{"id":6325,"date":"2026-11-02T18:52:29","date_gmt":"2026-11-02T09:52:29","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=6325"},"modified":"2026-11-02T18:52:16","modified_gmt":"2026-11-02T09:52:16","slug":"13-2_rand","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/13-2_rand\/","title":{"rendered":"13.2 \u7a2e\u3005\u306e\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570"},"content":{"rendered":"\n

13.2.1 \u6982\u8981<\/h3>\n

\u4e00\u69d8\u5206\u5e03\u4ee5\u5916\u306e\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u3092\u76f4\u63a5\u751f\u6210\u3059\u308b\u826f\u3044\u65b9\u6cd5\u306f\u307e\u3060\u306a\u304f, \u4e00\u69d8\u4e71\u6570\u3092\u751f\u6210\u3057\u3066\u305d\u308c\u304b\u3089\u9593\u63a5\u7684\u306b\u4ed6\u306e\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u3092\u6c42\u3081\u308b\u306e\u304c\u4e00\u822c\u7684\u3067\u3042\u308b.<\/p>\n

\u3053\u3053\u3067\u306f\u9023\u7d9a\u5206\u5e03\u306e\u5834\u5408\u306b\u3064\u3044\u3066, \u4ee3\u8868\u7684\u306a\u89e3\u6cd5\u3092\u8aac\u660e\u3059\u308b.<\/p>\n

\u96e2\u6563\u5206\u5e03\u3092\u542b\u3080\u8a73\u7d30\u306a\u78ba\u7387\u5206\u5e03\u4e71\u6570\u751f\u6210\u6cd5\u306b\u3064\u3044\u3066\u306f\u6587\u732e [3] \u3092\u53c2\u7167\u3059\u308b\u3068\u3088\u3044.<\/p>\n

13.2.1.2 \u78ba\u7387\u5206\u5e03\u95a2\u6570\u3068\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/h4>\n

\u78ba\u7387\u5909\u6570\u3092 \\(X\\) \u3068\u3059\u308b\u3068, \u5206\u5e03\u95a2\u6570 \\(F(x)\\) \u306f \\(X\\) \u304c \\(x\\) \u3092\u8d85\u3048\u306a\u3044\u78ba\u7387 (\\(P\\)) \u3092\u8868\u3059.
\n\\[
\nF(x) = P(X \\le x)
\n\\]\n\\(F(x)\\) \u306f\u5e83\u7fa9\u5358\u8abf\u5897\u52a0\u95a2\u6570 (\\(x_1 \\lt x_2\\) \u306a\u3089\u3070 \\(F(x_1) \\le F(x_2)\\)) \u3067, \\(0 \\le F(x) \\le 1\\) \u3067\u3042\u308b. <\/p>\n

\u6b21\u5f0f\u3092\u6e80\u305f\u3059\u95a2\u6570 \\(f(x)\\) \u3092\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u3044\u3046. \\(f(x) \\ge 0\\) \u3067\u3042\u308b.
\n\\[
\nF(x) = \\int_{-\\infty}^x f(t) dt
\n\\]\n

\u4f8b. \u6307\u6570\u5206\u5e03<\/h4>\n

\u6307\u6570\u5206\u5e03\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u6b21\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b\u5206\u5e03\u3067\u3042\u308b.
\n\\[
\nf(x) = \\frac{1}{\\theta} exp(-\\frac{x}{\\theta}), \\space (x \\ge 0, \\space \\theta \\gt 0)
\n\\]\n\u3053\u3053\u3067, θ \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3042\u308b. θ = 1 \u306e\u3068\u304d\u3092\u6a19\u6e96\u6307\u6570\u5206\u5e03\u3068\u3044\u3046.<\/p>\n

\u3053\u308c\u3092\u7a4d\u5206\u3057\u3066\u5206\u5e03\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u308b.
\n\\[
\nF(x) = \\int_0^x f(t) dt = 1 – exp(-\\frac{x}{\\theta}), \\space (x \\ge 0, \\space \\theta \\gt 0)
\n\\]\n

\u6a19\u6e96\u6307\u6570\u5206\u5e03\u306e \\(F(x)\\) \u3068 \\(f(x)\\) \u3092\u30b0\u30e9\u30d5\u3067\u8868\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n

\"\"<\/p>\n

13.2.2 \u4e00\u822c\u7684\u306a\u4e71\u6570\u751f\u6210\u6cd5<\/h3>\n

13.2.2.1 \u9006\u95a2\u6570\u6cd5<\/h4>\n

\u5206\u5e03\u95a2\u6570\u306e\u9006\u95a2\u6570 \\(F^{-1}(x)\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306a\u3089\u3070, \\(U\\) \u304c\u4e00\u69d8\u4e71\u6570\u3067\u3042\u308c\u3070\u6b21\u5f0f\u306b\u3088\u308a\u5206\u5e03\u95a2\u6570 \\(F(x)\\) \u3092\u3082\u3064\u4e71\u6570 \\(X\\) \u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
\n\\[
\nX = F^{-1}(U)
\n\\]\n\u5206\u5e03\u95a2\u6570\u306e\u9006\u95a2\u6570\u304c\u5bb9\u6613\u306b\u8a08\u7b97\u3067\u304d\u308b\u306a\u3089, \u3053\u306e\u65b9\u6cd5\u306b\u3088\u308a\u4e00\u69d8\u4e71\u6570\u3092\u4e00\u69d8\u5206\u5e03\u4ee5\u5916\u306e\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d, \u3053\u308c\u3092\u9006\u95a2\u6570\u6cd5\u3068\u3044\u3046.<\/p>\n

\u4f8b. \u6307\u6570\u5206\u5e03\u3078\u306e\u9069\u7528\u4f8b<\/h4>\n

\\(u = F(x)\\) \u3068\u304a\u3044\u3066 \\(x\\) \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\u9006\u95a2\u6570\u304c\u6c42\u3081\u3089\u308c\u308b.
\n\\[
\nF^{-1}(x) = -\\theta ln(1 – u), \\space (0 \\ge u \\gt 1)
\n\\]\n

\u4e00\u69d8\u4e71\u6570 U \u3092\u767a\u751f\u3055\u305b \\(X = -\\theta ln(1 – U)\\) \u3068\u3059\u308b\u3053\u3068\u3067, \u6307\u6570\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570 X \u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u3053\u308c\u3092\u56f3\u793a\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n

\"\"<\/p>\n

13.2.2.2 \u68c4\u5374\u63a1\u629e\u6cd5<\/h4>\n

\u6240\u671b\u306e\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092 \\(f(x)\\) \u3068\u3059\u308b. \u305d\u308c\u3068\u5b9a\u7fa9\u57df\u304c\u540c\u3058\u3067\u5bb9\u6613\u306b\u751f\u6210\u3067\u304d\u308b\u4e71\u6570\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092 \\(g(x)\\) \u3068\u3057, \u5b9a\u6570 c \u304c\u5b9a\u7fa9\u57df\u5185\u3067\u6b21\u5f0f\u3092\u6e80\u305f\u3059\u3082\u306e\u3068\u3059\u308b.
\n\\[
\nf(x) \\le c \\cdot g(x)
\n\\]\n\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u624b\u9806\u306b\u3088\u308a\u5206\u5e03 \\(f(x)\\) \u306b\u5f93\u3046\u4e71\u6570\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d, \u3053\u308c\u3092\u68c4\u5374\u63a1\u629e\u6cd5\u3068\u3044\u3046.<\/p>\n

    \n
  1. \\(g(x)\\) \u306b\u5f93\u3046\u4e71\u6570 x \u3092\u751f\u6210\u3059\u308b<\/li>\n
  2. \u533a\u9593 (0, 1) \u306e\u4e00\u69d8\u4e71\u6570 u \u3092\u751f\u6210\u3059\u308b<\/li>\n
  3. \\(u \\le f(x)\/(c \\cdot g(x))\\) \u3067\u3042\u308c\u3070 x \u3092\u63a1\u7528\u3059\u308b<\/li>\n
  4. \u305d\u3046\u3067\u306a\u3051\u308c\u3070, \u30b9\u30c6\u30c3\u30d7 1 \u304b\u3089\u3084\u308a\u76f4\u3059<\/li>\n<\/ol>\n

    \u305f\u3060\u3057, \u3053\u306e\u65b9\u6cd5\u3067\u306f\u63a1\u7528\u3055\u308c\u308b\u5272\u5408\u304c\u5c0f\u3055\u3044\u3068\u7cbe\u5ea6\u304c\u843d\u3061\u308b\u305f\u3081 \\(g(x)\\) \u306e\u9078\u3073\u65b9\u306b\u306f\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3042\u308b.<\/p>\n

    \u4f8b. \u6307\u6570\u5206\u5e03\u3078\u306e\u9069\u7528\u4f8b<\/h4>\n

    \u6307\u6570\u5206\u5e03\u3092\u8868\u3059\u4e0a\u56f3\u306b\u304a\u3044\u3066, \\(x = 5\\) \u306b\u304a\u3051\u308b \\(F(x)\\) \u306e\u5024\u306f 0.993 \u3068\u5341\u5206\u5927\u304d\u306a\u5024\u3067\u3042\u308b\u304b\u3089, \u5b9a\u7fa9\u57df\u3092 [0. 5] \u3067\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u306b\u3059\u308b. \u305d\u3057\u3066 \\(g(x)\\) \u3068\u3057\u3066 [0, 5] \u306e\u4e00\u69d8\u5206\u5e03\u3092\u4f7f\u3046\u3068, \\(c = 5\\) \u3068\u306a\u308b.<\/p>\n

    \u5f93\u3063\u3066, x \u3068 u \u3092\u751f\u6210\u3057, \\(u \\le f(x)\\) \u3067\u3042\u308c\u3070 x \u3092\u63a1\u7528\u3059\u308c\u3070\u3088\u3044.<\/p>\n

    13.2.2.3 \u5727\u643e\u6cd5<\/h4>\n

    \u68c4\u5374\u63a1\u629e\u6cd5\u3067\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f(x)\\) \u306e\u8a08\u7b97\u304c\u5fc5\u8981\u3067\u3042\u308b\u304c, \u305d\u306e\u8a08\u7b97\u306f\u8907\u96d1\u3067\u6642\u9593\u304c\u304b\u304b\u308b\u3053\u3068\u304c\u3042\u308b\u306e\u3067, \u305d\u308c\u304c\u5fc5\u8981\u306b\u306a\u308b\u983b\u5ea6\u3092\u6e1b\u3089\u3059\u3088\u3046\u306a\u6539\u826f\u3092\u52a0\u3048\u305f\u65b9\u6cd5\u3067\u3042\u308b.<\/p>\n

    \u751f\u6210\u3057\u305f\u3044\u4e71\u6570\u306e \\(f(x)\\) \u306b\u5bfe\u3057\u3066, \u5b9a\u7fa9\u57df\u3067\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3057\u8a08\u7b97\u304c\u5bb9\u6613\u306a\u95a2\u6570 \\(h_1, h_2\\) \u3092\u8003\u3048\u308b.
    \n\\[
    \n0 \\ge h_1(x) \\ge f(x) \\ge h_2(x) \\ge c \\cdot g(x)
    \n\\]\n\u305d\u3057\u3066, \u68c4\u5374\u63a1\u629e\u6cd5\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u4e2d\u306e\u5224\u5b9a \\(u \\le f(x)\/(c \\cdot g(x))\\) \u306e\u4ee3\u308f\u308a\u306b \\(u \\le h_1(x)\/(c \\cdot g(x))\\) \u3092\u6e80\u305f\u305b\u3070 x \u3092\u63a1\u7528\u3059\u308b\u3053\u3068\u306b\u3059\u308b. \u307e\u305f, \\(u \\gt h_2(x)\/(c \\cdot g(x))\\) \u3067\u3042\u308c\u3070 x \u3092\u68c4\u5374\u3059\u308b\u3053\u3068\u306b\u3059\u308b. \u3069\u3061\u3089\u3067\u3082\u306a\u3044\u3068\u304d\u306b\u3060\u3051\u5224\u5b9a \\(u \\le f(x)\/(c \\cdot g(x))\\) \u3092\u884c\u3046\u3053\u3068\u306b\u3059\u308b.<\/p>\n

    \u3053\u306e\u3088\u3046\u306b\u3059\u308b\u3068\u8a08\u7b97\u306b\u6642\u9593\u304c\u304b\u304b\u308b \\(f(x)\\) \u306e\u8a08\u7b97\u983b\u5ea6\u3092\u6e1b\u3089\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n

    13.2.3 \u4ee3\u8868\u7684\u306a\u5206\u5e03\u306e\u4e71\u6570\u751f\u6210\u6cd5<\/h3>\n

    \u3042\u308b\u5206\u5e03\u306e\u4e71\u6570\u3092\u751f\u6210\u3057\u305f\u3044\u3068\u304d\u306b, \u4e00\u822c\u7684\u306a\u4e71\u6570\u751f\u6210\u6cd5\u304c\u9069\u7528\u3067\u304d\u308c\u3070\u3088\u3044\u304c, \u9006\u95a2\u6570\u304c\u6c42\u3081\u3089\u308c\u306a\u3044\u5834\u5408\u3084\u68c4\u5374\u63a1\u629e\u6cd5\u304c\u3046\u307e\u304f\u4f7f\u3048\u306a\u3044\u5834\u5408\u306b\u306f\u4ed6\u306e\u3084\u308a\u65b9\u3092\u5de5\u592b\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b. \u307e\u305f, \u5206\u5e03\u3054\u3068\u306b\u7279\u6709\u306e\u52b9\u7387\u306e\u826f\u3044\u3084\u308a\u65b9\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u5834\u5408\u3082\u3042\u308b.<\/p>\n

    \u4ee5\u4e0b, \u4ee3\u8868\u7684\u306a\u5206\u5e03\u3054\u3068\u306b, \u4f7f\u7528\u3055\u308c\u308b\u4e71\u6570\u751f\u6210\u6cd5\u3092\u8aac\u660e\u3059\u308b.<\/p>\n

    13.2.3.1 \u6b63\u898f\u5206\u5e03<\/h4>\n

    \u6a19\u6e96\u6b63\u898f\u5206\u5e03 (\u5e73\u5747 0, \u6a19\u6e96\u504f\u5dee 1) \u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u6b21\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b.
    \n\\[
    \nf(x) = \\frac{1}{\\sqrt{2\\pi}} exp(\\frac{-x^2}{2})
    \n\\]\n\u5206\u5e03\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.
    \n\\[
    \n\\begin{align}
    \nF(x) & = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^x exp(\\frac{-t^2}{2}) dt \\\\
    \n& = \\frac{1}{2}(1 + erf(\\frac{x}{\\sqrt{2}})) \\\\
    \n\\end{align}
    \n\\]\n\u3053\u3053\u3067, erf() \u306f\u8aa4\u5dee\u95a2\u6570\u3067\u3042\u308b.<\/p>\n

    \u30b0\u30e9\u30d5\u3067\u8868\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n

    \"\"<\/p>\n

    \u3053\u306e\u5206\u5e03\u95a2\u6570\u306e\u9006\u95a2\u6570\u306e\u8a08\u7b97\u306f\u96e3\u3057\u304f, \u9006\u95a2\u6570\u6cd5\u306f\u4f7f\u3048\u306a\u3044. \u305d\u306e\u305f\u3081, \u3044\u304f\u3064\u304b\u306e\u65b9\u6cd5\u304c\u8003\u3048\u3089\u308c\u3066\u3044\u308b.<\/p>\n

    \u306a\u304a, \u6a19\u6e96\u6b63\u898f\u4e71\u6570 (\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570) X (\u5e73\u5747 0, \u6a19\u6e96\u504f\u5dee 1) \u306f, \u6b21\u5f0f\u306b\u3088\u308a\u5e73\u5747 μ, \u6a19\u6e96\u504f\u5dee σ \u306e\u4efb\u610f\u306e\u6b63\u898f\u4e71\u6570 (\u4efb\u610f\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570) Y \u306b\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
    \n\\[
    \n\uff39 = \\mu + \\sigma X
    \n\\]\n\u3059\u306a\u308f\u3061, \u6a19\u6e96\u6b63\u898f\u4e71\u6570\u304c\u751f\u6210\u3067\u304d\u308c\u3070\u3088\u3044.<\/p>\n

    13.2.3.1.1 Box-Muller \u6cd5<\/h4>\n

    \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u7c21\u4fbf\u3067\u3088\u304f\u4f7f\u308f\u308c\u308b\u65b9\u6cd5\u304c Box-Muller \u6cd5\u3067\u3042\u308b. \u3053\u306e\u65b9\u6cd5\u3067\u306f, 2 \u3064\u306e\u4e00\u69d8\u4e71\u6570 \\(U_1, U_2\\) \u3092\u751f\u6210\u3057\u3066\u6b21\u306e\u5909\u63db\u3092\u884c\u3046\u3053\u3068\u306b\u3088\u308a 2 \u3064\u306e\u72ec\u7acb\u3057\u305f\u6a19\u6e96\u6b63\u898f\u4e71\u6570 \\(Y_1, Y_2\\) \u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b. (\u8a73\u7d30\u306f\u6587\u732e [3] \u53c2\u7167)
    \n\\[
    \n\\begin{align}
    \n& Y_1 = \\sqrt{-2 ln(U_1)} cos(2\\pi U_2) \\\\
    \n& Y_2 = \\sqrt{-2 ln(U_1)} sin(2\\pi U_2) \\\\
    \n\\end{align}
    \n\\]\n\u3053\u306e\u65b9\u6cd5\u306b\u3064\u3044\u3066\u5bfe\u6570\u3084\u4e09\u89d2\u95a2\u6570\u306e\u8a08\u7b97\u3092\u907f\u3051\u308b\u306a\u3069\u306e\u6539\u826f\u3092\u884c\u3063\u305f\u3082\u306e\u304c Polar \u6cd5 \u3068\u3088\u3070\u308c\u3066\u3044\u308b. \u306a\u304a, Knuth \u306e\u6587\u732e [1] \u3067\u306f Algorithm P \u3068\u8868\u8a18\u3055\u308c\u3066\u3044\u308b.<\/p>\n

    13.2.3.1.2 Ziggurat \u6cd5<\/h4>\n

    \u68c4\u5374\u63a1\u629e\u6cd5\u306e\u4e00\u7a2e\u3067, \u9ad8\u901f\u3067\u7cbe\u5ea6\u304c\u3088\u3044\u65b9\u6cd5\u3067\u3042\u308b. (\u8a73\u7d30\u306f\u6587\u732e [3] \u53c2\u7167)<\/p>\n

    \u3053\u306e\u65b9\u6cd5\u306f Python \u306e Numpy \u3067\u3082\u63a1\u7528\u3055\u308c\u3066\u3044\u308b.<\/p>\n

    \u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f(x)\\) \u306e\u5de6\u5074\u3092\u8986\u3046 n \u500b\u306e\u7b49\u9762\u7a4d\u306e\u9577\u65b9\u5f62 (\u4e00\u756a\u4e0b\u3060\u3051\u306f\u53f3\u5074\u306e\u88fe\u3092\u542b\u3080) \u3092\u7a4d\u307f\u91cd\u306d\u3066\u5168\u4f53\u3092\u8986\u3046. \u56f3\u306b n = 8 \u306e\u4f8b\u3092\u793a\u3059 (\u5b9f\u7528\u7684\u306b\u306f n = 128, 256 \u306a\u3069\u304c\u4f7f\u308f\u308c\u308b).<\/p>\n

    \"\"<\/p>\n

    \u4e71\u6570\u3092\u767a\u751f\u3055\u305b\u308b\u306b\u306f, \u30e9\u30f3\u30c0\u30e0\u306b 1 \u3064\u306e\u9577\u65b9\u5f62\u3092\u9078\u3073, \u305d\u306e\u4e2d\u3067\u30e9\u30f3\u30c0\u30e0\u306b 1 \u70b9\u3092\u9078\u3073 \\(f(x)\\) \u306e\u4e0b\u306b\u5165\u308c\u3070\u305d\u306e x \u5ea7\u6a19\u3092\u76ee\u7684\u306e\u4e71\u6570\u3068\u3057\u3066\u63a1\u7528\u3059\u308b. \u88fe\u306e\u90e8\u5206\u3092\u542b\u3080\u4e00\u756a\u4e0b\u304c\u9078\u3070\u308c\u305f\u3068\u304d\u306f\u7279\u5225\u306a\u6271\u3044\u304c\u5fc5\u8981\u306b\u306a\u308b.<\/p>\n

    \u3053\u306e\u65b9\u6cd5\u306f x > 0 \u306b\u304a\u3044\u3066 \\(f(x)\\) \u304c\u5358\u8abf\u6e1b\u5c11\u3059\u308b\u5206\u5e03\u306b\u9069\u7528\u3067\u304d\u308b. \u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306f x = 0 \u306b\u3064\u3044\u3066\u5de6\u53f3\u5bfe\u79f0\u3067 x > 0 \u306b\u304a\u3044\u3066\u5358\u8abf\u6e1b\u5c11\u3059\u308b\u306e\u3067, \u3053\u306e\u65b9\u6cd5\u3092 x > 0 \u306b\u304a\u3044\u3066\u9069\u7528\u3057\u534a\u5206\u306e\u78ba\u7387\u3067\u7b26\u53f7\u304c\u8ca0\u306b\u306a\u308b\u3088\u3046\u306b\u3059\u308c\u3070\u3088\u3044.<\/p>\n

    13.2.3.2 \u6307\u6570\u5206\u5e03<\/h4>\n

    13.2.2.1 \u306e\u4f8b\u306b\u793a\u3057\u305f\u3088\u3046\u306b, \u9006\u95a2\u6570\u6cd5 (13.2.2.1) \u3092\u4f7f\u7528\u3057\u3066\u6307\u6570\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570 (\u6307\u6570\u4e71\u6570) \u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n

    \u3057\u304b\u3057, Ziggurat \u6cd5 (13.2.3.1.2) \u3092\u9069\u7528\u3059\u308b\u3053\u3068\u3082\u3067\u304d, \u9006\u95a2\u6570\u6cd5\u3088\u308a\u3082\u9ad8\u901f\u306b\u6307\u6570\u4e71\u6570\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n

    13.2.3.3 \u30ac\u30f3\u30de\u5206\u5e03<\/h4>\n

    \u30ac\u30f3\u30de\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u6b21\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b.
    \n\\[
    \nf(x) = \\frac{\\beta^{-\\alpha} x^{\\alpha -1} e^{-x\/\\beta}}{\\Gamma(\\alpha)} \\space (0 \\lt x \\lt \\infty, \\space \\alpha \\gt 0, \\space \\beta \\gt 0)
    \n\\]\n\u307e\u305f, \u5206\u5e03\u95a2\u6570\u306f\u6b21\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b.
    \n\\[
    \nF(x) = \\frac{\\gamma(\\alpha, x\/\\beta)}{\\Gamma(\\alpha)}
    \n\\]\n\\(\\Gamma()\\) \u306f\u30ac\u30f3\u30de\u95a2\u6570, \\(\\gamma()\\) \u306f\u7b2c1\u7a2e\u4e0d\u5b8c\u5168\u30ac\u30f3\u30de\u95a2\u6570\u3067\u3042\u308b. α \u306f\u5f62\u72b6\u6bcd\u6570, β \u306f\u5c3a\u5ea6\u6bcd\u6570\u3068\u3088\u3070\u308c\u308b\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3042\u308b.<\/p>\n

    \\(\\beta = 1\\) \u306e\u30ac\u30f3\u30de\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f(x)\\) \u304a\u3088\u3073\u5206\u5e03\u95a2\u6570 \\(F(x)\\) \u306e\u30b0\u30e9\u30d5\u3092 α = 0.5, 1, 1.5 \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306b\u793a\u3059.<\/p>\n

    \"\"<\/p>\n

    \"\"<\/p>\n

    \u56f3\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b, \u30ac\u30f3\u30de\u5206\u5e03\u306f α \u306e\u5024 (0 < α < 1, α = 1, α > 1 \u306e 3 \u30d1\u30bf\u30fc\u30f3) \u306b\u3088\u308a\u5206\u5e03\u306e\u5f62\u72b6\u304c\u7570\u306a\u308b.<\/p>\n

    \u30ac\u30f3\u30de\u5206\u5e03\u3092 \\(\\Gamma(\\alpha, \\beta)\\) \u3068\u8868\u8a18\u3059\u308b. \\(\\beta = 1\\) \u306e\u30ac\u30f3\u30de\u5206\u5e03 \\(\\Gamma(\\alpha, 1)\\) \u306f\u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u3068\u3088\u3070\u308c\u308b. \u30ac\u30f3\u30de\u5206\u5e03 \\(\\Gamma(\\alpha, \\beta)\\) \u306b\u5f93\u3046\u78ba\u7387\u5909\u6570 X \u306f\u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570 Y \u304b\u3089\u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
    \n\\[
    \nX = \\beta Y
    \n\\]\n\u5f93\u3063\u3066, \u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03 \\(\\Gamma(\\alpha, 1)\\) \u306b\u5f93\u3046\u4e71\u6570 X \u304c\u751f\u6210\u3067\u304d\u308c\u3070, \u30ac\u30f3\u30de\u5206\u5e03 \\(\\Gamma(\\alpha, \\beta)\\) \u306b\u5f93\u3046\u4e71\u6570 Y \u306b\u5909\u63db\u3067\u304d\u308b.<\/p>\n

    α = 1 \u306e\u3068\u304d, \u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03 \\(\\Gamma(1,1)\\) \u306f\u6a19\u6e96\u6307\u6570\u5206\u5e03\u306b\u4e00\u81f4\u3059\u308b.<\/p>\n

    0 < α < 1 \u306e\u5834\u5408\u306b\u306f, \\(\\Gamma(\\alpha + 1)\\) \u306b\u5f93\u3046\u4e71\u6570 \\(Y_{\\Gamma(\\alpha+1,1)}\\) \u3068\u305d\u308c\u3068\u306f\u72ec\u7acb\u306a (0, 1) \u306b\u304a\u3051\u308b\u4e00\u69d8\u4e71\u6570 \\(U\\) \u304b\u3089, \u6b21\u5f0f\u306b\u3088\u308a\u5fc5\u8981\u306a\u4e71\u6570\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
    \n\\[
    \nY_{\\Gamma(\\alpha,1)} = Y_{\\Gamma(\\alpha+1,1)} U^{1\/\\alpha}
    \n\\]\n

    \u4ee5\u4e0a\u304b\u3089, α > 1 \u306e\u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u304c\u751f\u6210\u3067\u304d\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308b.<\/p>\n

    13.2.3.3.1 \u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u306e\u751f\u6210<\/h4>\n

    \u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304b\u3089\u9006\u95a2\u6570\u3092\u6c42\u3081\u308b\u306e\u306f\u96e3\u3057\u3044\u306e\u3067, \u68c4\u5374\u63a1\u629e\u6cd5\u304c\u4f7f\u7528\u3055\u308c\u308b. \u591a\u304f\u306e\u6587\u732e\u3067\u751f\u6210\u6cd5\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b\u304c, \u6587\u732e [3] \u306b\u8a73\u3057\u304f\u89e3\u8aac\u3055\u308c\u3066\u3044\u308b α > 1 \u306e\u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u3092\u751f\u6210\u3059\u308b\u65b9\u6cd5 (\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0 3.60) \u3092\u7c21\u5358\u306b\u7d39\u4ecb\u3059\u308b. \u3053\u306e\u65b9\u6cd5\u306f XLPack \u3067\u63a1\u7528\u3057\u3066\u3044\u308b.<\/p>\n

    \u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f_{\\Gamma(\\alpha, 1)}(x)\\) \u306b\u5909\u6570\u5909\u63db \\(Y = h(Z)\\) \u3092\u65bd\u3059\u3068\u6b21\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f_Z\\) \u304c\u5f97\u3089\u308c\u308b.
    \n\\[
    \nf_Z(z) = f_{\\Gamma(\\alpha, 1)}(h(z))|h'(z)| = \\frac{(h(z))^{\\alpha-1}e^{-h(z)}|h'(z)|}{\\Gamma(\\alpha)}
    \n\\]\n\u5909\u63db h \u3068\u3057\u3066\u6b21\u5f0f\u3092\u63a1\u7528\u3059\u308b.
    \n\\[
    \nh(z) = d(1 + cz)^k, \\space (-1\/c \\lt z \\lt \\infty, d \\gt 0, c \\gt 0, k \\in N)
    \n\\]\n\u3053\u3053\u3067, \\(f_Z\\) \u304c\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f_{N(0,1)}(z) = (1\/\\sqrt{2\\pi})e^{-z^2\/2}\\) \u306b\u8fd1\u304f\u306a\u308b\u3088\u3046\u306b d, c, k \u3092\u6c7a\u3081\u308b. \u5177\u4f53\u7684\u306b\u306f, \\(d = \\alpha – 1\/3\\), \\(c = 1\/\\sqrt{9d}\\), k = 3 \u3068\u3059\u308b.<\/p>\n

    \u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570 Z \u3092\u751f\u6210\u3057\u3066\u68c4\u5374\u63a1\u629e\u6cd5\u3092\u9069\u7528\u3059\u308b. Z \u306f Ziggurat \u6cd5\u306a\u3069\u3067\u9ad8\u901f\u306b\u751f\u6210\u3067\u304d\u308b. \u68c4\u5374\u63a1\u629e\u306f, \\(Z \\gt -1\/c\\) \u306e\u3068\u304d\u306b, Z \u306b\u72ec\u7acb\u306a\u4e00\u69d8\u4e71\u6570 u \u304c \\(u e^{-Z^2\/2} \\le e^{g(Z)}\\) \u3092\u6e80\u305f\u3059\u304b\u3069\u3046\u304b\u3067\u5224\u5b9a\u3059\u308b. \u305f\u3060\u3057, \\(g(z) = d ln(1 + cz)^3 – d(a + cz)^3 + d\\) \u3067\u3042\u308b. \u3053\u306e\u3088\u3046\u306b\u3059\u308b\u3068, \u63a1\u629e\u7387\u304c\u975e\u5e38\u306b\u9ad8\u304f\u306a\u308a\u52b9\u7387\u304c\u3088\u3044.<\/p>\n

    \u3055\u3089\u306b, \\(g(z)\\) \u306e\u8a08\u7b97\u306b\u542b\u307e\u308c\u308b\u5bfe\u6570\u8a08\u7b97\u3092\u6e1b\u3089\u3059\u305f\u3081 \\(s(z) \\le e^{g(z)} \\le e^{-z^2\/2}\\) \u3068\u3057\u3066\u5727\u643e\u6cd5\u3092\u9069\u7528\u3059\u308b. \u3053\u3053\u3067, \\(s(z) = (1 – 0.0331z^4)e^{-z^2\/2}\\) \u304c\u4f7f\u7528\u3055\u308c\u308b.<\/p>\n

    13.2.4 \u7d71\u8a08\u7684\u691c\u5b9a<\/h3>\n

    \u3053\u3053\u306b n \u500b\u306e\u751f\u6210\u3055\u308c\u305f\u4e71\u6570 \\(X_1, X_2, \\dots, X_n\\) \u306e\u30b5\u30f3\u30d7\u30eb\u304c\u3042\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u308c\u304c\u3042\u308b\u78ba\u7387\u5206\u5e03\u306b\u5f93\u3046\u3068\u307f\u306a\u305b\u308b\u306e\u304b\u3069\u3046\u304b, \u305d\u306e\u9069\u5408\u5ea6\u3092\u691c\u5b9a\u3059\u308b\u65b9\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3059\u308b.<\/p>\n

    13.2.4.1 Kolmogorov-Smirnov \u691c\u5b9a (KS \u691c\u5b9a)<\/h4>\n

    \u9023\u7d9a\u5206\u5e03\u306e\u5834\u5408 (\u4f8b\u3048\u3070, 0 \u304b\u3089 1 \u306e\u5b9f\u6570\u4e71\u6570\u306e\u5834\u5408) \u306b\u306f Kolmogorov-Smirnov \u691c\u5b9a (KS \u691c\u5b9a) \u304c\u7528\u3044\u3089\u308c\u308b.<\/p>\n

    \u6b21\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b \\(F_n(x)\\) \u3092\u30b5\u30f3\u30d7\u30eb\u306e\u7d4c\u9a13\u5206\u5e03\u95a2\u6570\u3068\u3044\u3046.
    \n\\[
    \nF_n(x) = \\frac{1}{n} (x \u4ee5\u4e0b\u306e X_i \u306e\u6570)
    \n\\]\n\u3053\u308c\u3092\u7528\u3044\u3066\u6b21\u306e\u7d71\u8a08\u91cf\u3092\u8a08\u7b97\u3059\u308b.
    \n\\[
    \n\\begin{align}
    \n& K_n^+ = \\sqrt{n} \\space max_{-\\infty \\lt x \\lt +\\infty} (F_n(x) – F(x)) \\\\
    \n& K_n^- = \\sqrt{n} \\space max_{-\\infty \\lt x \\lt +\\infty} (F(x) – F_n(x)) \\\\
    \n\\end{align}
    \n\\]\n\u3053\u3053\u3067, \\(F_n(x)\\) \u306f \\(X_i\\) \u3092\u6607\u9806\u306b\u4e26\u3079\u66ff\u3048\u3066\u304b\u3089\u305d\u306e\u500b\u6570\u3092\u5c0f\u3055\u3044\u65b9\u304b\u3089\u6570\u3048\u3066\u7b97\u51fa\u3059\u308b\u306e\u3067, \u3053\u306e\u8a08\u7b97\u306f \\(X_i\\) \u304c\u6607\u9806\u306b\u4e26\u3093\u3067\u3044\u308b\u3068\u3057\u3066\u6b21\u306e\u3088\u3046\u306b\u7c21\u5358\u306a\u5f62\u306b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
    \n\\[
    \n\\begin{align}
    \n& K_n^+ = \\sqrt{n} \\space max_{1 \\le j \\le n} (\\frac{j}{n} – F(X_j)) \\\\
    \n& K_n^- = \\sqrt{n} \\space max_{1 \\le j \\le n} (F(X_j) – \\frac{j – 1}{n}) \\\\
    \n\\end{align}
    \n\\]\n\\(K_n^+\\) \u3068 \\(K_n^-\\) \u306e\u5024\u304c\u5c0f\u3055\u3044\u307b\u3069\u30b5\u30f3\u30d7\u30eb\u304c \\(F(x)\\) \u306e\u5206\u5e03\u306b\u5f93\u3046\u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u308b. \u305d\u306e\u5177\u4f53\u7684\u306a\u5024\u306f, \u5c0f\u3055\u3044 n \u306b\u3064\u3044\u3066\u306f, \u6587\u732e [1] \u306b\u8868\u304c\u793a\u3055\u308c\u3066\u3044\u308b. n \u304c\u5927\u304d\u3044\u5834\u5408\u306b\u306f\u6b21\u5f0f\u306b\u3088\u308a\u8fd1\u4f3c\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
    \n\\[
    \nK_n^+ = K_n^- = y_p – \\frac{1}{6}n^{-1\/2} + O(1\/n), \\space \u305f\u3060\u3057 \\space y_p^2 = \\frac{1}{2}ln(1\/(1 – p))
    \n\\]\np \u306f\u6709\u610f\u6c34\u6e96\u3092\u8868\u3059.\u3044\u304f\u3064\u304b\u306e p \u306b\u3064\u3044\u3066 \\(y_p\\) \u306e\u5024\u3092\u6c42\u3081\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n\n\n\n
    1 %<\/td>\n5 %<\/td>\n10 %<\/td>\n25 %<\/td>\n<\/tr>\n
    0.07089<\/td>\n0.1601<\/td>\n0.2295<\/td>\n0.3793<\/td>\n<\/tr>\n<\/table>\n

    \u4f8b. \u6307\u6570\u5206\u5e03<\/h4>\n

    XLPack \u306e GenrandExp() (Ziggurat \u6cd5, \u30e1\u30eb\u30bb\u30f3\u30cc\u30c4\u30a4\u30b9\u30bf\u30fc) \u3092\u4f7f\u3063\u3066 10,000 \u500b\u306e\u6a19\u6e96\u6307\u6570\u4e71\u6570\u3092\u751f\u6210\u3057, \u305d\u306e \\(F_n(x)\\) \u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n

    \"\"<\/p>\n

    KS \u691c\u5b9a\u306e\u5024\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.
    \n\\[
    \nK_n^+ = 0.00582, \\space K_n^- = 0.03413
    \n\\]\n\u3053\u308c\u306f, \u6709\u610f\u6c34\u6e96\u3092 1 % \u3088\u308a\u304b\u306a\u308a\u5c0f\u3055\u304f\u3068\u3063\u3066\u3082\u68c4\u5374\u3055\u308c\u306a\u3044\u5024\u3067\u3042\u308b.<\/p>\n

    \u4f8b. \u6b63\u898f\u5206\u5e03<\/h4>\n

    XLPack \u306e GenrandNorm() (Ziggurat \u6cd5, \u30e1\u30eb\u30bb\u30f3\u30cc\u30c4\u30a4\u30b9\u30bf\u30fc) \u3092\u4f7f\u3063\u3066 10,000 \u500b\u306e\u6a19\u6e96\u6b63\u898f\u4e71\u6570\u3092\u751f\u6210\u3057, \u305d\u306e \\(F_n(x)\\) \u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n

    \"\"<\/p>\n

    KS \u691c\u5b9a\u306e\u5024\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.
    \n\\[
    \nK_n^+ = 0.00984, \\space K_n^- = 0.01929
    \n\\]\n

    \u4f8b. \u30ac\u30f3\u30de\u5206\u5e03<\/h4>\n

    XLPack \u306e GenrandGam() (\u5727\u643e\u6cd5, \u30e1\u30eb\u30bb\u30f3\u30cc\u30c4\u30a4\u30b9\u30bf\u30fc) \u3092\u4f7f\u3063\u3066 10,000 \u500b\u306e\u6a19\u6e96\u30ac\u30f3\u30de\u4e71\u6570 (α = 1.5) \u3092\u751f\u6210\u3057, \u305d\u306e \\(F_n(x)\\) \u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n

    \"\"<\/p>\n

    KS \u691c\u5b9a\u306e\u5024\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.
    \n\\[
    \nK_n^+ = 0.01150, \\space K_n^- = 0.00666
    \n\\]\n","protected":false},"excerpt":{"rendered":"

    13.2.1 \u6982\u8981 \u4e00\u69d8\u5206\u5e03\u4ee5\u5916\u306e\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u3092\u76f4\u63a5\u751f\u6210\u3059\u308b\u826f\u3044\u65b9\u6cd5\u306f\u307e\u3060\u306a\u304f, \u4e00\u69d8\u4e71\u6570\u3092\u751f\u6210\u3057\u3066\u305d\u308c\u304b\u3089\u9593\u63a5\u7684\u306b\u4ed6\u306e\u5206\u5e03\u306b\u5f93\u3046\u4e71\u6570\u3092\u6c42\u3081\u308b\u306e\u304c\u4e00\u822c\u7684\u3067\u3042\u308b. \u3053\u3053\u3067\u306f\u9023\u7d9a\u5206\u5e03\u306e\u5834\u5408\u306b\u3064\u3044\u3066, \u4ee3\u8868\u7684\u306a\u89e3\u6cd5\u3092\u8aac\u660e\u3059\u308b […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-6325","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t