{"id":2614,"date":"2020-08-15T11:00:00","date_gmt":"2020-08-15T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2614"},"modified":"2020-07-31T08:10:21","modified_gmt":"2020-07-30T23:10:21","slug":"multivariate-normal-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/multivariate-normal-distribution","title":{"rendered":"\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03"},"content":{"rendered":"\n\n\n\n\n<p>\u3000\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3092\u30d9\u30af\u30c8\u30eb\u306e\u5f62\u3067\u307e\u3068\u3081\u305f\u3082\u306e\u306f\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\uff08multivariate normal distribution\uff09\u306b\u5f93\u3044\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u306f\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X_{i}\\sim N(\\mu_{i},\\sigma_{i}^{2})\\)\uff08\\(i=1,\\cdots,p\\)\uff09\u3092\u3001\u3072\u3068\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(X={}^{T}(X_{1},\\cdots,X_{p})\\)\u306b\u307e\u3068\u3081\u305f\u3082\u306e\u304c\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(X_{i},X_{j}\\)\uff08\\(i\\neq j\\)\uff09\u306f\u4e92\u3044\u306b\u72ec\u7acb\u3067\u306a\u304f\u3066\u3082\u69cb\u3044\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001\\(X_{i}\\)\u3068\\(X_{j}\\)\uff08\\(i\\neq j\\)\uff09\u306e\u5171\u5206\u6563\u3092\\(\\sigma_{ij}\\)\u3068\u304a\u304d\u307e\u3059\u3002\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306f\u671f\u5f85\u5024\\(\\mu_{i}\\)\u3001\u5206\u6563\\(\\sigma_{i}^{2}\\)\u3001\u5171\u5206\u6563\\(\\sigma_{ij}\\)\uff08\\(i\\neq j\\)\uff09\u3092\u307e\u3068\u3081\u305f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mu = \\left(\\begin{array}{c}<br>\\mu_{1} \\\\<br>\\vdots \\\\<br>\\mu_{p}<br>\\end{array}\\right),\\ \\ \\ \\ \\Sigma=\\left(\\begin{array}{cccc}<br>\\sigma_{1}^{2} &amp; \\sigma_{12} &amp; \\cdots &amp; \\sigma_{1p} \\\\<br>\\sigma_{21} &amp; \\sigma_{2}^{2} &amp; \\cdots &amp; \\sigma_{2p} \\\\<br>\\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\<br>\\sigma_{p1} &amp; \\sigma_{p2} &amp; \\cdots &amp; \\sigma_{p}^{2}<br>\\end{array}\\right)<br>\\end{align}<\/div><br>\u3092\u7528\u3044\u3066\u8868\u3055\u308c\u307e\u3059\u3002\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\\(p\\)\u6b21\u5909\u91cf\u78ba\u7387\u5909\u6570\u306f\u8a18\u53f7\u3067\\(X\\sim N_{p}(\\mu,\\Sigma)\\)\u3068\u8868\u3055\u308c\u307e\u3059\u3002\u7279\u306b\\(p=1\\)\u306e\u3068\u304d\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u3068\u540c\u3058\u3082\u306e\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/normal-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03bc\uff1d\uff10-100x100.png\" alt=\"\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\u3001normal distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6b63\u898f\u5206\u5e03\u306f\u6570\u5b66\u7684\u306b\u3082\u4fbf\u5229\u306a\u6027\u8cea\u3092\u305f\u304f\u3055\u3093\u6301\u3063\u3066\u3044\u308b\u305f\u3081\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">3<\/span> \u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u78ba\u7387\u5909\u6570\u306e\u72ec\u7acb\u6027<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">4<\/span> \u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7dda\u5f62\u95a2\u6570<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">5<\/span> \u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_1\">6<\/span> \u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">7<\/span> \u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u78ba\u7387\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/span><\/h2>\n\n\n\n<p>\u203b\u3000\u8868\u306f\u6a2a\u306b\u30b9\u30af\u30ed\u30fc\u30eb\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n<figure class=\"wp-block-table\">\n<table style=\"width: 100%; height: 343px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\">\u30d1\u30e9\u30e1\u30fc\u30bf<\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\mu,\\ \\ \\ 0 &lt; \\Sigma\\)<br \/>\u305f\u3060\u3057\u3001\\(\\Sigma &gt;0\\)\u306f\\(\\Sigma\\)\u304c\u6b63\u5b9a\u5024\u884c\u5217\u3092\u8868\u3057\u307e\u3059\u3002<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 56px;\">\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td>\n<td style=\"width: 82.2414%; height: 56px;\">\\(-\\infty&lt; x_{i} &lt;\\infty \\),\uff08\\(i=1,\\cdots,p\\)\uff09<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 35px;\">\\(\\displaystyle \\frac{1}{\\sqrt{(2\\pi)^{p}|\\Sigma|}}\\exp\\left[ -\\frac{1}{2}{}^{T}\\!(x-\\mu)\\Sigma^{-1}(x-\\mu) \\right] \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle \\exp\\left[ {}^{T}\\!\\mu t+\\frac{1}{2}{}^{T}\\!t\\Sigma t \\right] \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle \\exp\\left[ i{}^{T}\\!\\mu t-\\frac{1}{2}{}^{T}\\!t\\Sigma t \\right] \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle {}^{T}\\!\\mu t+\\frac{1}{2}{}^{T}\\!t\\Sigma t\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\\(r\\)\u6b21\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\kappa_{r}=0\uff08r&gt;2\uff09\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\( \\displaystyle \\mu \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563\u5171\u5206\u6563\u884c\u5217<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\( \\displaystyle \\Sigma \\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n<h2><span id=\"i-2\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u6b63\u898f\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(p\\)\u6b21\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{p}(\\mu,\\Sigma)\\)\u306b\u3064\u3044\u3066\u3001\\(p=1\\)\u3068\u3057\u305f\u3068\u304d\u306e\u5206\u5e03\\(X\\sim N(\\mu,\\sigma^{2})\\)\u306f\u6b63\u898f\u5206\u5e03\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\\(p=1\\)\u3068\u3059\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>f(x) &amp;= \\frac{1}{\\sqrt{2\\pi\\sigma^{2}}}\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2} \\right]<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u6b63\u898f\u5206\u5e03\u3068\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/normal-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03bc\uff1d\uff10-100x100.png\" alt=\"\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\u3001normal distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6b63\u898f\u5206\u5e03\u306f\u6570\u5b66\u7684\u306b\u3082\u4fbf\u5229\u306a\u6027\u8cea\u3092\u305f\u304f\u3055\u3093\u6301\u3063\u3066\u3044\u308b\u305f\u3081\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-3\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u78ba\u7387\u5909\u6570\u306e\u72ec\u7acb\u6027<\/span><\/h2>\n\n\n\n<p>\u3000\\(p\\)\u6b21\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{p}(\\mu,\\Sigma)\\)\u306b\u3064\u3044\u3066\u3001\u3082\u3057\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u306b\u3042\u305f\u308b\\(\\Sigma\\)\u304c\u5bfe\u89d2\u884c\u5217<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\Sigma &amp;= \\left(\\begin{array}{cccc}<br>\\sigma_{1}^{2} &amp; 0 &amp; \\cdots &amp; 0 \\\\<br>0 &amp; \\sigma_{2}^{2} &amp; \\cdots &amp; 0 \\\\<br>\\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\<br>0 &amp; 0 &amp; \\cdots &amp; \\sigma_{p}^{2}<br>\\end{array}\\right)<br>\\end{align}<\/div><br>\u3068\u306a\u3063\u3066\u3044\u308b\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X_{i}\\)\u3068\\(X_{j}\\)\uff08\\(i\\neq j\\)\uff09\u306e\u5171\u5206\u6563\u304c\\(0\\)\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u5171\u5206\u6563\u306e\u6027\u8cea\u304b\u3089\\(X_{i}\\)\u3068\\(X_{j}\\)\uff08\\(i\\neq j\\)\uff09\u306f\u4e92\u3044\u306b\u72ec\u7acb\u3068\u306a\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u5171\u5206\u6563\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/covariance\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/135-money-images-100x100.jpg\" alt=\"\u5171\u5206\u6563\u306e\u5b9a\u7fa9\u3010\u78ba\u7387\u5909\u6570\u306e\u5171\u5206\u6563\u3092\u77e5\u308b\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u5171\u5206\u6563\u306e\u5b9a\u7fa9\u3010\u78ba\u7387\u5909\u6570\u306e\u5171\u5206\u6563\u3092\u77e5\u308b\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606 \u5171\u5206\u6563\u306f\u3001\u30c7\u30fc\u30bf\u540c\u58eb\u306e\u76f8\u95a2\u95a2\u4fc2\u3092\u8868\u3057\u307e\u3059\u3002\u3053\u308c\u306f2\u7d44\u306e\u78ba\u7387\u5909\u6570\u3092\u5165\u624b\u3057\u305f\u6642\u30012\u7d44\u306e\u78ba\u7387\u5909\u6570\u306e\u9593\u306b\u3069\u306e\u3088\u3046\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-4\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7dda\u5f62\u95a2\u6570<\/span><\/h2>\n\n\n\n<ul><li>\u3000\\(p\\)\u6b21\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{p}(\\mu,\\Sigma)\\)\u306b\u3064\u3044\u3066\u3001<br>\u884c\u5217\\(A\\)\u3092\\(p\\times p\\)\u306e\u975e\u7279\u7570\u884c\u5217\u3001\u30d9\u30af\u30c8\u30eb\\(b\\)\u3092\\(p\\)\u6b21\u306e\u30d9\u30af\u30c8\u30eb\u3068\u3059\u308b\u3068\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Y = AX+b \\sim N_{p}(A\\mu+b,A\\Sigma\\ {}^{T}\\!A)<br>\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br>\u3000<\/li><li>\u3000\\(p\\)\u6b21\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{p}(\\mu,\\Sigma)\\)\u306b\u3064\u3044\u3066\u3001\u884c\u5217\\(A\\)\u3092\\(q\\times p\\)\u306e\\(\\mathrm{rank}(A)=q\\leq p\\)\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Z = AX \\sim N_{q}(A\\mu,A\\Sigma\\ {}^{T}\\!A)<br>\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002<\/li><\/ul>\n\n\n\n<h2><span id=\"i-5\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(p\\)\u6b21\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X_{i}\\sim N_{p}(0,\\Sigma)\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u306b\u3064\u3044\u3066\u3001\u305d\u308c\u305e\u308c\u306e\u78ba\u7387\u5909\u6570\u304c\u4e92\u3044\u306b\u72ec\u7acb\u3060\u3063\u305f\u5834\u5408<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{1}\\ {}^{T}X_{1} + \\cdots + X_{n}\\ {}^{T}X_{n} \\sim W_{p}(n,\\Sigma)<br>\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u3053\u3067\\(W_{p}(n,\\Sigma)\\)\u306f\u81ea\u7531\u5ea6\\(n\\)\u306e\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3092\u8868\u3057\u307e\u3059\u3002<br>\u3000\u671f\u5f85\u5024\\(0\\)\u3067\u306a\u3044\u5834\u5408\u306e\u78ba\u7387\u5909\u6570\\(X_{i}\\sim N_{p}(\\mu_{i},\\Sigma)\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u3067\u3082\u540c\u69d8\u306e\u3053\u3068\u304c\u6210\u308a\u7acb\u3061\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{1}\\ {}^{T}X_{1} + \\cdots + X_{n}\\ {}^{T}X_{n} \\sim W_{p}\\left(n\\ ,\\ \\Sigma\\ ;\\ \\sum_{i=1}^{n}\\mu_{i}\\ {}^{T}\\mu_{i}\\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\( W_{p}\\left(n\\ ,\\ \\Sigma\\ ;\\ \\sum_{i=1}^{n}\\mu_{i}\\ {}^{T}\\mu_{i}\\right) \\)\u306f\u975e\u5fc3\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-6\">\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(p\\)\u6b21\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3092\\(X = {}^{T}(X_{1}\\ X_{2})\\sim N_{p}(\\mu,\\Sigma)\\)\u3068\u5206\u5272\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(X_{1}\\)\u306f\\(p_{1}\\)\u6b21\u306e\u78ba\u7387\u5909\u6570\u3001\\(X_{2}\\)\u306f\\(p_{2}\\)\u6b21\u306e\u78ba\u7387\u5909\u6570\u3067\u3042\u308a\\(p_{1}+p_{2}=p\\)\u304c\u6210\u308a\u7acb\u3063\u3066\u3044\u307e\u3059\u3002<br>\u3000\\(X\\)\u304c\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u3001\\(X_{1}\\)\u3068\\(X_{2}\\)\u3082\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\u3088\u3063\u3066<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{1}\\sim N_{p_{1}}(\\mu_{1},\\Sigma_{11}),\\ \\ X_{2} \\sim N_{p_{2}}(\\mu_{2},\\Sigma_{22})<br>\\end{align}<\/div>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(\\mu\\)\u3068\\(\\mu_{1},\\ \\mu_{2}\\)\u3001\\(\\Sigma\\)\u3068\\(\\Sigma_{11},\\ \\Sigma_{22}\\)\u3068\u306f\u3064\u304e\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mu = \\left( \\begin{array}{c}<br>\\mu_{1} \\\\<br>\\mu_{2}<br>\\end{array}\\right), \\Sigma = \\left( \\begin{array}{cc}<br>\\Sigma_{11} &amp; \\Sigma_{12}\\\\<br>\\Sigma_{21} &amp; \\Sigma_{22}<br>\\end{array}\\right)<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3053\u306e\u3068\u304d\u3001\\(X_{1}\\)\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\\(X_{2}\\)\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03\u3082\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{2}\\,|\\,X_{1} \\sim N_{p_{2}} \\left( \\mu_{2}+\\Sigma_{21}\\Sigma_{11}^{-1}(x_{2}-\\mu_{2}),\\ \\Sigma_{22}-\\Sigma_{21}\\Sigma_{11}^{-1}\\Sigma_{12} \\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3001\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u304c<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{E}[X_{2}|X_{1}] &amp;= \\mu_{2}+\\Sigma_{21}\\Sigma_{11}^{-1}(x_{2}-\\mu_{2}) \\\\<br>\\mathrm{Var}[X_{2}|X_{1}] &amp;= \\Sigma_{22}-\\Sigma_{21}\\Sigma_{11}^{-1}\\Sigma_{12}<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u3001\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"background: #e6ffe6; padding: 15px; border-radius: 10px; border: 0px solid #99cc00; word-break: break-all;\">\u4e0a\u3067\u306f\\(X_{1}\\)\u3092\u4e0e\u3048\u305f\u4e0b\u3067\\(X_{2}\\)\u306e\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u3092\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u304c\u3001\u9006\u306b\\(X_{2}\\)\u3092\u4e0e\u3048\u305f\u4e0b\u3067\\(X_{1}\\)\u306e\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u3092\u4e0e\u3048\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3068\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306f<br \/><div style=\"overflow-x: auto;\">\\begin{align}<br data-rich-text-line-break=\"true\" \/>\\mathrm{E}[X_{1}|X_{2}] &amp;= \\mu_{1}+\\Sigma_{12}\\Sigma_{22}^{-1}(x_{1}-\\mu_{1}) \\\\<br data-rich-text-line-break=\"true\" \/>\\mathrm{Var}[X_{1}|X_{2}] &amp;= \\Sigma_{11}-\\Sigma_{12}\\Sigma_{22}^{-1}\\Sigma_{21}<br data-rich-text-line-break=\"true\" \/>\\end{align}<\/div><br \/>\u3068\u306a\u308a\u307e\u3059\u3002<\/div>\n\n\n\n<p>\u8a3c\u660e\u306f\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3067\u884c\u3044\u307e\u3059\uff08\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u8a3c\u660e\u306e\u306a\u304b\u3067\u3001\\(n=1\\)\u3067\u7f6e\u304d\u63db\u3048\u308b\u3068\u3053\u306e\u8a3c\u660e\u306f\u5b8c\u4e86\u3057\u307e\u3059\uff09\u3002<\/p>\n\n\n\n<h2><span id=\"i-7\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u78ba\u7387\u5206\u5e03<\/span><\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"738\" height=\"404\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09.png\" alt=\"\" class=\"wp-image-2555\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09.png 738w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09-300x164.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09-320x175.png 320w\" sizes=\"(max-width: 738px) 100vw, 738px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>\u3000\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3092\u30d9\u30af\u30c8\u30eb\u306e\u5f62\u3067\u307e\u3068\u3081\u305f\u3082\u306e\u306f\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\uff08multivariate normal distribution\uff09\u306b &#8230; <\/p>\n","protected":false},"author":1,"featured_media":1202,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[116,71],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/9fc17caf-9cad-4670-b857-eb43ec911ab9.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2614"}],"collection":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/comments?post=2614"}],"version-history":[{"count":38,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2614\/revisions"}],"predecessor-version":[{"id":2664,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2614\/revisions\/2664"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1202"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2614"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2614"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2614"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}