{"id":325,"date":"2021-11-26T19:06:55","date_gmt":"2021-11-26T10:06:55","guid":{"rendered":"https:\/\/www.avelio.co.jp\/math\/wordpress\/?p=325"},"modified":"2025-04-10T13:49:50","modified_gmt":"2025-04-10T04:49:50","slug":"%e3%80%90%e6%99%82%e7%b3%bb%e5%88%97%e8%a7%a3%e6%9e%90%e5%85%a5%e9%96%80%e3%80%912-%e5%9f%ba%e6%9c%ac%e7%b5%b1%e8%a8%88%e9%87%8f%e3%81%a8ar%e3%83%a2%e3%83%87%e3%83%ab","status":"publish","type":"post","link":"https:\/\/www.avelio.co.jp\/math\/wordpress\/?p=325","title":{"rendered":"\u3010\u6642\u7cfb\u5217\u89e3\u6790\u5165\u9580\u30112. AR\u30e2\u30c7\u30eb"},"content":{"rendered":"\n<p><strong>\u203b\u4eca\u56de\u304b\u3089\u306f\u3001\u306a\u3093\u306e\u65ad\u308a\u3082\u306a\u304f\u78ba\u7387\u7d71\u8a08\u306e\u8a18\u53f7\u3092\u4f7f\u7528\u3057\u307e\u3059\u304c\u3001\u6570\u5f0f\u304c\u610f\u5473\u304c\u5206\u304b\u3089\u306a\u304f\u3066\u3082\u3001\u306a\u3093\u3068\u306a\u304f\u96f0\u56f2\u6c17\u3092\u63b4\u3081\u308b\u3088\u3046\u306b\u5fc3\u304c\u3051\u307e\u3057\u305f\u3002\u6570\u5b66\u304c\u5206\u304b\u3089\u306a\u3044\u4eba\u306f\u3068\u308a\u3042\u3048\u305a\u6570\u5f0f\u90e8\u5206\u306f\u6df1\u304f\u8003\u3048\u305a\u8aad\u3093\u3067\u304f\u3060\u3055\u3044\u3002\u8208\u5473\u304c\u6e67\u3044\u305f\u3089\u3001\u6570\u5b66\u52c9\u5f37\u3057\u3066\u307f\u3066\u306d\uff01<\/strong><\/p>\n\n\n\n<p><a href=\"https:\/\/www.avelio.co.jp\/math\/wordpress\/?p=284\" data-type=\"URL\" data-id=\"https:\/\/www.avelio.co.jp\/math\/wordpress\/?p=284\">\u524d\u56de\u306e\u8a18\u4e8b<\/a><\/p>\n\n\n\n<h1 class=\"wp-block-heading\">1. AR\u30e2\u30c7\u30eb<\/h1>\n\n\n\n<p>AR\u30e2\u30c7\u30eb\u306f\u3001\u65e5\u672c\u8a9e\u3067\u306f\u81ea\u5df1\u56de\u5e30\u30e2\u30c7\u30eb\u3068\u547c\u3070\u308c\u3001\u4f8b\u3048\u308b\u306a\u3089\u3001\u4eca\u65e5\u306e\u6211\u304c\u8eab\u304c\u660e\u65e5\u306e\u6211\u304c\u8eab\u3092\u4f5c\u308a\u4e0a\u3052\u308b\u3068\u3044\u3046\u3001\u56e0\u679c\u5f8b\u3092\u8868\u73fe\u3057\u305f\u30e2\u30c7\u30eb\u3067\u3059\u3002\u6570\u5f0f\u306b\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>$$ y_t = c + \\phi_1 y_{t-1} + \\varepsilon_t,~~~\\varepsilon_t \\sim \\textrm{W.N.}(\\sigma^2) $$<\/p>\n\n\n\n<p>\u306e\u3088\u3046\u306b\u5b9a\u5f0f\u5316\u3055\u308c\u308b\u3002\u3064\u307e\u308b\u3068\u3053\u308d\u3001\u660e\u65e5\u306e\u81ea\u5206 \\( y_t \\) \u306f\u4eca\u65e5\u306e\u81ea\u5206\u304c\u660e\u65e5\u306e\u81ea\u5206\u306b\u4e0e\u3048\u308b\u5f71\u97ff \\( c + \\phi_1 y_{t-1} \\) \u3068\u660e\u65e5\u8d77\u304d\u308b\u5076\u7136 \\( \\varepsilon_t \\) \u3092\u6df7\u305c\u5408\u308f\u305b\u3066\u3067\u51fa\u6765\u4e0a\u304c\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.avelio.co.jp\/math\/wordpress\/wp-content\/uploads\/2021\/11\/image-17.png\" alt=\"\" class=\"wp-image-423\" width=\"840\" height=\"298\"\/><\/figure>\n\n\n\n<p>\u30a4\u30e1\u30fc\u30b8\u56f3\u3092\u898b\u3066\u3082\u3089\u3046\u3068\u5206\u304b\u308b\u306e\u3067\u3059\u304c\u3001\u5076\u7136\u6027\u3092\u8868\u3057\u3066\u3044\u308b\u6570\u5024 \\(\\varepsilon_t\\) \u3082\u300c\u6a2a\u4e26\u3073\u306e\u30ac\u30c1\u30e3\u30ac\u30c1\u30e3\u30de\u30b7\u30f3\u300d\u304b\u3089\u5410\u304d\u51fa\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u3053\u306e\u6a2a\u4e26\u3073\u306e\u30ac\u30c1\u30e3\u30ac\u30c1\u30e3\u81ea\u4f53\u3082\u6642\u7cfb\u5217\u30e2\u30c7\u30eb\u3068\u8a00\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p>AR\u30e2\u30c7\u30eb\u3067\u4f7f\u308f\u308c\u3066\u3044\u308b\u3001\u3053\u306e\u300c\u6a2a\u4e26\u3073\u30ac\u30c1\u30e3\u30ac\u30c1\u30e3\u30de\u30b7\u30f3\u300d\u306e\u3053\u3068\u3092<strong>\u30db\u30ef\u30a4\u30c8\u30ce\u30a4\u30ba<\/strong>\u3068\u3044\u3044\u307e\u3059\u3002\u4e0a\u8a18\u306e\u6570\u5f0f\u5185\u306e<\/p>\n\n\n\n<p>$$  \\varepsilon_t \\sim \\textrm{W.N.}(\\sigma^2)  $$<\/p>\n\n\n\n<p>\u306e\u90e8\u5206\u306f\u3001\u300c\u30e9\u30f3\u30c0\u30e0\u9805 \\(\\varepsilon_t \\) \u306f\u30db\u30ef\u30a4\u30c8\u30ce\u30a4\u30ba\u304b\u3089\u5410\u304d\u51fa\u3055\u308c\u308b\u5024\u3060\u3088\uff01\u300d\u3068\u3044\u3046\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3058\u3083\u3042\u3001\u30db\u30ef\u30a4\u30c8\u30ce\u30a4\u30ba\u306f\u3069\u3093\u306a\u30de\u30b7\u30f3\u306a\u3093\u3067\u3057\u3087\u3046\u304b\uff1f\u304a\u786c\u304f\u3001\u5b9a\u7fa9\u3092\u66f8\u304f\u3068\u3001\u3053\u3046\u3060\uff01<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u6027\u8cea\u2460\uff1a\\( \\textrm{E}[\\varepsilon_t] = 0,~~~\\forall t \\)<\/li><li>\u6027\u8cea\u2461\uff1a\\( \\textrm{Var}[\\varepsilon_t] = \\sigma^2,~~~\\forall t \\)<\/li><li>\u6027\u8cea\u2462\uff1a\\( \\textrm{Cov}[\\varepsilon_t, \\varepsilon_{t-k}] = 0~~~\\forall  t,~k\\)<\/li><\/ul>\n\n\n\n<p>\u3092\u6e80\u305f\u3059\u78ba\u7387\u904e\u7a0b\u306e\u3053\u3068\u3092\u6307\u3057\u307e\u3059\u3002\u306f\u3044\u3001\u7ffb\u8a33\u3057\u307e\u3059\u306d\u3002<\/p>\n\n\n\n<p>\u300c\u6a2a\u4e26\u3073\u306e\u30de\u30b7\u30f3\u304c\u3069\u308c\u3082\uff10\u3092\u4e2d\u5fc3\u306b (\u2190\u6027\u8cea\u2460) \u6563\u3089\u3070\u308a\u5177\u5408 \\(\\sigma^2\\) \u3067 (\u2190\u6027\u8cea\u2461) \u6570\u5024\u3092\u5410\u304d\u51fa\u3057\u3066\u3001\u306a\u304a\u304b\u3064\u3001\u3069\u306e\u30de\u30b7\u30f3\u3082\u4ed6\u306e\u30de\u30b7\u30f3\u304c\u5410\u304d\u51fa\u3057\u305f\u5024\u306e\u5f71\u97ff\u3092\u53d7\u3051\u306a\u3044 (\u2190\u6027\u8cea\u2462) \u30de\u30b7\u30f3\u306e\u4e26\u3073\u300d\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u8981\u306f\u3001\u540c\u3058\u578b\u306e\u30de\u30b7\u30f3\u304c\u4e92\u3044\u306e\u5e72\u6e09\u3092\u53d7\u3051\u305a\u306b\u6a2a\u4e26\u3073\u306b\u7f6e\u3044\u3066\u3042\u308b\u3060\u3051\u306e\u72b6\u614b\u3067\u3059\u3002(\u672c\u5f53\u306f\u3001\u30de\u30b7\u30f3\u540c\u58eb\u304c\u5168\u304f\u95a2\u4fc2\u3057\u3066\u3044\u306a\u3044\u3068\u3044\u3046\u308f\u3051\u3067\u306a\u3044\u306e\u3067\u3059\u304c\u3001\u3068\u308a\u3042\u3048\u305a\u306f\u95a2\u4fc2\u306a\u3044\u3068\u8003\u3048\u3066\u3082\u3089\u3063\u3066\u5dee\u3057\u652f\u3048\u306a\u3044\u3067\u3059\u3002\u672c\u5f53\u306b\u5168\u304f\u5f71\u97ff\u3057\u5408\u308f\u306a\u3044\u540c\u3058\u578b\u306e\u30de\u30b7\u30f3\u306e\u4e26\u3073\u3092<strong>\u72ec\u7acb\u540c\u5206\u5e03<\/strong>\u3068\u3044\u3044\u307e\u3059\u3002)<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">2. \u5b9a\u5e38\u6027\u3068\u30a8\u30eb\u30b4\u30fc\u30c9\u6027<\/h1>\n\n\n\n<p> \\( | \\phi_1 | &lt; 1 \\) \u306e\u3068\u304d\u3001AR(1) \u30e2\u30c7\u30eb\u306e\u5e73\u5747 \\( \\mu_t \\)\u3001\u5206\u6563 \\( \\sigma_t^2 \\)\u3001\u81ea\u5df1\u5171\u5206\u6563 \\( \\gamma_{k, t} \\)\u3001\u81ea\u5df1\u76f8\u95a2\u4fc2\u6570 \\( \\rho_{k, t} \\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\begin{align*}<br>    \\mu_t &amp;= \\frac{1}{1 &#8211; \\phi_1} c \\\\<br>    \\sigma_t^2 &amp;= \\frac{\\sigma^2}{1 &#8211; \\phi_1^2} \\\\<br>    \\gamma_{k, t} &amp;= \\phi_1^k \\gamma_0 \\\\<br>    \\rho_{k, t} &amp;= \\phi_1^k<br>\\end{align*} <br>$$<\/p>\n\n\n\n<p>\u7740\u76ee\u3059\u3079\u304d\u306f\u3001\u5e73\u5747\u30fb\u5206\u6563\u304c\u6642\u523b \\( t \\) \u306b\u4f9d\u3089\u305a\u3001\u81ea\u5df1\u5171\u5206\u6563\u3068\u81ea\u5df1\u76f8\u95a2\u4fc2\u6570\u306f\u6642\u9593\u5dee \\( k \\) \u3060\u3051\u3067\u6c7a\u307e\u308b\u3068\u3044\u3046\u6027\u8cea\u3067\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u6027\u8cea\u3092 <strong>(\u5f31) \u5b9a\u5e38\u6027<\/strong>\u3068\u547c\u3073\u3001(\u5f31) \u5b9a\u5e38\u6027\u3092\u6301\u3064\u78ba\u7387\u904e\u7a0b\u3092 <strong>(\u5f31) \u5b9a\u5e38\u904e\u7a0b<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3055\u3089\u306b\u3001\u30e9\u30b0 \\( k \\) \u304c\u5927\u304d\u304f\u306a\u308b\u306b\u3064\u308c\u3066\u81ea\u5df1\u5171\u5206\u6563 \\( \\gamma_k \\) \u304c\u5341\u5206\u306b\u65e9\u304f \\( 0 \\) \u306b\u53ce\u675f\u3059\u308b\u306a\u3089\u3070\u3001\u5b9a\u5e38\u904e\u7a0b\u306e\u6642\u9593\u5e73\u5747 (\u89b3\u6e2c\u5217\u306e\u5e73\u5747) \\( \\frac{1}{T} \\sum_{t=1}^{T} y_t \\) \u304c\u7a7a\u9593\u5e73\u5747 \\( \\mu_t = \\textrm{E}(y_t) \\) \u306b\u78ba\u7387\u53ce\u675f\u3057\u307e\u3059\u3002 \u3053\u306e\u6027\u8cea\u3092<strong>\u30a8\u30eb\u30b4\u30fc\u30c9\u6027<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002<br><\/p>\n\n\n\n<p>\u30a8\u30eb\u30b4\u30fc\u30c9\u6027\u306e\u304a\u304b\u3052\u3067\u3001\u7a7a\u9593\u5e73\u5747\u3092 (\u89b3\u6e2c\u5024\u306e) \u6642\u9593\u5e73\u5747\u3067\u63a8\u5b9a\u3067\u304d\u308b\u3053\u3068\u304c\u4fdd\u8a3c\u3055\u308c\u3001\u305d\u308c\u306b\u3088\u3063\u3066\u5207\u7247 \\( c \\) \u3084\u81ea\u5df1\u56de\u5e30\u4fc2\u6570 \\(  \\phi_1 \\) \u306e\u63a8\u5b9a\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059\u3002\u89b3\u6e2c\u5024\u306e\u6642\u9593\u5e73\u5747\u3092 \\( \\hat{\\mu} \\) \u3067\u8868\u305b\u3070\u3001\u5207\u7247\u3068\u81ea\u5df1\u56de\u5e30\u4fc2\u6570\u306e\u63a8\u5b9a\u91cf \\( \\hat{c}\u3001 ~\\hat{\\phi}_1 \\) \u306f<\/p>\n\n\n\n<p>$$<br>\\begin{align*}<br>    \\hat{\\phi}_1 &amp;= \\frac{  \\sum_{t=2}^T   ( y_t &#8211;  \\hat{\\mu}  ) ( y_{t-1} &#8211;  \\hat{\\mu}  )   }{ \\sum_{t=1}^T   ( y_t &#8211;  \\hat{\\mu}  )^2 } \\\\<br>    \\hat{c} &amp;=  ( 1 &#8211; \\hat{\\phi}_1 ) \\hat{\\mu}<br>\\end{align*}<br>$$<\/p>\n\n\n\n<p>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002(\u3053\u306e\u63a8\u5b9a\u91cf\u306f\u53b3\u5bc6\u306b\u306f\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308a\u5c0e\u51fa\u3055\u308c\u307e\u3059\u3002\u6700\u5c24\u6cd5\u3084Yule-Walker\u6cd5\u306a\u3069\u3001\u4ed6\u306e\u65b9\u6cd5\u3067\u69cb\u7bc9\u3055\u308c\u308b\u63a8\u5b9a\u91cf\u3082\u3042\u308a\u307e\u3059\u3002\u63a8\u5b9a\u91cf\u306e\u69cb\u7bc9\u306a\u3069\u306e\u7406\u8ad6\u7684\u306f\u96e3\u6240\u306b\u306f\u8e0f\u307f\u8fbc\u307e\u306a\u3044..!)<\/p>\n\n\n\n<p>\u96e3\u3057\u3044\u8a00\u8449\u3092\u30dd\u30f3\u30dd\u30f3\u4e26\u3079\u3066\u3057\u307e\u3044\u307e\u3057\u305f\u306e\u3067\u3001\u3082\u3046\u5c11\u3057\u565b\u307f\u7815\u3044\u3066\u30a4\u30e1\u30fc\u30b8\u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u307e\u305a\u3001\u5b9a\u5e38\u6027\u3068\u30a8\u30eb\u30b4\u30fc\u30c9\u6027\u306e\u6761\u4ef6\u3092\u8ab2\u3059\u80cc\u666f\u306b\u306f\u3001\u6642\u7cfb\u5217\u30c7\u30fc\u30bf\u306e\u7279\u6b8a\u6027\u304c\u3042\u308a\u307e\u3059\u3002\u901a\u5e38\u306e\u30c7\u30fc\u30bf\u306f\u3001\u3042\u308b\u78ba\u7387\u5909\u6570\u306b\u3064\u3044\u3066\u8907\u6570\u306e\u5b9f\u73fe\u5024\u304c\u3042\u308b\u305f\u3081\u3001\u305d\u308c\u3089\u3092\u7528\u3044\u3066\u5e73\u5747\u3084\u5206\u6563\u306a\u3069\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u304c\u3001\u6642\u7cfb\u5217\u30c7\u30fc\u30bf\u306e\u5834\u5408\u30011\u3064\u306e\u6642\u523b\u306b\u5bfe\u3057\u30661\u3064\u306e\u5b9f\u73fe\u5024\u3057\u304b\u5b58\u5728\u3057\u307e\u305b\u3093\u3002\u3059\u306a\u308f\u3061\u3001\u6628\u65e5\u306e\u6c17\u6e29\u3092\u751f\u6210\u3059\u308b\u78ba\u7387\u5909\u6570\u306e\u5e73\u5747\u3084\u5206\u6563\u3092\u63a8\u6e2c\u3057\u305f\u3044\u304b\u3089\u3068\u3044\u3063\u3066\u3001\u6628\u65e5\u306e\u6c17\u6e29\u3092\u4f55\u5ea6\u3082\u89b3\u6e2c\u3059\u308b\u3053\u3068\u306f\u30bf\u30a4\u30e0\u30ea\u30fc\u30d7\u304c\u3067\u304d\u305f\u308a\u3057\u306a\u3044\u9650\u308a\u539f\u7406\u7684\u306b\u4e0d\u53ef\u80fd\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002\u3042\u304f\u307e\u3067\u5b9f\u73fe\u5024\u306f\uff11\u3064\u3057\u304b\u5f97\u3089\u308c\u307e\u305b\u3093\u30021\u3064\u306e\u78ba\u7387\u5909\u6570\u306b\u5bfe\u3057\u3066\u30011\u3064\u306e\u5b9f\u73fe\u5024\u3057\u304b\u306a\u3044\u305f\u3081\u3001\u305d\u306e\u307e\u307e\u3067\u306f\u5206\u6563\u3067\u3059\u3089\u63a8\u5b9a\u3067\u304d\u306a\u304f\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u904e\u53bb100\u65e5\u5206\u306b\u5f97\u3089\u308c\u305f\u6c17\u6e29\u30c7\u30fc\u30bf\u3092\u3001\u4eca\u65e5\u306e\u6c17\u6e29\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a\u306b\u5229\u7528\u3067\u304d\u308b\u3053\u3068\u3092\u4fdd\u8a3c\u3057\u3066\u304f\u308c\u308b\u6027\u8cea\u304c\u3001\u5b9a\u5e38\u6027\u3068\u30a8\u30eb\u30b4\u30fc\u30c9\u6027\u3068\u3044\u3046\u308f\u3051\u3067\u3059\u3002<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">3. AR(p)\u30e2\u30c7\u30eb<\/h1>\n\n\n\n<p>\u4eca\u307e\u3067\u306f\u30011\u6642\u523b\u524d\u306e\u81ea\u5206\u81ea\u8eab\u3092\u8aac\u660e\u5909\u6570\u3068\u3057\u3066\u304d\u307e\u3057\u305f\u304c\u3001\u3088\u308a\u4e00\u822c\u306b \\( p \\) \u6642\u523b\u524d\u307e\u3067\u306e\u81ea\u5206\u81ea\u8eab\u3092\u8aac\u660e\u5909\u6570\u3068\u3057\u305fAR\u30e2\u30c7\u30eb\u3092<strong> \\( p \\) \u6b21\u306eAR\u30e2\u30c7\u30eb<\/strong>\u3068\u547c\u3073\u3001<strong>AR(p)\u30e2\u30c7\u30eb<\/strong>\u3068\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>y_t = c + \\sum_{i=1}^p \\phi_i y_{t-i} + \\varepsilon_t,~~~~\\varepsilon_t \\sim \\textrm{W.N.}(\\sigma^2)<br>$$<\/p>\n\n\n\n<p>\u6027\u8cea\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\begin{align*}<br>\\mu &amp;= \\textrm{E}[y_t] = \\frac{c}{1-\\sum_{i=1}^p \\phi_i} \\\\<br>\\gamma_0 &amp;= \\textrm{Var}[y_t] =  \\frac{\\sigma^2}{1-\\sum_{i=1}^p \\phi_i \\rho_i} \\\\ <br>\\gamma_k &amp;= \\sum_{I=1}^{p} \\phi_i \\gamma_{k-i},~~~k \\ge 1 \\\\<br>\\rho_k &amp;= \\sum_{I=1}^{p} \\phi_i \\rho_{k-i},~~~k \\ge 1 \\\\ <br>\\end{align*}<br>$$<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">4. python\u306b\u3088\u308bAR\u30e2\u30c7\u30eb\u306e\u5b9f\u88c5<\/h1>\n\n\n\n<p>\u4eca\u56de\u306f\u3001\u8fb2\u6797\u6c34\u7523\u7701\u304c\u767a\u8868\u3057\u3066\u3044\u308b\u725b\u4e73\u751f\u7523\u91cf\u306e\u30c7\u30fc\u30bf\u3092\u53d6\u5f97\u3057\u3001\u3053\u306e\u30c7\u30fc\u30bf\u3092AR\u30e2\u30c7\u30eb\u3092\u7528\u3044\u3066\u5206\u6790\u3057\u3066\u3044\u304d\u307e\u3059\u3002(\u30b5\u30a6\u30ca\u4e0a\u304c\u308a\u306e\u30b3\u30fc\u30d2\u30fc\u725b\u4e73\u304c\u7f8e\u5473\u3057\u304b\u3063\u305f\u306e\u3067\u3001\u3053\u306e\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306b\u3057\u307e\u3057\u305f\u3002)<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>\u30c7\u30fc\u30bf\u51fa\u5178\uff1a<a href=\"https:\/\/www.e-stat.go.jp\/stat-search\/files?page=1&amp;layout=datalist&amp;toukei=00500225&amp;tstat=000001015114&amp;cycle=0&amp;year=20200&amp;month=0&amp;tclass1=000001037133\">https:\/\/www.e-stat.go.jp\/stat-search\/files?page=1&amp;layout=datalist&amp;toukei=00500225&amp;tstat=000001015114&amp;cycle=0&amp;year=20200&amp;month=0&amp;tclass1=000001037133<\/a><\/p><\/blockquote>\n\n\n\n<pre class=\"wp-block-preformatted\">1985-01-31    591919\n1985-02-28    546518\n1985-03-31    622400\n1985-04-30    628679\n1985-05-31    661391\n               ...  \n2018-08-31    606470\n2018-09-30    560308\n2018-10-31    596228\n2018-11-30    579820\n2018-12-31    609506\nFreq: M, Length: 408, dtype: int64<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"394\" height=\"264\" src=\"https:\/\/www.avelio.co.jp\/math\/wordpress\/wp-content\/uploads\/2021\/12\/image-2.png\" alt=\"\" class=\"wp-image-512\"\/><figcaption>\u51fa\u5178\uff1a<a href=\"https:\/\/www.maff.go.jp\/j\/tokei\/kouhyou\/gyunyu\/\" data-type=\"URL\" data-id=\"https:\/\/www.maff.go.jp\/j\/tokei\/kouhyou\/gyunyu\/\">\u725b\u4e73\u4e73\u88fd\u54c1\u7d71\u8a08\u8abf\u67fb(\u8fb2\u6797\u6c34\u7523\u7701)<\/a><\/figcaption><\/figure>\n\n\n\n<p>AR\u30e2\u30c7\u30eb\u306f\u5b9a\u5e38\u904e\u7a0b\u306b\u3057\u304b\u9069\u7528\u3067\u304d\u306a\u3044\u305f\u3081\u3001\u307e\u305a\u306f\u6642\u7cfb\u5217\u30c7\u30fc\u30bf\u304c\u5b9a\u5e38\u904e\u7a0b\u306b\u5f93\u3046\u304b\u3069\u3046\u304b\u3092\u691c\u5b9a\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u306f\u3001<strong>ADF\u691c\u5b9a<\/strong> (augmented Dicker-Fuller test) \u3092\u3059\u308b\u3053\u3068\u3067\u5b9a\u5e38\u904e\u7a0b\u304b\u3069\u3046\u304b\u306e\u78ba\u8a8d\u304c\u884c\u3048\u307e\u3059\u3002\u672c\u8a18\u4e8b\u3067\u306f\u3001ADF\u691c\u5b9a\u306e\u8a73\u7d30\u306f\u7701\u304d\u307e\u3059\u304c\u3001\u691c\u5b9a\u306e\u7d50\u679c\u30011\u6b21\u5dee\u5206\u7cfb\u5217 \\( \\Delta y_t ~\\small{(= y_t &#8211; y_{t-1})} \\) \u304c\u5b9a\u5e38\u904e\u7a0b\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u6b21\u306b\u3001AR\u30e2\u30c7\u30eb\u306e\u6b21\u6570\u3092\u6c7a\u5b9a\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u4e00\u822c\u306b\u8aac\u660e\u5909\u6570\u304c\u591a\u3051\u308c\u3070\u591a\u3044\u307b\u3069\u3001\u30e2\u30c7\u30eb\u306f\u89b3\u6e2c\u30c7\u30fc\u30bf\u306b\u3074\u3063\u305f\u308a\u3068\u30d5\u30a3\u30c3\u30c8\u3057\u307e\u3059\u304c\u3001\u305d\u306e\u4e00\u65b9\u3067\u904e\u5270\u9069\u5408\u306e\u554f\u984c\u304c\u767a\u751f\u3057\u307e\u3059\u3002\u904e\u5270\u9069\u5408\u304c\u8d77\u3053\u308b\u3068\u3001\u4eca\u5ea6\u306f\u4e88\u6e2c\u7cbe\u5ea6\u304c\u843d\u3061\u307e\u3059\u3002\u3064\u307e\u308b\u3068\u3053\u308d\u3001&#8221;\u305f\u307e\u305f\u307e\u5f97\u3089\u308c\u305f\u3060\u3051\u306e&#8221;\u89b3\u6e2c\u5024\u3092\u9811\u5f35\u3063\u3066\u8aac\u660e\u3057\u3088\u3046\u3068\u3057\u3059\u304e\u308b\u3042\u307e\u308a\u3001\u65b0\u3057\u304f\u5f97\u3089\u308c\u308b\u30c7\u30fc\u30bf\u306e\u4e88\u6e2c\u304c\u304a\u3056\u306a\u308a\u306b\u306a\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u306f\u3001<strong>\u8d64\u6c60\u60c5\u5831\u91cf\u57fa\u6e96 <\/strong>(AIC: Akaike information criterion) \u3068\u3044\u3046\u91cf\u3092\u6bd4\u8f03\u3057\u3066\u6b21\u6570\u3092\u6c7a\u5b9a\u3057\u307e\u3059\u3002AIC\u306f\u3001\u30e2\u30c7\u30eb\u306e\u9069\u5408\u5ea6\u306b\u3088\u308b\u52a0\u70b9\u8981\u7d20\u3068\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u591a\u3055\u306b\u3088\u308b\u6e1b\u70b9\u8981\u7d20\u3092\u540c\u6642\u306b\u52a0\u5473\u3057\u305f\u91cf\u3067\u3042\u308a\u3001AIC\u306e\u5024\u304c\u5c0f\u3055\u3044\u307b\u3069\u9069\u5207\u306a\u30e2\u30c7\u30eb\u3068\u8a00\u3048\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u8a73\u3057\u3044\u7406\u8ad6\u306e\u8aac\u660e\u306f\u7701\u7565\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u306f\u3001python\u306e\u7d71\u8a08\u89e3\u6790\u7528\u30e9\u30a4\u30d6\u30e9\u30eaStatsModels\u3092\u4f7f\u7528\u3057\u3066\u3001\u5206\u6790\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code># statsmodels\u306etsa(time series analysis)\u30e2\u30b8\u30e5\u30fc\u30eb\u304b\u3089 ar_model\u30af\u30e9\u30b9\u3092\u30a4\u30f3\u30dd\u30fc\u30c8\nfrom statsmodels.tsa import ar_model\n\n# \u8fb2\u6797\u6c34\u7523\u7701\u306e\u30b5\u30a4\u30c8\u304b\u3089\u30c7\u30fc\u30bf\u3092\u53d6\u5f97\u3057\u3066\u3001\u3044\u3044\u611f\u3058\u306b\u52a0\u5de5\u3057\u305f\u539f\u7cfb\u5217\u3092 y \u306b\u683c\u7d0d\u3057\u3066\u3044\u307e\u3059\u3002\n# 1\u6b21\u5dee\u5206\u7cfb\u5217\uff1ay_diff\ny_diff = y.diff().dropna()\n\n# y_diff_train\uff1a2016\u5e74\u307e\u3067\u306e\u30c7\u30fc\u30bf\u3092\u8a13\u7df4\u30c7\u30fc\u30bf\u3068\u3059\u308b\n# y_diff_test\uff1a2017\u5e74\u4ee5\u964d\u306e\u30c7\u30fc\u30bf\u3092\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u3068\u3059\u308b\ny_diff_train = y_diff&#91;:'2016']\ny_diff_test = y_diff&#91;'2017':]\n\n# AR\u30e2\u30c7\u30eb\u306e\u69cb\u7bc9\nmodel = ar_model.AR(y_diff)\n\n# \u30e2\u30c7\u30eb\u306e\u30d5\u30a3\u30c3\u30c6\u30a3\u30f3\u30b0\n# maxlag(=20)\u307e\u3067\u306e\u6b21\u6570\u306e\u4e2d\u3067AIC\u3092\u6700\u5c0f\u306b\u3059\u308b\u6b21\u6570\u3092\u81ea\u52d5\u9078\u629e\u3057\u3066\u304f\u308c\u308b\nresults = model.fit(maxlag=20, ic='aic')<\/code><\/pre>\n\n\n\n<p><\/p>\n\n\n\n<p><code>results.k_ar<\/code> \u3067\u9078\u629e\u3055\u308c\u305f\u6b21\u6570\u3092\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u6b21\u657013\u304c\u9078\u629e\u3055\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3055\u3066\u3001\u69cb\u7bc9\u3055\u308c\u305fAR(13)\u30e2\u30c7\u30eb\u306e\u826f\u3057\u60aa\u3057\u3092\u5224\u65ad\u3059\u308b\u5834\u5408\u3001\u6b8b\u5dee (residual error) \\( \\varepsilon_t \\) \u304c\u30db\u30ef\u30a4\u30c8\u30ce\u30a4\u30ba\u304b\u3069\u3046\u304b\u304c\u91cd\u8981\u3067\u3059\u3002\u6b8b\u5dee\u306f\u5b9f\u969b\u5024\u3068\u30e2\u30c7\u30eb\u4e88\u6e2c\u5024\u306e\u8aa4\u5dee\u3067\u3042\u308a\u3001\u3082\u3057\u3042\u308b\u6642\u7cfb\u5217\u30e2\u30c7\u30eb\u304c\u6642\u7cfb\u5217\u30c7\u30fc\u30bf\u306e\u81ea\u5df1\u76f8\u95a2\u6027\u8cea\u3092\u3046\u307e\u304f\u8aac\u660e\u3067\u304d\u305f\u3089\u3001\u6b8b\u5dee\u306e\u81ea\u5df1\u76f8\u95a2\u306f\u7121\u76f8\u95a2\u306b\u306a\u308b\u306f\u3059\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u6642\u7cfb\u5217\u304c\u30db\u30ef\u30a4\u30c8\u30ce\u30a4\u30ba\u306e\u5834\u5408\u3001\u30c7\u30fc\u30bf\u6570\u304c \\( n \\) \u306e\u3068\u304d\u3001\u81ea\u5df1\u76f8\u95a2\u4fc2\u6570\u306e\u6a19\u6e96\u504f\u5dee\u306f \\( \\frac{1}{\\sqrt{n}} \\) \u3068\u306a\u308b\u306e\u3067\u3001\u6b8b\u5dee\u306e\u81ea\u5df1\u76f8\u95a2\u4fc2\u6570\u304c \\( \\pm \\frac{1.96}{\\sqrt{n}} \\) \u306e95\uff05\u4fe1\u983c\u533a\u9593\u306b\u53ce\u307e\u3063\u3066\u3044\u308c\u3070\u3001\u826f\u3044\u30e2\u30c7\u30eb\u304c\u69cb\u7bc9\u3067\u304d\u3066\u3044\u308b\u3068\u8a00\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3067\u306f\u3001\u4eca\u56de\u306e\u7d50\u679c\u306e\u6b8b\u5dee\u3092\u78ba\u8a8d\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>from statsmodels.graphics import tsaplots\n\n# \u6b8b\u5dee\nresid = results.resid\n# \u6b8b\u5dee\u306e(\u504f)\u81ea\u5df1\u76f8\u95a2\u4fc2\u6570\u3092\u6b21\u657050\u307e\u3067\u63cf\u753b\ntsaplots.plot_pacf(resid, lags=50)<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"380\" height=\"264\" src=\"https:\/\/www.avelio.co.jp\/math\/wordpress\/wp-content\/uploads\/2021\/12\/image-3.png\" alt=\"\" class=\"wp-image-517\"\/><\/figure>\n\n\n\n<p>\u6c34\u8272\u90e8\u5206\u306f\u300195\uff05\u4fe1\u983c\u533a\u9593\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002\u30e9\u30b012\u306e\u504f\u81ea\u5df1\u76f8\u95a2\u306f\u4fe1\u983c\u533a\u9593\u304b\u3089\u5916\u308c\u3066\u3044\u308b\u306e\u3067\u7121\u76f8\u95a2\u3067\u306f\u306a\u3055\u305d\u3046\u3067\u3059\u3002\u3053\u306e\u30e2\u30c7\u30eb\u304c\uff11\u6b21\u968e\u5dee\u7cfb\u5217\u306b\u3042\u308b12\u30f5\u6708\u306e\u5faa\u74b0\u6210\u5206\u3092\u5341\u5206\u306b\u8868\u73fe\u3067\u304d\u3066\u3044\u308b\u3068\u306f\u8a00\u3048\u308b\u3067\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<p>\u6700\u5f8c\u306b\u3001\u3053\u306e\u30e2\u30c7\u30eb\u306e\u4e88\u6e2c\u3068\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf (2017\u5e74\u30682018\u5e74) \u3092\u91cd\u306d\u3066\u3001\u3044\u3044\u611f\u3058\u3067\u4e88\u6e2c\u3067\u304d\u3066\u3044\u308b\u306e\u304b\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002\u9752\u7dda\u304c\u5b9f\u969b\u306e\u5024\u3067\u3001\u30aa\u30ec\u30f3\u30b8\u8272\u306e\u70b9\u7dda\u304c\u4e88\u6e2c\u3067\u3059\u3002 <\/p>\n\n\n\n<pre class=\"wp-block-code\"><code># \u4e88\u6e2c\ny_diff_pred = results.predict('2017','2018')\n\n# \u4e88\u6e2c\u3092\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306b\u91cd\u306d\u3066\u63cf\u753b\nplt.plot(y_diff_test)\nplt.plot(y_diff_pred, '--')<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"402\" height=\"248\" src=\"https:\/\/www.avelio.co.jp\/math\/wordpress\/wp-content\/uploads\/2021\/12\/image-4.png\" alt=\"\" class=\"wp-image-521\"\/><\/figure>\n\n\n\n<p>\u5927\u307e\u304b\u306b12\u30f5\u6708\u306e\u5468\u671f\u3092\u4e88\u6e2c\u304c\u3067\u304d\u3066\u3044\u305d\u3046\u3067\u3059\u304c\u3001\u6b8b\u5dee\u306e\u81ea\u5df1\u76f8\u95a2\u3067\u78ba\u8a8d\u3057\u305f\u3088\u3046\u306b\u3001\u5468\u671f\u6027\u306e\u8868\u73fe\u304c\u5341\u4e8c\u5206\u3068\u306f\u8a00\u3048\u306a\u3044\u3088\u3046\u3067\u3059\u3002<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">5. \u307e\u3068\u3081<\/h1>\n\n\n\n<p>\u4eca\u56de\u306fAR\u30e2\u30c7\u30eb\u306e\u6027\u8cea\u3092\u5177\u4f53\u4f8b\u306b\u6319\u3052\u306a\u304c\u3089\u3001\u5b9a\u5e38\u6027\u3084\u30a8\u30eb\u30b4\u30fc\u30c9\u6027\u3068\u3044\u3063\u305f\u91cd\u8981\u306a\u6642\u7cfb\u5217\u30e2\u30c7\u30eb\u306e\u6027\u8cea\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3057\u305f\u3002\u307e\u305f\u5f8c\u534a\u3067\u306f\u3001\u5b9f\u30c7\u30fc\u30bf\u3092\u5206\u6790\u3057\u306a\u304c\u3089\u3001\u6642\u7cfb\u5217\u89e3\u6790\u306b\u304a\u3051\u308b\u91cd\u8981\u306a\u30d7\u30ed\u30bb\u30b9 (\u5b9a\u5e38\u6027\u306e\u691c\u5b9a\u3001AIC\u306b\u3088\u308b\u6b21\u6570\u6c7a\u5b9a\u3001\u6b8b\u5dee\u306e\u81ea\u5df1\u76f8\u95a2\u306b\u3088\u308b\u30e2\u30c7\u30eb\u306e\u826f\u3057\u60aa\u3057\u306e\u5224\u65ad&#8230;.etc) \u3092\u8aac\u660e\u3057\u307e\u3057\u305f\u3002\u6b21\u56de\u306f\u3001MA\u30e2\u30c7\u30eb\u3001\u305d\u3057\u3066\u3001AR\u30e2\u30c7\u30eb\u3068MA\u30e2\u30c7\u30eb\u3092\u30df\u30c3\u30af\u30b9\u3057\u305fARMA\u30e2\u30c7\u30eb\u3092\u8aac\u660e\u3059\u308b\u4e88\u5b9a\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u203b\u4eca\u56de\u304b\u3089\u306f\u3001\u306a\u3093\u306e\u65ad\u308a\u3082\u306a\u304f\u78ba\u7387\u7d71\u8a08\u306e\u8a18\u53f7\u3092\u4f7f\u7528\u3057\u307e\u3059\u304c\u3001\u6570\u5f0f\u304c\u610f\u5473\u304c\u5206\u304b\u3089\u306a\u304f\u3066\u3082\u3001\u306a\u3093\u3068\u306a\u304f\u96f0\u56f2\u6c17\u3092\u63b4\u3081\u308b\u3088\u3046\u306b\u5fc3\u304c\u3051\u307e\u3057\u305f\u3002\u6570\u5b66\u304c\u5206\u304b\u3089\u306a\u3044\u4eba\u306f\u3068\u308a\u3042\u3048\u305a\u6570\u5f0f\u90e8\u5206\u306f\u6df1\u304f\u8003\u3048\u305a\u8aad\u3093\u3067\u304f\u3060\u3055\u3044\u3002\u8208\u5473\u304c\u6e67\u3044\u305f\u3089\u3001\u6570\u5b66\u52c9 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":423,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[],"class_list":["post-325","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5","category-6"],"_links":{"self":[{"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/325","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=325"}],"version-history":[{"count":92,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/325\/revisions"}],"predecessor-version":[{"id":595,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/325\/revisions\/595"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=\/wp\/v2\/media\/423"}],"wp:attachment":[{"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.avelio.co.jp\/math\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}