{"id":2710,"date":"2020-08-17T11:00:00","date_gmt":"2020-08-17T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2710"},"modified":"2020-08-03T11:36:14","modified_gmt":"2020-08-03T02:36:14","slug":"mnormd-cnorm","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mnormd-cnorm","title":{"rendered":"\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011"},"content":{"rendered":"\n\n\n\n\n

\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\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03<\/h2>\n\n\n\n

\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
\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

\\begin{align}
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})
\\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
\\begin{align}
M=\\left( \\begin{array}{cc}
M_{1} & M_{2}
\\end{array}\\right),\\ \\ V = \\left( \\begin{array}{cc}
V_{11} & V_{12}\\\\
V_{21} & V_{22}
\\end{array}\\right),\\ \\ V_{ij}\u306fp_{i}\\times p_{j}\u306e\u884c\u5217\u3067\u3059
\\end{align}<\/div><\/p>\n\n\n\n

\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

\u6761\u4ef6\u4ed8\u304d\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/strong><\/div>\n
\\(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
\\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
\\begin{align}\n\\mathrm{E}[X_{2}|X_{1}] &= M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12} \\\\\n\\mathrm{Var}[X_{2}|X_{1}] &= 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

\u3000\u7279\u306b\\(n=1\\)\u306e\u3068\u304d\u3001\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306f\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u3053\u306e\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u3082\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3060\u3063\u305f\u5834\u5408\u306e\u6761\u4ef6\u4ed8\u304d\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u203b\u884c\u5217\u5909\u91cf\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

\n \n
\"\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\"<\/div>\n
\n
\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/div>\n
\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\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n\n
\n \n
\"\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\"<\/div>\n
\n
\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03<\/div>\n
\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>\n <\/a>\n <\/div>\n\n\n\n

\u8a3c\u660e<\/h3>\n\n\n\n

\u3000\u884c\u5217\\(E\\)\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002

\\begin{align}
E = \\left(\\begin{array}{cc}
I_{p_{1}} & O \\\\
-V_{21}V_{11}^{-1} & I_{p_{2}}
\\end{array}\\right)
\\end{align}<\/div>
\u3053\u306e\u884c\u5217\u306b\u3064\u3044\u3066\u3001\u884c\u5217\u5f0f\u304c\\(|E|=1\\)\u3068\u306a\u308a\u3001\u9006\u884c\u5217\u304c
\\begin{align}
E^{-1} = \\left(\\begin{array}{cc}
I_{p_{1}} & O \\\\
V_{21}V_{11}^{-1} & I_{p_{2}}
\\end{array}\\right)
\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u3092\u7528\u3044\u3066\u3001\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u66f8\u304d\u63db\u3048\u3066\u3044\u304d\u307e\u3059\u3002
\\begin{align}
f(x) &= \\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] \\\\
&= \\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)\\ {}^{T}\\!E\\ {}^{T}\\!E^{-1}V^{-1}E^{-1}E\\,{}^{T}\\!(X-M) \\right\\} \\right] \\\\
&= \\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)\\ {}^{T}\\!E\\} ( {}^{T}\\!EVE)^{-1}\\ {}^{T}\\{ (X-M)\\ {}^{T}\\!E \\} \\right\\} \\right] \\\\
\\end{align}<\/div>\u3053\u3053\u3067\u3001\u3053\u306e\u5f0f\u3092\u3082\u3046\u5c11\u3057\u8a73\u3057\u304f\u5206\u89e3\u3057\u3066\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u305f\u3060\u3057\\(M_{2}^{\\prime}=M_{2}-(X_{1}-M_{1})V_{11}^{-1}V_{12}\\)\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305a\u6307\u6570\u90e8\u5206\u306b\u3064\u3044\u3066\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002

\\begin{align}
EV\\ {}^{T}E &= \\left(\\begin{array}{cc}
I_{p_{1}} & O \\\\
-V_{21}V_{11}^{-1} & I_{p_{2}}
\\end{array}\\right)\\left(\\begin{array}{cc}
V_{11} & V_{12} \\\\
V_{21} & V_{22}
\\end{array}\\right)\\left(\\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\\\
O & I_{p_{2}}
\\end{array}\\right) \\\\
&= \\left(\\begin{array}{cc}
V_{11} & V_{12} \\\\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\\end{array}\\right)\\left(\\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\\\
O & I_{p_{2}}
\\end{array}\\right) \\\\
&= \\left(\\begin{array}{cc}
V_{11} & O \\\\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\\end{array}\\right)
\\end{align}<\/div>\u3055\u3089\u306b\u3001\u3082\u3046\u3072\u3068\u3064\u306e\u90e8\u5206\\((X-M)\\ {}^{T}\\!E\\)\u306b\u3064\u3044\u3066\u8a08\u7b97\u3057\u3066\u3044\u304f\u3068
\\begin{align}
(X-M)\\ {}^{T}\\!E &= \\left(\\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}
\\end{array}\\right)\\left(\\begin{array}{cc}
I_{p_{1}} & -V_{11}^{-1}V_{12} \\\\
O & I_{p_{2}}
\\end{array}\\right) \\\\
&= \\left(\\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}^{\\prime}
\\end{array}\\right)
\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u3092\u307e\u3068\u3081\u308b\u3068
\\begin{align}
\\{ (X-M)\\ {}^{T}\\!E\\} ( {}^{T}\\!EVE)^{-1}\\ {}^{T}\\{ (X-M)\\ {}^{T}\\!E \\} &= \\left(\\begin{array}{cc}
X_{1}-M_{1} & X_{2}-M_{2}
\\end{array}\\right)\\left(\\begin{array}{cc}
V_{11} & O \\\\
O & -V_{21}V_{11}^{-1}V_{12}+V_{22}
\\end{array}\\right)^{-1}\\left(\\begin{array}{c}
{}^{T}(X_{1}-M_{1}) \\\\
{}^{T}(X_{2}-M_{2})
\\end{array}\\right) \\\\
&= (X_{1}-M_{1})V_{11}^{-1}\\ {}^{T}\\!(X_{1}-M_{1}) + (X_{2}-M_{2}^{\\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\\ {}^{T}\\!(X_{2}-M_{2}^{\\prime})
\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u52a0\u3048\u3066\u3001\u5206\u6bcd\u90e8\u5206\u306b\u3042\u308b\\(|U|^{\\frac{1}{2}p}|V|^{\\frac{1}{2}n}\\)\u306b\u3064\u3044\u3066\u3082\u5909\u5f62\u3057\u3066\u304a\u304f\u3068\u3001
\\begin{align}
|U|^{\\frac{1}{2}p}|V|^{\\frac{1}{2}n} &= |U|^{\\frac{1}{2}p}|EV\\ {}^{T}\\!E|^{\\frac{1}{2}n} \\\\
&= |U|^{\\frac{1}{2}(p_{1}+p_{2})}( |V_{11}||V_{22}-V_{21}V_{11}^{-1}V_{12}| )^{\\frac{1}{2}n} \\\\
&= |U|^{\\frac{1}{2}p_{1}}|V_{11}|^{n}\\times |U|^{\\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\\frac{1}{2}n}
\\end{align}<\/div>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u4e0a\u3067\u5909\u5f62\u3057\u3066\u3044\u305f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f
\\begin{align}
f(x) &= \\frac{1}{(2\\pi)^{\\frac{1}{2}np_{1}}|U|^{\\frac{1}{2}p_{1}}|V_{11}|^{\\frac{1}{2}n}}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr}\\left\\{ U^{-1}(X_{1}-M_{1})V_{11}^{-1}\\ {}^{T}\\!(X_{1}-M_{1}) \\right\\} \\right] \\\\
&\\ \\ \\times \\frac{1}{(2\\pi)^{\\frac{1}{2}np_{2}}|U|^{\\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\\frac{1}{2}n}}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr}\\left\\{ U^{-1}(X_{2}-M_{2}^{\\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\\ {}^{T}\\!(X_{2}-M_{2}^{\\prime}) \\right\\} \\right] \\\\
\\end{align}<\/div>\u3068\u5909\u5f62\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\uff1c\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u5b9a\u7fa9\uff1e<\/a>\u304b\u3089\\(X_{1}\\)\u3092\u4e0e\u3048\u305f\u4e0b\u3067\u3001\\(X_{2}\\)\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u3001\u4e0a\u3067\u5909\u5f62\u3057\u3066\u3044\u305f\\(X_{1}\\)\u3068\\(X_{2}\\)\u306e\u540c\u6642\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u3001\\(X_{1}\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u5272\u308b\u3068\u3001
\\begin{align}
f(X_{2}|X_{1}) &= \\frac{1}{(2\\pi)^{\\frac{1}{2}np_{2}}|U|^{\\frac{1}{2}p_{2}}|V_{22}-V_{21}V_{11}^{-1}V_{12}|^{\\frac{1}{2}n}}\\exp\\left[ -\\frac{1}{2}\\mathrm{tr}\\left\\{ U^{-1}(X_{2}-M_{2}^{\\prime})(V_{22}-V_{21}V_{11}^{-1}V_{12})^{-1}\\ {}^{T}\\!(X_{2}-M_{2}^{\\prime}) \\right\\} \\right]
\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u671f\u5f85\u5024\\(M_{2}^{\\prime}M_{2}+(X_{1}-M_{1})V_{11}^{-1}V_{12}\\)\u5206\u6563\\(U\\otimes(V_{22}-V_{21}V_{11}^{-1}V_{12}) \\)\u306e\u884c\u5217\u5909\u91cf\u6b63\u898f\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

\u25a1<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

\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 … <\/p>\n","protected":false},"author":1,"featured_media":618,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[59,71],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/publicdomainq-0011362.jpg","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2710"}],"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=2710"}],"version-history":[{"count":29,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2710\/revisions"}],"predecessor-version":[{"id":2747,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2710\/revisions\/2747"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/618"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2710"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2710"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2710"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}