{"id":2662,"date":"2020-08-16T11:00:00","date_gmt":"2020-08-16T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2662"},"modified":"2020-08-03T11:32:12","modified_gmt":"2020-08-03T02:32:12","slug":"matrix-normal-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/matrix-normal-distribution","title":{"rendered":"\u884c\u5217\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\u884c\u5217\u306e\u5f62\u3067\u307e\u3068\u3081\u305f\u3082\u306e\u306f\u3001\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\uff08matrix normal distribution\uff09\u306b\u5f93\u3044\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306b\u3064\u3044\u3066<div style=\"overflow-x: auto;\">\\begin{align}<br>X=\\left(\\begin{array}{ccc}<br>X_{11} &amp; \\cdots &amp; X_{1p} \\\\<br>\\vdots &amp; \\ddots &amp; \\vdots \\\\<br>X_{n1} &amp; \\cdots &amp; X_{np}<br>\\end{array}\\right)<br>\\end{align}<\/div>\u3068\u3057\u305f\u3068\u304d\u3001\u4efb\u610f\u306e\u5217\u30d9\u30af\u30c8\u30eb\\(x_{i} = \\,{}^{t}\\!(x_{1i},\\cdots,x_{ni})\\)\uff08\\(i=1,\\cdots,p\\)\uff09\u306e\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u306f\\(U\\)\u3001\u4efb\u610f\u306e\u884c\u30d9\u30af\u30c8\u30eb\\(x_{j} = \\,{}^{t}\\!(x_{j1},\\cdots,x_{jp})\\)\uff08\\(j=1,\\cdots,n\\)\uff09\u306e\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u306f\\(V\\)\u3067\u4e0e\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3082\u306e\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306f\u3001\u8a18\u53f7\\(X\\sim N_{n\\times p}(M,\\ U,\\ V)\\)\u3067\u8868\u3055\u308c\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(n\\times p\\)\u306e\u884c\u5217\\(M\\)\u306f\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u671f\u5f85\u5024\u3001\\(n\\times n\\)\u306e\u884c\u5217\\(U\\)\u306f\u6b63\u5b9a\u5024\u884c\u5217\u3001\\(p\\times p\\)\u306e\u884c\u5217\\(V\\)\u306f\u6b63\u5b9a\u5024\u884c\u5217\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{n\\times p}(M,\\ U,\\ V)\\)\u306b\u3064\u3044\u3066\u3001\\(p=1\\)\u306e\u3068\u304d\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u5206\u6563\u5171\u5206\u6563\u884c\u5217\\(U\\)\u306b\u306a\u308b\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\\(X\\sim N_{n}(M,U)\\)\u306b\u5f93\u3044\u307e\u3059\u3002\u3055\u3089\u306b\\(p=1,\\ n=1\\)\u306e\u3068\u304d\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u6b63\u898f\u5206\u5e03\u3001\u591a\u5909\u91cf\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\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<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><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u8a18\u53f7\u306e\u5b9a\u7fa9<\/a><\/li><\/ul><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">2<\/span> \u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">3<\/span> \u884c\u5217\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\">4<\/span> \u884c\u5217\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\">5<\/span> \u6761\u4ef6\u4ed8\u304d\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">6<\/span> \u884c\u5217\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: 235px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 19.6552%; height: 36px;\">\u30d1\u30e9\u30e1\u30fc\u30bf<\/td>\n<td style=\"width: 80.1724%; height: 36px;\">\\(M,\\ \\ \\ 0 &lt; U,\\ V\\)<br \/>\u305f\u3060\u3057\u3001\\(U,\\ V &gt;0\\)\u306f\\(U,\\ V\\)\u304c\u6b63\u5b9a\u5024\u884c\u5217\u3092\u8868\u3057\u307e\u3059\u3002<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 19.6552%; height: 56px;\">\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td>\n<td style=\"width: 80.1724%; height: 56px;\">\\(-\\infty&lt; x_{ij} &lt;\\infty \\),\uff08\\(i=1,\\cdots,n,\\ \\ j=1,\\cdots,p\\)\uff09<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 19.6552%; 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: 80.1724%; height: 35px;\">\\(\\displaystyle \\frac{1}{(2\\pi)^{\\frac{1}{2}np}|U|^{\\frac{1}{2}p}|V|^{\\frac{1}{2}n}}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr} \\left\\{ U^{-1}(X-M)V^{-1}\\,{}^{T}\\!(X-M) \\right\\} \\right] \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 19.6552%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 80.1724%; height: 36px;\">\\(\\displaystyle \\exp\\left[ \\mathrm{tr}\\left( {}^{T}\\!MT \\right)+\\frac{1}{2}\\mathrm{tr}\\left( {}^{T}\\!TUTV \\right) \\right] \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 19.6552%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 80.1724%; height: 36px;\">\\( \\displaystyle M \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 19.6552%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563\u5171\u5206\u6563\u884c\u5217<\/a><br \/>\uff08\u78ba\u7387\u5909\u6570\u3092\u30d9\u30af\u30c8\u30eb\u5c55\u958b\u3057\u305f\u3082\u306e\uff09<\/td>\n<td style=\"width: 80.1724%; height: 36px;\">\\( \\displaystyle U\\otimes V \\)<br \/>\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u5217\u30d9\u30af\u30c8\u30eb\u3092\u3072\u3068\u3064\u306b\u4e26\u3079\u305f\u30d9\u30af\u30c8\u30eb\u306e<br \/>\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u3092\u8868\u3057\u307e\u3059\u3002<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n<h3><span id=\"i-2\">\u8a18\u53f7\u306e\u5b9a\u7fa9<\/span><\/h3>\n\n\n\n<ul><li>\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u7a4d\\(\\otimes\\)<br>\\(p\\times q\\)\u306e\u884c\u5217\\(A\\)\u3068\u3001\\(s\\times t\\)\u306e\u884c\u5217\\(B\\)\u306e\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u7a4d\\(A\\otimes B\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}<br>(A\\otimes B)_{ij} = (a_{ij}B)_{ij},\\ \\ \\ (i=1,\\cdots,p,\\ \\ \\ j=1,\\cdots,q)<br>\\end{align}<\/div><br><\/li><li>\u30d9\u30af\u30c8\u30eb\u5c55\u958b\\(\\mathrm{vec}\\)<br>\u884c\u5217\\(X=(x_{1},\\cdots,x_{n})\\)\uff08\\(x_{1},\\cdots,x_{n}\\)\u306f\u884c\u5217\\(X\\)\u306e\u5217\u30d9\u30af\u30c8\u30eb\u3092\u8868\u3057\u307e\u3059\uff09\u306b\u3064\u3044\u3066\u3001\u30d9\u30af\u30c8\u30eb\u5c55\u958b\\(\\mathrm{vec}(X)\\)\u3068\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{vec}(X) = \\left(\\begin{array}{c}<br>x_{1} \\\\<br>\\vdots \\\\<br>x_{n}<br>\\end{array}\\right)<br>\\end{align}<\/div><\/li><\/ul>\n\n\n\n<h2><span id=\"i-3\">\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u591a\u5909\u91cf\u6b63\u898f\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>\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{n\\times p}(M,U,V)\\)\u306f\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\\(N_{np}(\\mathrm{vec}(M),U\\otimes V)\\)\u3068\u540c\u5024\u3068\u306a\u308a\u307e\u3059\u3002\uff08\u3053\u308c\u306f\u9006\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\uff09\n<\/div>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u3000\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u66f8\u304d\u63db\u3048\u3066\u3044\u304d\u307e\u3059\u3002\u307e\u305a\u6307\u6570\u90e8\u5206\u306b\u3064\u3044\u3066\u8a08\u7b97\u3059\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>-\\frac{1}{2}\\mathrm{tr}\\left[ U^{-1}(X-M)V^{-1}\\,{}^{T}\\!(X-M) \\right] &amp;= -\\frac{1}{2}\\mathrm{vec}\\,{}^{T}\\!(X-M)\\mathrm{vec}\\left( U^{-1}(X-M)V^{-1} \\right) \\\\<br>&amp;= -\\frac{1}{2}\\mathrm{vec}\\,{}^{T}\\!(X-M)\\left( U^{-1}\\otimes V^{-1} \\right)\\mathrm{vec}\\left( X-M)V \\right) \\\\<br>&amp;= -\\frac{1}{2}\\,{}^{T}\\!\\left\\{\\mathrm{vec}(X)-\\mathrm{vec}(M)\\right\\}\\left( U^{-1}\\otimes V^{-1} \\right)\\left\\{\\mathrm{vec}(X)-\\mathrm{vec}(M)\\right\\}) \\\\<br>\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u7a4d\u306e\u6027\u8cea\u3092\u7528\u3044\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>|U\\otimes V| &amp;= |U|^{p}|V|^{n}<br>\\end{align}<\/div>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3053\u306e\uff12\u3064\u3092\u7528\u3044\u3066\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u66f8\u304d\u63db\u3048\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{1}{(2\\pi)^{\\frac{1}{2}np}|U|^{\\frac{1}{2}p}|V|^{\\frac{1}{2}n}}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr} \\left\\{ U^{-1}(X-M)V^{-1}\\,{}^{T}!(X-M) \\right\\} \\right] &amp;= \\frac{1}{(2\\pi)^{\\frac{1}{2}np}|U\\otimes V|^{\\frac{1}{2}}}\\exp\\left[ -\\frac{1}{2}\\,{}^{T}\\!\\left\\{\\mathrm{vec}(X)-\\mathrm{vec}(M)\\right\\}\\left( U^{-1}\\otimes V^{-1} \\right)\\left\\{\\mathrm{vec}(X)-\\mathrm{vec}(M)\\right\\} \\right]<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u4e00\u81f4\u3059\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\u591a\u5909\u91cf\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\/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-4\">\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\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>\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7dda\u5f62\u95a2\u6570<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N_{n\\times p}(M,U,V)\\)\u306f\u3068\u3057\u3066\u3001\u884c\u5217\\(A,B,C\\)\u3092\u30b5\u30a4\u30ba\\(m\\times n,\\ p\\times \\ell,\\ m\\times \\ell\\)\u306e\u975e\u7279\u7570\u884c\u5217\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\n<div style=\"overflow-x: auto;\">\\begin{align}\nAXB+C \\sim N_{m\\times \\ell}(AMB+C,\\ AU\\,{}^{T}\\!A,\\ BV\\,{}^{T}\\!B)\n\\end{align}<\/div>\n<\/div>\n\n\n\n<h2><span id=\"i-5\">\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u30a6\u30a3\u30b7\u30e3\u30fc\u30c8\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X_{i}\\sim N_{n\\times p}(O,\\ I_{n},\\ V)\\)\u3068\u3057\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u671f\u5f85\u5024\\(O\\)\u306e\u4efb\u610f\u306e\u884c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306f\u4e92\u3044\u306b\u72ec\u7acb\u306b\u306a\u3063\u3066\u3044\u308b\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>{}^{T}\\!XX \\sim W_{p}(n,\\ V)<br>\\end{align}<\/div>\u3053\u3053\u3067\\(W_{p}(n,\\ V)\\)\u306f\u81ea\u7531\u5ea6\\(n\\)\u306e\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\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(n\\times p\\)\u306e\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3092\\(X = (X_{1}\\ X_{2})\\sim N_{p}(M,\\ U,\\ V)\\)\u3068\u5206\u5272\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(X_{1}\\)\u306f\\(n\\times p_{1}\\)\u306e\u78ba\u7387\u5909\u6570\u3001\\(X_{2}\\)\u306f\\(n\\times p_{2}\\)\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\u884c\u5217\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u3001\\(X_{1}\\)\u3068\\(X_{2}\\)\u3082\u884c\u5217\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_{n\\times p_{1}}(M_{1},\\ U,\\ V_{11}),\\ \\ X_{2} \\sim N_{n\\times p_{2}}(M_{2},\\ U,\\ V_{22})<br>\\end{align}<\/div>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(M\\)\u3068\\(M_{1},\\ M_{2}\\)\u3001\\(V\\)\u3068\\(V_{11},\\ V_{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>M=\\left( \\begin{array}{cc}<br>M_{1} &amp; M_{2} <br>\\end{array}\\right), \\ \\ \\ \\\u00a0V = \\left( \\begin{array}{cc}<br>V_{11} &amp; V_{12}\\\\<br>V_{21} &amp; V_{22}<br>\\end{array}\\right),\\ \\ \\ \\ V_{ij}\u306fp_{i}\\times p_{j}\u306e\u884c\u5217\u3067\u3059<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3053\u306e\u3068\u304d\u3001\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u6761\u4ef6\u4ed8\u304d\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\\(X_{1}\\)\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\\(X_{2}\\)\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03\u3082\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u3001\n<div style=\"overflow-x: auto;\">\\begin{align}\nX_{2}\\,|\\,X_{1} \\sim N_{n\\times p_{2}} \\left( M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12},\\ U\\otimes(V_{22}-V_{21}V_{11}^{-1}V_{12}) \\right)\n\\end{align}<\/div>\n\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\n<div style=\"overflow-x: auto;\">\\begin{align}\n\\mathrm{E}[X_{2}|X_{1}] &amp;= M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12} \\\\\n\\mathrm{Var}[X_{2}|X_{1}] &amp;= U\\otimes(V_{22}-V_{21}V_{11}^{-1}V_{12})\n\\end{align}<\/div>\n\u3068\u306a\u308b\u3001\u6761\u4ef6\u4ed8\u304d\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\n<\/div>\n\n\n\n<p>\u3053\u306e\u6761\u4ef6\u4ed8\u304d\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u3001\\(n=1)\\\u3068\u3057\u3066\u8ee2\u7f6e\u3092\u3068\u3063\u305f\u3082\u306e\u306f\u6761\u4ef6\u4ed8\u304d\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u4e00\u81f4\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<p>\u8a3c\u660e\u306f\u5225\u30da\u30fc\u30b8\u3067\u7d39\u4ecb\u3057\u307e\u3059\uff08\u8a08\u7b97\u304c\u5927\u5909\u306a\u306e\u3067\u3042\u308b\u7a0b\u5ea6\u306e\u8a08\u7b97\u77e5\u8b58\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\uff09\u3002\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u3069\u3046\u305e\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mnormd-cnorm\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/publicdomainq-0011362-100x100.jpg\" alt=\"\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u8a3c\u660e\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u8a3c\u660e\u306f\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3067\u884c\u3044\u307e\u3059\u304c\u3001\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u8a3c\u660e\u306f\u8a3c\u660e\u4e2d\u3067\\(n=1\\)\u3068\u3057\u305f\u3082\u306e\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\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 \/>\\mathrm{E}[X_{1}|X_{2}] &amp;= M_{1}+(X_{2}-M_{2})V_{22}^{-1}V_{21} \\\\<br \/>\\mathrm{Var}[X_{1}|X_{2}] &amp;= U\\otimes(V_{11}-V_{12}V_{22}^{-1}V_{21})<br \/>\\end{align}<\/div><br \/>\u3068\u306a\u308a\u307e\u3059\u3002<\/div>\n\n\n\n<h2><span id=\"i-7\">\u884c\u5217\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\u884c\u5217\u306e\u5f62\u3067\u307e\u3068\u3081\u305f\u3082\u306e\u306f\u3001\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\uff08matrix normal distribution\uff09\u306b\u5f93\u3044\u307e\u3059\u3002\u305f\u3060 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":2511,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[71,117],"jetpack_featured_media_url":"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.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2662"}],"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=2662"}],"version-history":[{"count":48,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2662\/revisions"}],"predecessor-version":[{"id":2751,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2662\/revisions\/2751"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2511"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2662"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2662"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2662"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}