{"id":5735,"date":"2019-12-08T20:02:41","date_gmt":"2019-12-08T11:02:41","guid":{"rendered":"https:\/\/pandanote.info\/?p=5735"},"modified":"2022-08-07T12:44:15","modified_gmt":"2022-08-07T03:44:15","slug":"%e6%95%b4%e6%95%b0%e6%ac%a1%e3%81%ae%e5%a4%89%e5%bd%a2bessel%e9%96%a2%e6%95%b0%e3%81%ae%e6%af%8d%e9%96%a2%e6%95%b0%e3%81%a8von-mises%e5%88%86%e5%b8%83%e3%80%82","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=5735","title":{"rendered":"\u6574\u6570\u6b21\u306e\u5909\u5f62Bessel\u95a2\u6570\u306e\u6bcd\u95a2\u6570\u3068Von Mises\u5206\u5e03\u306e\u95a2\u4fc2\u3002"},"content":{"rendered":"<h2>\u306f\u3058\u3081\u306b<\/h2>\n<p>\u78ba\u7387\u5909\u6570\u306e\u5909\u6570\u5909\u63db\u306e\u65b9\u6cd5\u3092\u8abf\u3079\u3066\u3044\u308b\u3046\u3061\u306b\u3001Python3\u306e\u95a2\u6570\u3067\u78ba\u7387\u5bc6\u5ea6\u5206\u5e03\u304cVon Mises\u5206\u5e03\u306b\u306a\u308b\u4e71\u6570\u3092\u751f\u6210\u3059\u308b\u95a2\u6570\u306a\u308b\u3082\u306e\u3092\u898b\u3064\u3051\u3066\u3057\u307e\u3044\u3001\u3055\u3089\u306bVon Mises\u5206\u5e03\u306e\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3092\u8a08\u7b97\u3059\u308b\u969b\u306b(\u7b2c1\u7a2e)\u5909\u5f62Bessel\u95a2\u6570\u304c\u3069\u3053\u304b\u3089\u3068\u3082\u306a\u304f\u767b\u5834\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\u3001\u306a\u3093\u3067\u305d\u3046\u306a\u308b\u306e\u304b\u306b\u3064\u3044\u3066\u306e\u52c9\u5f37\u3082\u517c\u306d\u3066\u3001\u6574\u6570\u6b21\u306e(\u7b2c1\u7a2e)\u5909\u5f62Bessel\u95a2\u6570\u304b\u3089\u306a\u308b\u95a2\u6570\u5217\u306e\u6bcd\u95a2\u6570\u3092Taylor\u5c55\u958b\u3057\u3001Von Mises\u5206\u5e03\u3068\u306e\u95a2\u4fc2\u3092\u63a2\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n<h2>\u6bcd\u95a2\u6570\u306eTaylor\u5c55\u958b<\/h2>\n<h3>\u3068\u308a\u3042\u3048\u305a\u5c55\u958b\u3002<\/h3>\n<p>\u4ee5\u4e0b\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001(\u7b2c1\u7a2e)\u5909\u5f62Bessel\u95a2\u6570\u3092\u5358\u306b\u300c\u5909\u5f62Bessel\u95a2\u6570\u300d\u3068\u66f8\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n<p>$k$\u3092\u6574\u6570\u3068\u3057\u3066\u3001$k$\u6b21\u306e\u5909\u5f62Bessel\u95a2\u6570\u306e\u95a2\u6570\u5217$I_k(x)$\u306e\u6bcd\u95a2\u6570$G(x;w)$\u306f(\\ref{eq:besselgenerate})\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n<p>\u6700\u521d\u306f\u300c\u6bcd\u95a2\u6570\u300d\u304c\u300c\u6bcd\u8266\u6570\u300d\u3068\u304b\u5909\u63db\u3055\u308c\u3066\u3057\u307e\u3063\u305f\u4e0a\u306b\u305d\u308c\u304c\u8f9e\u66f8\u306b\u5b66\u7fd2\u3055\u308c\u3066\u3057\u307e\u3063\u3066\u3001\u306a\u304b\u306a\u304b\u5909\u63db\u3067\u304d\u307e\u305b\u3093\u3067\u3057\u305f\u3002\u56f0\u3063\u305f\u3082\u306e\u3067\u3059\u3002\ud83e\udd14<br \/>\n\\begin{align}<br \/>\nG(x;w) &#038;= e^{\\frac{x}{2}(w+\\frac{1}{w})}\u3000\\label{eq:besselgenerate}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:besselgenerate})\u5f0f\u306e\u53f3\u8fba\u3092$w$\u306b\u3064\u3044\u3066Taylor\u5c55\u958b\u3057\u307e\u3059\u304c\u3001\u3044\u304d\u306a\u308a\u5c55\u958b\u305b\u305a\u306b\u3001<br \/>\n\\begin{align}<br \/>\nG(x;w) &#038;= e^{\\frac{xw}{2}}\\cdot e^{\\frac{x}{2w}} \\nonumber\\cr<br \/>\n&#038;= \\left[ \\sum_{n=0}^{\\infty} \\frac{1}{n!}\\left(\\frac{xw}{2}\\right)^n\\right]\\left[ \\sum_{m=0}^{\\infty} \\frac{1}{m!}\\left(\\frac{x}{2w}\\right)^m\\right] \\nonumber\\cr<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\sum_{m=0}^{\\infty} \\frac{1}{n!m!} \\left(\\frac{x}{2}\\right)^{n+m} w^{n-m} \\label{eq:firstform}<br \/>\n\\end{align}<br \/>\n\u3068\u5c55\u958b\u3057\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:firstform})\u5f0f\u53f3\u8fba\u306e\u9805\u306e\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u9806\u5e8f(\u30a4\u30e1\u30fc\u30b8)\u3092\u56f3\u306b\u3057\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n<p><a href=\"https:\/\/pandanote.info\/?attachment_id=5743\" rel=\"attachment wp-att-5743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pandanote.info\/wordpress\/wp-content\/uploads\/2019\/12\/modified_bessel_function_scene1.svg\" alt=\"\" width=\"300\" height=\"300\" class=\"alignnone size-medium wp-image-5743\" \/><\/a><\/p>\n<p>\u3044\u3063\u305f\u3093$n=0$\u306b\u56fa\u5b9a\u3057\u3066$m$\u3092\u4e0a\u56f3\u306e(a)\u306e\u65b9\u5411\u306b\u79fb\u52d5\u3055\u305b\u3066\u5404\u9805\u306e\u5024\u3092\u8a08\u7b97\u3057\u3066\u305d\u306e\u548c\u3092\u6c42\u3081\u3001\u6b21\u306b$n$\u3092\u4e0a\u56f3\u306e(b)\u306e\u65b9\u5411\u306b+1\u79fb\u52d5\u3055\u305b\u3066\u304b\u3089\u518d\u5ea6\u4e0a\u56f3\u306e(a)\u306e\u65b9\u5411\u306b\u79fb\u52d5\u3055\u305b\u3066\u5404\u9805\u306e\u5024\u3092\u6c42\u3081\u308b\u2026 \u3068\u3044\u3046\u624b\u9806\u3067\u8a08\u7b97\u3092\u884c\u3046\u3053\u3068\u3092(\\ref{eq:firstform})\u5f0f\u306e\u53f3\u8fba\u306f\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<h3>\u3061\u3087\u3063\u3068\u5909\u5f62\u3002<\/h3>\n<p>\u6b21\u306b\u3001(\\ref{eq:firstform})\u5f0f\u306e\u53f3\u8fba\u306e$w$\u306e\u5404\u9805\u306e\u4fc2\u6570\u3092\u8abf\u3079\u308b\u305f\u3081\u3001$n-m=k$\u3068\u304a\u3044\u3066(\\ref{eq:firstform})\u5f0f\u306e\u53f3\u8fba\u306b\u4ee3\u5165\u3057\u3001$m$\u3092\u6d88\u53bb\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u3001$k$\u306f\u3059\u3079\u3066\u306e\u6574\u6570\u5024\u3092\u3068\u308a\u5f97\u307e\u3059\u304c\u3001$n$\u306e\u5024\u306b\u3064\u3044\u3066\u306f$n$\u53ca\u3073$n+k(=m)$\u304c\u8ca0\u3067\u306a\u3044\u6574\u6570\u5024\u306e\u307f\u3092\u3068\u308a\u5f97\u308b\u3053\u3068\u3001\u3059\u306a\u308f\u3061$n = \\max(k,0)$\u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n<p>\u307e\u305f\u3001$n+k$\u304c\u6574\u6570\u306e\u5834\u5408\u306b\u306f$(n+k)! = \\Gamma(n+k+1)$\u3067\u3042\u308b\u306e\u3067\u3001\u3053\u308c\u3082\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n<p>\u3059\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\nG(x;w) &#038;= \\sum_{k=-\\infty}^{\\infty} \\sum_{n=\\max(k,0)}^{\\infty} \\frac{1}{n!\\Gamma(n+k+1)} \\left(\\frac{x}{2}\\right)^{2n+k} w^{k} \\label{eq:secondform}<br \/>\n\\end{align}<\/p>\n<p>(\\ref{eq:secondform})\u5f0f\u306e\u53f3\u8fba\u306e\u8a08\u7b97\u306e\u9806\u5e8f(\u30a4\u30e1\u30fc\u30b8)\u3092\u56f3\u306b\u3057\u3066\u307f\u307e\u3057\u305f\u2193<\/p>\n<p><a href=\"https:\/\/pandanote.info\/?attachment_id=5749\" rel=\"attachment wp-att-5749\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pandanote.info\/wordpress\/wp-content\/uploads\/2019\/12\/modified_bessel_function_scene2.svg\" alt=\"\" width=\"300\" height=\"300\" class=\"alignnone size-medium wp-image-5749\" \/><\/a><\/p>\n<p>\u307e\u305a\u3001$k$\u3092\u56fa\u5b9a\u3057\u3066$n$\u306e\u5024\u3092\u4e0a\u56f3\u306e(a)\u306e\u3088\u3046\u306b\u79fb\u52d5\u3055\u305b\u3001\u5bfe\u5fdc\u3059\u308b\u5404\u9805\u306e\u8a08\u7b97\u3092\u884c\u3044\u307e\u3059\u3002\u4e0a\u56f3\u306e$n = m+k$\u306e\u76f4\u7dda\u72b6\u306e\u683c\u5b50\u70b9\u306b\u5bfe\u5fdc\u3059\u308b$(m,n)=(n-k,n)$\u306e\u5024\u306b\u5bfe\u5fdc\u3059\u308b\u9805\u306e\u8a08\u7b97\u3092\u884c\u3044\u3001\u305d\u306e\u7dcf\u548c\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002\u6b21\u306b\u3001$k$\u306e\u5024\u3092\u79fb\u52d5(\u5897\u52a0)\u3055\u305b\u308b\u3068\u76f4\u7dda$n = m+k$\u304c\u4e0a\u56f3\u306e(b)\u306e\u5411\u304d\u306b\u79fb\u52d5\u3059\u308b\u306e\u3067\u3001\u79fb\u52d5\u3057\u305f\u5148\u306e\u76f4\u7dda\u4e0a\u306b\u3042\u308b\u683c\u5b50\u70b9\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306e\u8a08\u7b97\u3092\u884c\u3046\u3068(\\ref{eq:secondform})\u5f0f\u306e\u53f3\u8fba\u306e\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u5bf8\u6cd5\u3067\u3059\u3002<\/p>\n<h3>\u5909\u6570\u306e\u7f6e\u304d\u63db\u3048\u3002<\/h3>\n<p>(\\ref{eq:secondform})\u5f0f\u306e\u53f3\u8fba\u306b\u5909\u5f62Bessel\u95a2\u6570\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u767b\u5834\u3057\u307e\u3059\u304c\u3001\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u7bc4\u56f2\u304c\u5fae\u5999\u306b\u7570\u306a\u308a\u307e\u3059\u3002\u7279\u306b$n$\u306e\u7bc4\u56f2\u304c$k$\u306e\u5024\u306b\u4f9d\u5b58\u3057\u3066\u3044\u308b\u90e8\u5206\u306b\u3064\u3044\u3066\u306f\u306a\u3093\u3068\u304b\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u307e\u3059\u3002<\/p>\n<p>\u5177\u4f53\u7684\u306b\u306f\u524d\u7bc0\u306e\u56f3\u306e\u767d\u629c\u304d\u306e\u9752\u4e38\u306e\u683c\u5b50\u70b9(\u4ee5\u4e0b\u3001\u3053\u306e\u7bc0\u3067\u306f\u5358\u306b\u300c\u767d\u629c\u304d\u306e\u9752\u4e38\u306e\u683c\u5b50\u70b9\u300d\u3068\u66f8\u304d\u307e\u3059\u3002)\u3092\u8a08\u7b97\u306e\u5bfe\u8c61\u306b\u542b\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070\u3001\u6700\u521d\u306e\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u7bc4\u56f2\u3092$n \\ge 0$\u306e\u6574\u6570\u306b\u3067\u304d\u307e\u3059\u306e\u3067\u3001\u30b4\u30fc\u30eb\u306b\u8fd1\u3065\u304f\u3053\u3068\u304c\u3067\u304d\u305d\u3046\u3067\u3059\u3002<\/p>\n<p>\u767d\u629c\u304d\u306e\u9752\u4e38\u306e\u683c\u5b50\u70b9\u306e$m(=n+k)$\u53ca\u3073$n$\u306b\u3064\u3044\u3066\u306f$m \\lt 0, n \\gt 0$\u3068\u306a\u308a\u307e\u3059\u3002$m \\lt 0$\u306e\u5834\u5408\u306b\u306f\u3001$\\displaystyle\\lim_{x \\to m+0}{\\Gamma(x)}$\u53ca\u3073$\\displaystyle\\lim_{x \\to m-0}{\\Gamma(x)}$\u306f\u6b63\u8ca0\u3069\u3061\u3089\u304b\u306e$\\infty$\u306b\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u305d\u306e\u9006\u6570\u306f\u2026<br \/>\n\\begin{align}<br \/>\n\\frac{1}{\\Gamma(m)} &#038;= \\frac{1}{\\Gamma(n+k)} \\nonumber\\cr<br \/>\n&#038;= 0 \\label{eq:gammam}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001\u767d\u629c\u304d\u306e\u9752\u4e38\u306e\u683c\u5b50\u70b9\u306b\u304a\u3044\u3066\u306f\u3001(\\ref{eq:gammamzero})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\n\\frac{1}{n!\\Gamma(n+k+1)} \\left(\\frac{x}{2}\\right)^{2n+k} &#038;= 0 \\label{eq:gammamzero}<br \/>\n\\end{align}<\/p>\n<p>\u3088\u3063\u3066\u3001$m \\lt 0, n \\gt 0$\u306e\u5834\u5408\u3082$n$\u306b\u3064\u3044\u3066\u306e\u7dcf\u548c\u306e\u8a08\u7b97\u306b\u542b\u3081\u3066\u3082\u8a08\u7b97\u306e\u7d50\u679c\u306f\u5909\u308f\u308a\u307e\u305b\u3093\u306e\u3067\u3001\u8a08\u7b97\u306e\u5bfe\u8c61\u306b\u8ffd\u52a0\u3057\u3066\u3057\u307e\u3046\u3053\u3068\u306b\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001(\\ref{eq:secondform})\u5f0f\u306e$n$\u306b\u3064\u3044\u3066\u306e\u7dcf\u548c\u306e\u8a08\u7b97\u306f$k$\u306e\u5024\u306b\u95a2\u4fc2\u306a\u304f\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u3066\u3001<\/p>\n<p>\\begin{align}<br \/>\nG(x;w) &#038;= \\sum_{k=-\\infty}^{\\infty} \\sum_{n=0}^{\\infty} \\frac{1}{n!\\Gamma(n+k+1)} \\left(\\frac{x}{2}\\right)^{2n+k} w^{k}<br \/>\n&#038;= \\sum_{k=-\\infty}^{\\infty}I_k(x) w^k\\label{eq:thirdform}<br \/>\n\\end{align}<\/p>\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3053\u3053\u307e\u3067\u306e\u8a08\u7b97\u3067\u3001$G(x;w)$\u304c$k$\u6b21\u306e\u5909\u5f62Bessel\u95a2\u6570\u306e\u95a2\u6570\u5217$I_k(x)$\u306e\u6bcd\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002$\\blacksquare$<\/p>\n<h2>\u78ba\u7387\u5206\u5e03\u306e\u8a08\u7b97\u3078\u306e\u5fdc\u7528<\/h2>\n<p>(\\ref{eq:thirdform})\u5f0f\u306e$w$\u3092$w = e^{i\\theta}$\u3068\u7f6e\u304d\u307e\u3059\u3002\u3059\u308b\u3068\u3001(\\ref{eq:thirdform})\u5f0f\u306e\u5de6\u8fba\u306f\u3001<br \/>\n\\begin{align}<br \/>\ne^{\\frac{x}{2}(e^{i\\theta}+\\frac{1}{e^{i\\theta}})} &#038;= e^{\\frac{x}{2}(e^{i\\theta}+e^{-i\\theta})} \\nonumber\\cr<br \/>\n&#038;= e^{x\\cos\\theta} \\label{eq:eitheta}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:thirdform})\u5f0f\u306e\u53f3\u8fba\u306f\u3001$k$\u304c\u8ca0\u3067\u306a\u3044\u6574\u6570\u306e\u3068\u304d\u306b\u306f$I_k=I_{-k}$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304b\u3089\u3001<br \/>\n\\begin{align}<br \/>\n\\sum_{k=-\\infty}^{\\infty}I_k(x) e^{ik\\theta} &#038;= I_0(x) + \\sum_{k=1}^{\\infty}I_k(x) e^{ik\\theta} + \\sum_{k=-\\infty}^{-1}I_k(x) e^{ik\\theta} \\nonumber\\cr<br \/>\n&#038;= I_0(x) + \\sum_{k=1}^{\\infty}I_k(x) e^{ik\\theta} + \\sum_{k=1}^{\\infty}I_{-k}(x) e^{-ik\\theta} \\nonumber\\cr<br \/>\n&#038;= I_0(x) + \\sum_{k=1}^{\\infty}I_k(x) e^{ik\\theta} + \\sum_{k=1}^{\\infty}I_k(x) e^{-ik\\theta} \\nonumber\\cr<br \/>\n&#038;= I_0(x) + \\sum_{k=1}^{\\infty}I_k(x)(e^{ik\\theta} + e^{-ik\\theta}) \\nonumber\\cr<br \/>\n&#038;= I_0(x) + 2\\sum_{k=1}^{\\infty}I_k(x)\\cos k\\theta \\label{eq:cosktheta}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066(\\ref{eq:eitheta})\u5f0f\u53ca\u3073(\\ref{eq:cosktheta})\u5f0f\u3092\u307e\u3068\u3081\u308b\u3068(\\ref{eq:coscos})\u5f0f\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\ne^{x\\cos\\theta} &#038;= I_0(x) + 2\\sum_{k=1}^{\\infty}I_k(x)\\cos k\\theta \\label{eq:coscos}<br \/>\n\\end{align}<br \/>\n\u307e\u305f\u3001$\\theta$\u3092$\\theta &#8211; \\mu$\u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001<br \/>\n\\begin{align}<br \/>\ne^{x\\cos(\\theta &#8211; \\mu)} &#038;= I_0(x) + 2\\sum_{k=1}^{\\infty}I_k(x)\\cos k(\\theta &#8211; \\mu) \\label{eq:coscosmu}<br \/>\n\\end{align}<br \/>\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3082\u5c0e\u3051\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:coscosmu})\u5f0f\u306e\u53f3\u8fba\u306f\u3001\u5e73\u5747$\\mu$\u306evon Mises\u5206\u5e03(\u78ba\u7387\u5909\u6570\u306f$\\theta$\u306b\u306a\u308a\u307e\u3059\u3002)<br \/>\n\\begin{align}<br \/>\np(\\theta | \\mu, x) &#038;= \\frac{e^{x\\cos(\\theta &#8211; \\mu)}}{2\\pi I_0(x)} \\label{eq:vonMises}<br \/>\n\\end{align}<br \/>\n\u306e\u5206\u5b50\u3068\u4e00\u81f4\u3057\u307e\u3059\u306e\u3067\u3001(\\ref{eq:vonMises})\u5f0f\u306f(\\ref{eq:vonMisessecond})\u5f0f\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\np(\\theta | \\mu, x) &#038;= \\frac{1}{2\\pi}\\left[ 1 + 2\\sum_{k=1}^{\\infty}\\frac{I_k(x)}{I_0(k)}\\cos k(\\theta &#8211; \\mu) \\right] \\label{eq:vonMisessecond}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:vonMisessecond})\u5f0f\u306f\u4e0d\u5b9a\u7a4d\u5206\u304c\u8a08\u7b97\u3067\u304d\u3066\u3001\u7a4d\u5206\u5b9a\u6570\u4ee5\u5916\u306e\u9805\u306b\u3064\u3044\u3066\u306f(\\ref{eq:vonMisesint})\u5f0f\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\nP(\\theta | \\mu, x) &#038;= \\frac{1}{2\\pi}\\left[ \\theta + 2\\sum_{k=1}^{\\infty}\\frac{I_k(x)}{I_0(k)}\\frac{\\sin k(\\theta &#8211; \\mu)}{k} \\right] \\label{eq:vonMisesint}<br \/>\n\\end{align}<\/p>\n<h2>\u307e\u3068\u3081<\/h2>\n<p>(\\ref{eq:secondform})\u5f0f\u304b\u3089(\\ref{eq:thirdform})\u5f0f\u3078\u306e\u8b70\u8ad6\u306b\u3064\u3044\u3066\u306f\u56de\u308a\u304f\u3069\u3044\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u672cWeb\u30b5\u30a4\u30c8\u306e\u7ba1\u7406\u4eba\u305f\u308bpanda\u304c\u7406\u89e3\u3057\u305f\u3068\u3053\u308d\u3092\u305d\u306e\u307e\u307e\u66f8\u3044\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n<p>Von Mises\u5206\u5e03\u3068Von Mises-Fisher\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\u3044\u307e\u3044\u3061\u7406\u89e3\u3067\u304d\u3066\u3044\u306a\u3044\u306e\u3067\u3001\u7406\u89e3\u3067\u304d\u305f\u3068\u3053\u308d\u3067\u307e\u305f\u66f8\u304d\u307e\u3059\u3002<\/p>\n<p>\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u306f\u3058\u3081\u306b \u78ba\u7387\u5909\u6570\u306e\u5909\u6570\u5909\u63db\u306e\u65b9\u6cd5\u3092\u8abf\u3079\u3066\u3044\u308b\u3046\u3061\u306b\u3001Python3\u306e\u95a2\u6570\u3067\u78ba\u7387\u5bc6\u5ea6\u5206\u5e03\u304cVon Mises\u5206\u5e03\u306b\u306a\u308b\u4e71\u6570\u3092\u751f\u6210\u3059\u308b\u95a2\u6570\u306a\u308b\u3082\u306e\u3092\u898b\u3064\u3051\u3066\u3057\u307e\u3044\u3001\u3055\u3089\u306bVon Mises\u5206\u5e03\u306e\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3092\u8a08\u7b97\u3059\u308b\u969b\u306b(\u7b2c1\u7a2e)\u5909\u5f62Bessel\u95a2\u6570\u304c\u3069\u3053\u304b\u3089\u3068\u3082\u306a\u304f\u767b\u5834\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\u3001\u306a\u3093\u3067\u305d\u3046\u306a\u308b\u306e\u304b\u306b\u3064\u3044\u3066\u306e\u52c9\u5f37\u3082\u517c\u306d\u3066\u3001\u6574\u6570\u6b21\u306e(\u7b2c1\u7a2e)\u5909\u5f62Bessel\u95a2\u6570\u304b\u3089\u306a\u308b\u95a2\u6570\u5217\u306e\u6bcd\u95a2\u6570\u3092T\u2026 <span class=\"read-more\"><a href=\"https:\/\/pandanote.info\/?p=5735\">Read More &raquo;<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":5761,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-5735","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-13"],"_links":{"self":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/5735","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5735"}],"version-history":[{"count":28,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/5735\/revisions"}],"predecessor-version":[{"id":9385,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/5735\/revisions\/9385"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/5761"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5735"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5735"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5735"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}