\n\n如果使用限制极大似然估计，会发现参数估计结果与之相距甚远，而对数似然函数值相差无几。参数估计结果 \\((\\sigma,\\phi) = (128.8275, 25.47324)\\) 。\n\nfit_topo_reml <- gls(z ~ 1,\n data = topo, method = \"REML\",\n correlation = corExp(value = 65, form = ~ x + y)\n)\nsummary(fit_topo_reml)\n\n#> Generalized least squares fit by REML\n#> Model: z ~ 1 \n#> Data: topo \n#> AIC BIC logLik\n#> 485.1558 490.9513 -239.5779\n#> \n#> Correlation Structure: Exponential spatial correlation\n#> Formula: ~x + y \n#> Parameter estimate(s):\n#> range \n#> 25.47324 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 877.8956 116.7163 7.521619 0\n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -1.45850506 -0.70167923 -0.37178079 -0.03800119 0.63732032 \n#> \n#> Residual standard error: 128.8275 \n#> Degrees of freedom: 52 total; 51 residual", + "text": "21.3 高斯过程回归\n高斯过程回归模型如下：\n\\[\n\\boldsymbol{y}(x) = D\\boldsymbol{\\beta} + S(x)\n\\]\n其中，\\(\\boldsymbol{\\beta}\\) 是一个 \\(p\\times 1\\) 维列向量，随机过程 \\(S(x)\\) 是均值为零，协方差为 \\(V_{\\boldsymbol{\\theta}}\\) 的平稳高斯过程，协方差矩阵 \\(V_{\\boldsymbol{\\theta}}\\) 的元素如下：\n\\[\n\\mathsf{Cov}\\{S(x_i), S(x_j)\\} = \\sigma^2 \\exp(-\\|x_i - x_j\\| / \\phi)\n\\]\n其中， \\(\\boldsymbol{\\theta} = (\\sigma^2,\\phi)\\) 表示与协方差矩阵相关的参数，随机过程 \\(S(x)\\) 的一个实现服从多元正态分布 \\(\\mathrm{MVN}(\\boldsymbol{0},V_{\\boldsymbol{\\theta}})\\) ，则 \\(\\boldsymbol{y}(x)\\) 也服从多元正态分布 \\(\\mathrm{MVN}(D\\boldsymbol{\\beta},V_{\\boldsymbol{\\theta}})\\) 。参数 \\(\\boldsymbol{\\beta}\\) 的广义最小二乘估计为 \\(\\hat{\\boldsymbol{\\beta}}(\\boldsymbol{\\theta}) = (D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}D)^{-1} D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}\\boldsymbol{y}\\) ，关于参数 \\(\\boldsymbol{\\theta}\\) 的剖面对数似然函数如下：\n\\[\n\\log \\mathcal{L}(\\boldsymbol{\\theta}) = -\\frac{n}{2}\\log (2\\pi) - \\frac{1}{2}\\log (\\det V_{\\boldsymbol{\\theta}}) -\\frac{1}{2}\\boldsymbol{y}^{\\top}V_{\\boldsymbol{\\theta}}^{-1}\\big(I - D(D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}D)^{-1}D^{\\top}V_{\\boldsymbol{\\theta}}^{-1}\\big)\\boldsymbol{y}\n\\]\n下面考虑一个来自 MASS 包真实数据 topo。topo 数据集最初来自 John C. Davis （1973年）所著的书《Statistics and Data Analysis in Geology》。后来， J. J. Warnes 和 B. D. Ripley （1987年）以该数据集为例指出空间高斯过程的协方差函数的似然估计中存在的问题(Warnes 和 Ripley 1987)，并将其作为数据集 topo 放在 MASS 包里。Paulo J. Ribeiro Jr 和 Peter J. Diggle （2001年）将该数据集打包成自定义的 geodata 数据类型，放在 geoR 包里，并在他俩合著的书《Model-based Geostatistics》中多次出现。topo 是空间地形数据集，包含有 52 行 3 列，数据点是 310 平方英尺范围内的海拔高度数据，x 坐标每单位 50 英尺，y 坐标单位同 x 坐标，海拔高度 z 单位是英尺。\n\nlibrary(MASS)\ndata(topo)\nstr(topo)\n\n#> 'data.frame': 52 obs. of 3 variables:\n#> $ x: num 0.3 1.4 2.4 3.6 5.7 1.6 2.9 3.4 3.4 4.8 ...\n#> $ y: num 6.1 6.2 6.1 6.2 6.2 5.2 5.1 5.3 5.7 5.6 ...\n#> $ z: int 870 793 755 690 800 800 730 728 710 780 ...\n\n\n根据 topo 数据集， \\(D = \\boldsymbol{1}\\) 是一个 \\(52 \\times 1\\) 的列向量，\\(\\boldsymbol{\\beta} = \\beta\\) 是一个截距项。设置参数初值 \\((\\sigma,\\phi) = (65,2)\\) 。为了与 Ripley 的论文中的图比较，下面扔掉了对数似然函数中常数项，用 R 语言编码的似然函数如下：\n\nlog_lik <- function(x) {\n n <- nrow(topo)\n D <- t(t(rep(1, n)))\n Sigma <- x[1]^2 * exp(-as.matrix(dist(topo[, c(\"x\", \"y\")])) / x[2])\n inv_Sigma <- solve(Sigma)\n P <- diag(1, n) - D %*% solve(t(D) %*% solve(Sigma, D), t(D)) %*% inv_Sigma\n as.vector(-1 / 2 * log(det(Sigma)) - 1 / 2 * t(topo[, \"z\"]) %*% inv_Sigma %*% P %*% topo[, \"z\"])\n}\nlog_lik(x = c(65, 2))\n\n#> [1] -207.1364\n\n\n关于参数的偏导计算复杂，就不计算梯度了，下面调用 R 软件内置的 nlminb 优化器。发现，对不同的初始值，收敛到不同的位置，目标函数值非常接近。\n\nop <- OP(\n objective = F_objective(log_lik, n = 2L),\n bounds = V_bound(lb = c(55, 5), ub = c(75, 8)),\n maximum = TRUE\n)\nnlp <- ROI_solve(op, solver = \"nlminb\", start = c(65, 2))\nnlp$solution\n\n#> [1] 65 5\n\nnlp$objval\n\n#> [1] -197.4197\n\n\n如果初始值靠近局部极值点，则就近收敛到该极值点，比如初值 \\((65, 7)\\) ， \\((70, 7.5)\\) 。\n\nnlp <- ROI_solve(op, solver = \"nlminb\", start = c(65, 7))\nnlp$solution\n\n#> [1] 65 7\n\nnlp$objval\n\n#> [1] -196.9407\n\nnlp <- ROI_solve(op, solver = \"nlminb\", start = c(70, 7.5))\nnlp$solution\n\n#> [1] 70.0 7.5\n\nnlp$objval\n\n#> [1] -196.8441\n\n\n尝试调用来自 nloptr 包的全局优化求解器 nloptr.directL ，大大小小的坑都跳过去了，结果还是比较满意的。\n\nnlp <- ROI_solve(op, solver = \"nloptr.directL\")\nnlp$solution\n\n#> [1] 63.934193 6.121331\n\nnlp$objval\n\n#> [1] -196.8158\n\n\n目标区域网格化，计算格点处的似然函数值，然后绘制似然函数图像。\n\ndat <- expand.grid(\n sigma = seq(from = 55, to = 75, length.out = 41),\n phi = seq(from = 5, to = 8, length.out = 31)\n)\ndat$fn <- apply(dat, 1, log_lik)\n\n似然函数关于参数 \\((\\sigma,\\phi)\\) 的三维曲面见 图 21.3 。\n\n代码wireframe(\n data = dat, fn ~ sigma * phi,\n shade = TRUE, drape = FALSE,\n xlab = expression(sigma), ylab = expression(phi),\n zlab = list(expression(\n italic(log-lik) ~ group(\"(\", list(sigma, phi), \")\")\n ), rot = 90),\n scales = list(arrows = FALSE, col = \"black\"),\n shade.colors.palette = custom_palette,\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = -0.5, units = \"inches\"),\n right.padding = list(x = -1.0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -1.5, units = \"inches\"),\n top.padding = list(x = -1.5, units = \"inches\")\n )\n ),\n par.settings = list(axis.line = list(col = \"transparent\")),\n screen = list(z = 30, x = -65, y = 0)\n)\n\n\n\n\n\n\n图 21.3: 对数似然函数的曲面图\n\n\n\n\n等高线图呈现一道非常长且平滑的山岭 long flat ridge，山岭上布满许多局部极大值，普通的数值优化求解器常常陷入其中，只有全局优化求解器才可能找到全局极大值点。高斯过程回归模型的对数似然函数是非凸的，多模态的。\n\n代码levelplot(fn ~ sigma * phi,\n data = dat, aspect = 1,\n xlim = c(54.5, 75.5), ylim = c(4.9, 8.1),\n xlab = expression(sigma), ylab = expression(phi),\n col.regions = cm.colors, contour = TRUE,\n scales = list(\n x = list(alternating = 1, tck = c(1, 0)),\n y = list(alternating = 1, tck = c(1, 0))\n ),\n # 减少三维图形的边空\n lattice.options = list(\n layout.widths = list(\n left.padding = list(x = 0, units = \"inches\"),\n right.padding = list(x = 0, units = \"inches\")\n ),\n layout.heights = list(\n bottom.padding = list(x = -.5, units = \"inches\"),\n top.padding = list(x = -.5, units = \"inches\")\n )\n )\n)\n\n\n\n\n\n\n图 21.4: 对数似然函数的等高线图\n\n\n\n\n上图中没有看到许多局部极小值，与作者论文中的图 1 似乎不符。原因是什么？似然函数中涉及到的矩阵运算不精确，应该设计精度更高的运算方式？lattice 包绘图引擎无法展示更加细微的差异？还有一种解释，上图是对的，算法迭代时，对不同的初值，常常收敛到不同的结果，而这些不同的结果都位于岭上不同位置，对应的对数似然值却又几乎一样。\n作为验证，下面调用 nlme 包的 gls() 函数拟合数据，参数的极大似然估计结果与全局优化求解器的结果比较一致。参数估计结果 \\((\\sigma, \\phi)= (63.93429, 6.121352)\\) ，对数似然函数值为 -244.6006 ，自编的似然函数 log_lik() 在最优解处的值为 -196.8158，再加上之前扔掉的常数项 -52 / 2 * log(2 * pi) ，就是 -244.6006 ，丝毫不差。\n\nlibrary(nlme)\nfit_topo_ml <- gls(z ~ 1,\n data = topo, method = \"ML\",\n correlation = corExp(value = 65, form = ~ x + y)\n)\nsummary(fit_topo_ml)\n\n#> Generalized least squares fit by maximum likelihood\n#> Model: z ~ 1 \n#> Data: topo \n#> AIC BIC logLik\n#> 495.2012 501.055 -244.6006\n#> \n#> Correlation Structure: Exponential spatial correlation\n#> Formula: ~x + y \n#> Parameter estimate(s):\n#> range \n#> 6.121352 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 863.708 45.49859 18.98318 0\n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -2.7169766 -1.1919732 -0.5272282 0.1453374 1.5061096 \n#> \n#> Residual standard error: 63.93429 \n#> Degrees of freedom: 52 total; 51 residual\n\n\n如果使用限制极大似然估计，会发现参数估计结果与之相距甚远，而对数似然函数值相差无几。参数估计结果 \\((\\sigma,\\phi) = (128.8275, 25.47324)\\) 。\n\nfit_topo_reml <- gls(z ~ 1,\n data = topo, method = \"REML\",\n correlation = corExp(value = 65, form = ~ x + y)\n)\nsummary(fit_topo_reml)\n\n#> Generalized least squares fit by REML\n#> Model: z ~ 1 \n#> Data: topo \n#> AIC BIC logLik\n#> 485.1558 490.9513 -239.5779\n#> \n#> Correlation Structure: Exponential spatial correlation\n#> Formula: ~x + y \n#> Parameter estimate(s):\n#> range \n#> 25.47324 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 877.8956 116.7163 7.521619 0\n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -1.45850506 -0.70167923 -0.37178079 -0.03800119 0.63732032 \n#> \n#> Residual standard error: 128.8275 \n#> Degrees of freedom: 52 total; 51 residual", "crumbs": [ "优化建模", "21  优化问题" @@ -1410,7 +1410,7 @@ "href": "optimization-problems.html#sec-poisson-mixture-distributions", "title": "21  优化问题", "section": "\n21.4 泊松混合分布", - "text": "21.4 泊松混合分布\n有限混合模型（Finite Mixtures of Distributions）的应用非常广泛，本节参考 BB 包 (Varadhan 和 Gilbert 2009) 的帮助手册，以泊松混合分布为例，介绍其参数的极大似然估计。更多详细的理论和算法介绍从略，感兴趣的读者可以查阅相关文献 (Hasselblad 1969)。BB 包比内置函数 optim() 功能更强，可以求解大规模非线性方程组，也可以求解带简单约束的非线性优化问题，还可以从多个初始值出发寻找全局最优解。\n两个泊松分布以一定比例 \\(p\\) 混合，以概率 \\(p\\) 服从泊松分布 \\(\\mathrm{Poisson}(\\lambda_1)\\) ，而以概率 \\(1-p\\) 服从泊松分布 \\(\\mathrm{Poisson}(\\lambda_1)\\) 。\n\\[\np\\times \\mathrm{Poisson}(\\lambda_1) + (1 - p)\\times \\mathrm{Poisson}(\\lambda_2)\n\\]\n泊松混合分布的概率密度函数 \\(f(x;p,\\lambda_1,\\lambda_2)\\) 如下：\n\\[\nf(x;p,\\lambda_1,\\lambda_2) = p \\times \\frac{\\lambda_1^x \\exp(-\\lambda_1)}{x!} + (1 - p) \\times \\frac{\\lambda_2^x \\exp(-\\lambda_2)}{x!}\n\\]\n随机变量 \\(X\\) 服从参数为 \\(p\\) 的伯努利分布 \\(X \\sim \\mathrm{Bernoulli}(1, p)\\) ，随机变量 \\(Y\\) 服从泊松混合分布，在伯努利分布的基础上，泊松混合分布也可作如下定义：\n\\[\n\\begin{array}{l}\nY \\sim \\left\\{\n\\begin{array}{l}\n\\mathrm{Poisson}(\\lambda_1), \\quad \\text{当} ~ X = 1 ~ \\text{时},\\\\\n\\mathrm{Poisson}(\\lambda_2), \\quad \\text{当} ~ X = 0 ~ \\text{时}.\n\\end{array} \\right.\n\\end{array}\n\\]\n对数似然函数如下：\n\\[\n\\ell(p,\\lambda_1,\\lambda_2) = \\sum_{i=0}^{n}y_i \\log\\big(p\\times \\exp(-\\lambda_1) \\times\\frac{\\lambda_1^{x_i}}{x_i!} + (1 - p)\\times \\exp(-\\lambda_2) \\times\\frac{\\lambda_2 ^{x_i}}{x_i!} \\big)\n\\]\n下 表格 21.1 数据来自 1947 年 Walter Schilling 发表在 JASA 的一篇文章 (Schilling 1947)。连续三年搜集伦敦《泰晤士报》刊登的死亡告示，每天的告示发布 80 岁及以上女性死亡人数。经过汇总统计，发现，在三年里，没有人死亡的告示出现 162 次，死亡 1 人的告示出现 267 次。\n\n\n表格 21.1: 死亡人数的统计\n\n\n\n死亡人数\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n\n\n发生频次\n162\n267\n271\n185\n111\n61\n27\n8\n3\n1\n\n\n\n\n\n考虑到夏季和冬季对老人死亡率的影响是不同的，因此，引入泊松混合分布来对数据建模。\n\n# 对数似然函数\n# p 是一个长度为 3 的向量\n# y 是观测数据向量\npoissmix_loglik <- function(p, y) {\n i <- 0:(length(y) - 1)\n loglik <- y * log(p[1] * exp(-p[2]) * p[2]^i / exp(lgamma(i + 1)) +\n (1 - p[1]) * exp(-p[3]) * p[3]^i / exp(lgamma(i + 1)))\n sum(loglik)\n}\n# lgamma(i + 1) 表示整数 i 的阶乘的对数\n# 参数的下限\nlo <- c(0, 0, 0)\n# 参数的上限\nhi <- c(1, Inf, Inf)\n# 随机生成一组参数初始值\np0 <- runif(3, c(0.2, 1, 1), c(0.8, 5, 8)) \n# 汇总统计出来的死亡人数的频次分布\ny <- c(162, 267, 271, 185, 111, 61, 27, 8, 3, 1)\n\n调用 BB 包的函数 BBoptim() 求解多元非线性箱式约束优化问题。\n\nlibrary(BB)\n# 参数估计\nans <- BBoptim(\n par = p0, fn = poissmix_loglik, y = y,\n lower = lo, upper = hi, \n control = list(maximize = TRUE)\n)\n\n#> iter: 0 f-value: -2968.501 pgrad: 7.572203 \n#> iter: 10 f-value: -1992.18 pgrad: 3.503607 \n#> iter: 20 f-value: -1991.277 pgrad: 2.365923 \n#> iter: 30 f-value: -1990.325 pgrad: 1.405494 \n#> iter: 40 f-value: -1989.979 pgrad: 1.709452 \n#> iter: 50 f-value: -1989.946 pgrad: 0.02604793 \n#> Successful convergence.\n\nans\n\n#> $par\n#> [1] 0.3598808 1.2560870 2.6633987\n#> \n#> $value\n#> [1] -1989.946\n#> \n#> $gradient\n#> [1] 1.818989e-05\n#> \n#> $fn.reduction\n#> [1] -978.5554\n#> \n#> $iter\n#> [1] 59\n#> \n#> $feval\n#> [1] 61\n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> [1] \"Successful convergence\"\n#> \n#> $cpar\n#> method M \n#> 2 50\n\n\nnumDeriv::hessian 计算极大似然点的黑塞矩阵，然后计算参数估计的标准差。\n\n# 黑塞矩阵\nhess <- numDeriv::hessian(x = ans$par, func = poissmix_loglik, y = y)\nhess\n\n#> [,1] [,2] [,3]\n#> [1,] -907.1131 270.22803 341.25563\n#> [2,] 270.2280 -113.47865 -61.68173\n#> [3,] 341.2556 -61.68173 -192.78329\n\n# 标准差\nse <- sqrt(diag(solve(-hess)))\nse\n\n#> [1] 0.1946826 0.3500306 0.2504754\n\n\nmultiStart 从不同初始值出发寻找全局最大值，先找一系列局部极大值，通过比较获得全局最大值。\n\n# 随机生成 10 组初始值\np0 <- matrix(runif(30, c(0.2, 1, 1), c(0.8, 8, 8)), \n nrow = 10, ncol = 3, byrow = TRUE)\nans <- multiStart(\n par = p0, fn = poissmix_loglik, action = \"optimize\",\n y = y, lower = lo, upper = hi, quiet = TRUE,\n control = list(maximize = TRUE, trace = FALSE)\n)\n# 筛选出迭代收敛的解\npmat <- round(cbind(ans$fvalue[ans$conv], ans$par[ans$conv, ]), 4)\ndimnames(pmat) <- list(NULL, c(\"fvalue\", \"parameter 1\", \n \"parameter 2\", \"parameter 3\"))\n# 去掉结果一样的重复解\npmat[!duplicated(pmat), ]\n\n#> fvalue parameter 1 parameter 2 parameter 3\n#> [1,] -1989.946 0.6401 2.6634 1.2561\n#> [2,] -1989.946 0.3599 1.2561 2.6634\n#> [3,] -1999.873 0.3187 1.9075 2.2792", + "text": "21.4 泊松混合分布\n有限混合模型（Finite Mixtures of Distributions）的应用非常广泛，本节参考 BB 包 (Varadhan 和 Gilbert 2009) 的帮助手册，以泊松混合分布为例，介绍其参数的极大似然估计。更多详细的理论和算法介绍从略，感兴趣的读者可以查阅相关文献 (Hasselblad 1969)。BB 包比内置函数 optim() 功能更强，可以求解大规模非线性方程组，也可以求解带简单约束的非线性优化问题，还可以从多个初始值出发寻找全局最优解。\n两个泊松分布以一定比例 \\(p\\) 混合，以概率 \\(p\\) 服从泊松分布 \\(\\mathrm{Poisson}(\\lambda_1)\\) ，而以概率 \\(1-p\\) 服从泊松分布 \\(\\mathrm{Poisson}(\\lambda_1)\\) 。\n\\[\np\\times \\mathrm{Poisson}(\\lambda_1) + (1 - p)\\times \\mathrm{Poisson}(\\lambda_2)\n\\]\n泊松混合分布的概率密度函数 \\(f(x;p,\\lambda_1,\\lambda_2)\\) 如下：\n\\[\nf(x;p,\\lambda_1,\\lambda_2) = p \\times \\frac{\\lambda_1^x \\exp(-\\lambda_1)}{x!} + (1 - p) \\times \\frac{\\lambda_2^x \\exp(-\\lambda_2)}{x!}\n\\]\n随机变量 \\(X\\) 服从参数为 \\(p\\) 的伯努利分布 \\(X \\sim \\mathrm{Bernoulli}(1, p)\\) ，随机变量 \\(Y\\) 服从泊松混合分布，在伯努利分布的基础上，泊松混合分布也可作如下定义：\n\\[\n\\begin{array}{l}\nY \\sim \\left\\{\n\\begin{array}{l}\n\\mathrm{Poisson}(\\lambda_1), \\quad \\text{当} ~ X = 1 ~ \\text{时},\\\\\n\\mathrm{Poisson}(\\lambda_2), \\quad \\text{当} ~ X = 0 ~ \\text{时}.\n\\end{array} \\right.\n\\end{array}\n\\]\n对数似然函数如下：\n\\[\n\\ell(p,\\lambda_1,\\lambda_2) = \\sum_{i=0}^{n}y_i \\log\\big(p\\times \\exp(-\\lambda_1) \\times\\frac{\\lambda_1^{x_i}}{x_i!} + (1 - p)\\times \\exp(-\\lambda_2) \\times\\frac{\\lambda_2 ^{x_i}}{x_i!} \\big)\n\\]\n下 表格 21.1 数据来自 1947 年 Walter Schilling 发表在 JASA 的一篇文章 (Schilling 1947)。连续三年搜集伦敦《泰晤士报》刊登的死亡告示，每天的告示发布 80 岁及以上女性死亡人数。经过汇总统计，发现，在三年里，没有人死亡的告示出现 162 次，死亡 1 人的告示出现 267 次。\n\n\n表格 21.1: 死亡人数的统计\n\n\n\n死亡人数\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n\n\n发生频次\n162\n267\n271\n185\n111\n61\n27\n8\n3\n1\n\n\n\n\n\n考虑到夏季和冬季对老人死亡率的影响是不同的，因此，引入泊松混合分布来对数据建模。\n\n# 对数似然函数\n# p 是一个长度为 3 的向量\n# y 是观测数据向量\npoissmix_loglik <- function(p, y) {\n i <- 0:(length(y) - 1)\n loglik <- y * log(p[1] * exp(-p[2]) * p[2]^i / exp(lgamma(i + 1)) +\n (1 - p[1]) * exp(-p[3]) * p[3]^i / exp(lgamma(i + 1)))\n sum(loglik)\n}\n# lgamma(i + 1) 表示整数 i 的阶乘的对数\n# 参数的下限\nlo <- c(0, 0, 0)\n# 参数的上限\nhi <- c(1, Inf, Inf)\n# 随机生成一组参数初始值\np0 <- runif(3, c(0.2, 1, 1), c(0.8, 5, 8)) \n# 汇总统计出来的死亡人数的频次分布\ny <- c(162, 267, 271, 185, 111, 61, 27, 8, 3, 1)\n\n调用 BB 包的函数 BBoptim() 求解多元非线性箱式约束优化问题。\n\nlibrary(BB)\n# 参数估计\nans <- BBoptim(\n par = p0, fn = poissmix_loglik, y = y,\n lower = lo, upper = hi, \n control = list(maximize = TRUE)\n)\n\n#> iter: 0 f-value: -2301.554 pgrad: 4.703557 \n#> iter: 10 f-value: -1990.644 pgrad: 2.361774 \n#> iter: 20 f-value: -1989.947 pgrad: 0.2183447 \n#> iter: 30 f-value: -1989.946 pgrad: 0.005798029 \n#> Successful convergence.\n\nans\n\n#> $par\n#> [1] 0.6401137 2.6634054 1.2560965\n#> \n#> $value\n#> [1] -1989.946\n#> \n#> $gradient\n#> [1] 0.0002660272\n#> \n#> $fn.reduction\n#> [1] -311.6079\n#> \n#> $iter\n#> [1] 39\n#> \n#> $feval\n#> [1] 41\n#> \n#> $convergence\n#> [1] 0\n#> \n#> $message\n#> [1] \"Successful convergence\"\n#> \n#> $cpar\n#> method M \n#> 2 50\n\n\nnumDeriv::hessian 计算极大似然点的黑塞矩阵，然后计算参数估计的标准差。\n\n# 黑塞矩阵\nhess <- numDeriv::hessian(x = ans$par, func = poissmix_loglik, y = y)\nhess\n\n#> [,1] [,2] [,3]\n#> [1,] -907.1064 -341.25220 -270.22960\n#> [2,] -341.2522 -192.78034 -61.68217\n#> [3,] -270.2296 -61.68217 -113.48058\n\n# 标准差\nse <- sqrt(diag(solve(-hess)))\nse\n\n#> [1] 0.1946849 0.2504791 0.3500301\n\n\nmultiStart 从不同初始值出发寻找全局最大值，先找一系列局部极大值，通过比较获得全局最大值。\n\n# 随机生成 10 组初始值\np0 <- matrix(runif(30, c(0.2, 1, 1), c(0.8, 8, 8)), \n nrow = 10, ncol = 3, byrow = TRUE)\nans <- multiStart(\n par = p0, fn = poissmix_loglik, action = \"optimize\",\n y = y, lower = lo, upper = hi, quiet = TRUE,\n control = list(maximize = TRUE, trace = FALSE)\n)\n# 筛选出迭代收敛的解\npmat <- round(cbind(ans$fvalue[ans$conv], ans$par[ans$conv, ]), 4)\ndimnames(pmat) <- list(NULL, c(\"fvalue\", \"parameter 1\", \n \"parameter 2\", \"parameter 3\"))\n# 去掉结果一样的重复解\npmat[!duplicated(pmat), ]\n\n#> fvalue parameter 1 parameter 2 parameter 3\n#> [1,] -1989.946 0.3599 1.2561 2.6634\n#> [2,] -1995.912 0.3990 2.5910 1.8588\n#> [3,] -1989.946 0.6401 2.6634 1.2561", "crumbs": [ "优化建模", "21  优化问题" diff --git a/wrangling-objects.html b/wrangling-objects.html index a0739d44..2b28d6f0 100644 --- a/wrangling-objects.html +++ b/wrangling-objects.html @@ -506,7 +506,7 @@ 

# 数据结构
 str(x)
-
 Time-Series [1:100] from 2017 to 2017: 1.31088 -0.00582 0.48111 -1.56845 -0.03549 ...
+
 Time-Series [1:100] from 2017 to 2017: 0.777 -1.4823 0.1598 0.4565 -0.0718 ...

函数 start() 和 end() 查看开始和结束的时间点。