{"id":2858,"date":"2020-08-21T11:00:00","date_gmt":"2020-08-21T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2858"},"modified":"2020-08-19T10:06:15","modified_gmt":"2020-08-19T01:06:15","slug":"wishart-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wishart-distribution","title":{"rendered":"\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03"},"content":{"rendered":"\n\n\n\n\n<p>\u3000\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\uff08wishart distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306f\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u3092\u591a\u6b21\u5143\u306b\u62e1\u5f35\u3057\u305f\u3082\u306e\u3068\u3057\u3066\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\u3002<br>\u3000\\(p\\times p\\)\u6b21\u306e\u78ba\u7387\u5909\u6570\u884c\u5217\\(W\\)\u304c\u3001\\(n\\)\u500b\u306e\u72ec\u7acb\u306a\\(p\\)\u6b21\u306e\u78ba\u7387\u30d9\u30af\u30c8\u30eb\\(z_{i}\\sim N_{p}(0,\\ \\Sigma),\\ \\ i=1,\\cdots,n\\)\u3092\u7528\u3044\u3066<div style=\"overflow-x: auto;\">\\begin{align}W=\\sum_{i=1}^{n}z_{i}\\ {}^{T}\\!z_{i}\\end{align}<\/div>\u3068\u8868\u305b\u308b\u3068\u304d\u3001\\(W\\)\u306f\u81ea\u7531\u5ea6\\(n\\)\u3001\u5171\u5206\u6563\u884c\u5217\\(\\Sigma\\)\u306e\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3044\u3001\u8a18\u53f7\\(W\\sim W_{p}(n,\\ \\Sigma)\\)\u3067\u8868\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(n\\geq p\\)\u3067\u3059\u3002<\/p>\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> \u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3068\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">3<\/span> \u6a19\u672c\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">4<\/span> \u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3068\u7dda\u5f62\u95a2\u6570<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">5<\/span> \u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u518d\u751f\u6027<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_1\">6<\/span> \u5206\u5272\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">7<\/span> \u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u78ba\u7387\u5206\u5e03<\/a><\/li><\/ul><\/div>\n\n\n\n\n<h2><span id=\"i\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\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: 252px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\">\u30d1\u30e9\u30e1\u30fc\u30bf<\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(p,\\ n,\\ \\Sigma\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle \\frac{|W|^{\\frac{1}{2}(n-p-1)}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr}\\Sigma^{-1}W \\right]}{2^{\\frac{pn}{2}}\\pi^{\\frac{p(p-1)}{4}}|\\Sigma|^{\\frac{n}{2}}\\prod_{i=1}^{p}\\Gamma\\left( \\frac{n-i+1}{2} \\right)} \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle |I_{p}-2\\Sigma T|^{-\\frac{n}{2}} \\)<br \/>\u305f\u3060\u3057\u3001\\(T\\)\u306f\u5bfe\u79f0\u884c\u5217<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle |I_{p}-2i\\Sigma T|^{-\\frac{n}{2}} \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\\(r\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><br \/>\uff08\u539f\u70b9\u307e\u308f\u308a\uff09<\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle \\frac{ 2^{pr}|\\Sigma|^{r}\\prod_{i=1}^{p}\\Gamma\\left( \\frac{n+1-i}{2}+r \\right) }{ \\prod_{i=1}^{n}\\Gamma\\left( \\frac{n+1-i}{2} \\right) } \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\( \\displaystyle n\\Sigma \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\( \\displaystyle 2n\\Sigma\\otimes\\Sigma \\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n<h3>\u8a3c\u660e\u4e00\u89a7<\/h3>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cwisd-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/fe5298b70d9972a91424adc0358d5ef6_t-100x100.jpeg\" alt=\"\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u671f\u5f85\u5024\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u671f\u5f85\u5024\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u671f\u5f85\u5024\u306e\u6c42\u3081\u65b9\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002\u5206\u6563\u306e\u6c42\u3081\u65b9\u306f\u5c11\u3005\u9762\u5012\uff08\u30d9\u30af\u30c8\u30eb\u5c55\u958b\u304a\u3088\u3073\u8981\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-2\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3068\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3068\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u81ea\u7531\u5ea6\\(n\\)\u3001\u5206\u6563\u5171\u5206\u6563\u884c\u5217\\(I\\)\u306e\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim W_{p}(n,\\ I)\\)\u306f\u3001\\(p=1\\)\u306e\u3068\u304d\u81ea\u7531\u5ea6\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\n<\/div>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u3053\u306e\u3053\u3068\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u66f8\u304d\u63db\u3048\u308c\u3070\u3059\u3050\u306b\u5206\u304b\u308a\u307e\u3059\u3002\u81ea\u7531\u5ea6(n)\u3001\u5206\u6563\u5171\u5206\u6563\u884c\u5217(I)\u306e\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim W_{p}(n,\\ I)\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f<div style=\"overflow-x: auto;\">\\begin{align}f(X)=\\frac{|X|^{\\frac{1}{2}(n-p-1)}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr}X \\right]}{2^{\\frac{pn}{2}}\\pi^{\\frac{p(p-1)}{4}}\\prod_{i=1}^{p}\\Gamma\\left( \\frac{n-i+1}{2} \\right)}\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u5f0f\u3092\\(p=1\\)\u3067\u66f8\u304d\u63db\u3048\u308b\u3068\u6b21\u306e\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}f(X)=\\frac{X^{\\frac{1}{2}n-1}\\exp\\left[ -\\frac{1}{2}X \\right]}{2^{\\frac{n}{2}}\\Gamma\\left( \\frac{n}{2} \\right)}\\end{align}<\/div>\u3053\u3053\u3067\u3001\u6ce8\u610f\u3057\u306a\u3044\u3068\u3044\u3051\u306a\u3044\u3053\u3068\u306f\u3001\\(p=1\\)\u306e\u3068\u304d\\(X\\)\u306f\u884c\u5217\u3067\u306f\u306a\u3044\u306e\u3067\\(X\\)\u306e\u884c\u5217\u5f0f\u306f\u305d\u306e\u307e\u307e\\(X\\)\u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002\u3053\u306e\u5f0f\u306f\u81ea\u7531\u5ea6\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<p>\u203b\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\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\/chi-square-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/\u30ab\u30a42\u4e57\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u30ab\u30a42\u4e57\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30ab\u30a42\u4e57\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u30ab\u30a42\u4e57\u5206\u5e03\uff08chi square distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u591a\u304f\u306e\u66f8\u7c4d\u3067\u306f\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u5e73\u65b9\u548c\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u30ab\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-3\">\u6a19\u672c\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u6a19\u672c\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u6bcd\u96c6\u56e3\u304b\u3089\u306e\\(n\\)\u500b\u306ep\u6b21\u5143\u89b3\u6e2c\u5024\u30d9\u30af\u30c8\u30eb\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\\(x_{i}\\sim N_{p}(\\mu,\\Sigma)\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u3001\u3053\u306e\u89b3\u6e2c\u5024\u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u5206\u6563\u5171\u5206\u6563\u884c\u5217\n<div style=\"overflow-x: auto;\">\\begin{align}\\widehat{\\Sigma}=\\frac{1}{n}\\sum_{i=1}^{n}(x_{i}-\\bar{x})\\ {}^{T}\\!(x_{i}-\\bar{x})\\end{align}<\/div>\u306f\\(W_{p}\\left( n-1,\\ \\frac{1}{n}\\Sigma \\right)\\)\u306b\u5f93\u3044\u307e\u3059\u3002\n<\/div>\n\n\n\n<p>\u3000\u3053\u306e\u6027\u8cea\u306f\u3001\u6a19\u672c\u304b\u3089\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u3092\u63a8\u5b9a\u3059\u308b\u969b\u3001\u4e0d\u504f\u63a8\u5b9a\u91cf\u3092\u8003\u3048\u308b\u3068\u304d\u306b\u5229\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u8a3c\u660e\u306b\u306f\u30b3\u30af\u30e9\u30f3\u306e\u5b9a\u7406\u3092\u7528\u3044\u307e\u3059\u3002\u8a18\u4e8b\u306e\u6e96\u5099\u304c\u3067\u304d\u6b21\u7b2c\u3001\u4ee5\u4e0b\u306b\u30ea\u30f3\u30af\u3092\u8a2d\u7f6e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-4\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u3068\u7dda\u5f62\u95a2\u6570<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u975e\u7279\u7570\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim W_{p}(n,\\ \\Sigma)\\)\u306b\u975e\u7279\u7570\u884c\u5217\\(C\\)\u3092\u639b\u3051\u305f\u3082\u306e\u3082\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306b\u5f93\u3044\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}CX\\ {}^{T}\\!C\\sim W_{p}(n,\\ C\\Sigma\\ {}^{T}\\!C)\\end{align}<\/div>\n<\/div>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u3000\u8a3c\u660e\u306f\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u5b9a\u7fa9\u3068\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea\u3092\u7528\u3044\u307e\u3059\u3002<br>\u307e\u305a\u3001\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim W_{p}(n,\\ \\Sigma)\\)\u306f\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\\(N_{p}(0,\\ \\Sigma)\\)\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(x_{1},\\cdots,x_{n}\\)\u3092\u7528\u3044\u3066\u3001<div style=\"overflow-x: auto;\">\\begin{align}X=\\sum_{i=1}^{n}x_{i}\\ {}^{T}\\!x_{i}\\sim W_{p}(n,\\ \\Sigma)\\end{align}<\/div>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089<div style=\"overflow-x: auto;\">\\begin{align}CX\\ {}^{T}\\!C = C\\sum_{i=1}^{n}x_{i}\\ {}^{T}\\!x_{i}\\ {}^{T}\\!C = \\sum_{i=1}^{n}(Cx_{i})\\ {}^{T}\\!(Cx_{i})\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea\u304b\u3089<div style=\"overflow-x: auto;\">\\begin{align}Cx_{i}\\sim N_{p}(0,\\ C\\Sigma\\ {}^{T}\\!C),\\ \\ \\ \\ i=1,\\cdots,n\\end{align}<\/div>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001<div style=\"overflow-x: auto;\">\\begin{align}\\sum_{i=1}^{n}(Cx_{i})\\ {}^{T}\\!(Cx_{i})\\sim W_{p}(n,\\ C\\Sigma\\ {}^{T}\\!C)\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<p>\u203b\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea\u306b\u3064\u3044\u3066\u306f\u6b21\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\/multivariate-normal-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/9fc17caf-9cad-4670-b857-eb43ec911ab9-100x100.png\" alt=\"\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\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\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-5\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u518d\u751f\u6027<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u518d\u751f\u6027<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\\(W_{1}\\sim W_{p}(n_{1},\\Sigma)\u3001W_{2}\\sim W_{p}(n_{2},\\Sigma)\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u304d\u3001\\(W_{1}+W_{2}\\sim W_{p}(n_{1}+n_{2},\\Sigma)\\)\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\n<\/div>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u3000\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u4f7f\u3063\u3066\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}M_{W_{1}+W_{2}}(T) &amp;= \\mathrm{E}\\left[\\exp[(W_{1}+W_{2})T]  \\right]\\\\<br>&amp;= \\mathrm{E}\\left[\\exp[W_{1}T] \\right]\\mathrm{E}\\left[\\exp[W_{2}T] \\right]\\\\<br>&amp;= |I_{p}-2\\Sigma T|^{-\\frac{n_{1}}{2}}|I_{p}-2\\Sigma T|^{-\\frac{n_{2}}{2}} \\\\<br>&amp;=|I_{p}-2\\Sigma T|^{-\\frac{n_{1}+n_{2}}{2}}<br>\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304b\u3089\u3001\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306e\u518d\u751f\u6027\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<h2><span id=\"i-6\">\u5206\u5272\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u5206\u5272\u884c\u5217\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u78ba\u7387\u5909\u6570\\(A\\)\u304c\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\\(W_{p}(n,\\ \\Sigma)\\)\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u304d\u3092\u8003\u3048\u307e\u3059\u3002\u884c\u5217\\(A\\)\u3092<div style=\"overflow-x: auto;\">\\begin{align}A=\\left(\\begin{array}{cc}\nA_{11} &#038; A_{12} \\\\\nA_{21} &#038; A_{22}\n\\end{array}\\right)\\end{align}<\/div>\u3068\u5206\u5272\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(A_{11}\\)\u306f\\(q\\times q\\)\uff08\\(1 \\leq q < p\\)\uff09\u306e\u884c\u5217\u3067\u3059\u3002\u3053\u306e\u5206\u5272\u306b\u5bfe\u5fdc\u3057\u3066<div style=\"overflow-x: auto;\">\\begin{align}\\Sigma=\\left(\\begin{array}{cc}\n\\Sigma_{11} &#038; \\Sigma_{12} \\\\\n\\Sigma_{21} &#038; \\Sigma_{22}\n\\end{array}\\right)\\end{align}<\/div>\u3068\u5206\u5272\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/br>\n\uff08\uff11\uff09\\(A_{11}-A_{12}A_{22}^{-1}A_{21}\\sim W_{q}(n-(p-q),\\ \\Sigma_{11}-\\Sigma_{12}\\Sigma_{22}^{-1}\\Sigma_{21})\\)<\/br>\n\uff08\uff12\uff09\\(A_{22}\\sim W_{p-q}(n,\\ \\Sigma_{22})\\)<\/br>\n\uff08\uff13\uff09\\(A_{22}\\)\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\\(A_{12}A_{22}^{-1}\\)\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}A_{12}A_{22}^{-1}\\sim N(\\Sigma_{12}\\Sigma_{22}^{-1},\\ (\\Sigma_{11}-\\Sigma_{12}\\Sigma_{22}^{-1}\\Sigma_{21})\\otimes A_{22}^{-1})\\end{align}<\/div>\n\uff08\uff14\uff09\\(A_{11}-A_{12}A_{22}^{-1}A_{21}\\)\u3068\\((A_{12}\\ A_{22})\\)\u306f\u4e92\u3044\u306b\u72ec\u7acb\u3067\u3042\u308b\n<\/div>\n\n\n\n<h2><span id=\"i-7\">\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\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\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\uff08wishart distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03\u306f\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u3092\u591a\u6b21\u5143\u306b\u62e1\u5f35\u3057\u305f\u3082\u306e &#8230; <\/p>\n","protected":false},"author":1,"featured_media":226,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[122,71],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/55b6b183f316a51ae4fdb6c509cb1835_t.jpeg","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2858"}],"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=2858"}],"version-history":[{"count":36,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2858\/revisions"}],"predecessor-version":[{"id":2914,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2858\/revisions\/2914"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/226"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2858"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2858"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2858"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}