{"id":262,"date":"2022-08-23T22:35:41","date_gmt":"2022-08-23T13:35:41","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/kernel-density-estimation\/"},"modified":"2024-09-21T10:31:47","modified_gmt":"2024-09-21T01:31:47","slug":"kernel-density-estimation","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/kernel-density-estimation\/","title":{"rendered":"\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u306e\u7c21\u5358\u306a\u8aac\u660e\u3068\u30d0\u30f3\u30c9\u5e45\u306e\u8a2d\u5b9a\u65b9\u6cd5"},"content":{"rendered":"\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u3068\u306f\u3069\u3093\u306a\u63a8\u5b9a\u3067\u3001\u3069\u3093\u306a\u6642\u306b\u4f7f\u308f\u308c\u308b\u306e\u304b\uff1f<\/p>\n\n\n\n

\u30d0\u30f3\u30c9\u5e45\u306e\u6c7a\u3081\u65b9\u3082\u7d39\u4ecb<\/p>\n\n\n\n\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u3068\u306f\u4f55\u304b\uff1f<\/h2>\n\n\n\n

\u3068\u3066\u3082\u5206\u304b\u308a\u3084\u3059\u3044\u8a18\u4e8b\u3092\u898b\u3064\u3051\u305f\u306e\u3067\u3001\u305d\u306e\u307e\u307e\u5f15\u7528\u3059\u308b\u3002<\/p>\n\n\n\n

\n

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\u300c\u78ba\u7387\u5bc6\u5ea6\u304c\u5168\u7136\u308f\u304b\u3089\u306a\u3044\u78ba\u7387\u5206\u5e03\u304b\u3089\u6a19\u672c\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u306b\u3001\u305d\u306e\u6a19\u672c\u304b\u3089\u3067\u304d\u308b\u3060\u3051\u3044\u3044\u611f\u3058\u306b\u5143\u306e\u78ba\u7387\u5bc6\u5ea6\u3092\u63a8\u5b9a\u3059\u308b\u65b9\u6cd5\u300d\u3067\u3059\u3002
\u5143\u306e\u78ba\u7387\u5bc6\u5ea6\u304c\u6b63\u898f\u5206\u5e03\u3060\u3068\u304b\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3060\u3068\u304b\u3060\u3051\u3067\u3082\u308f\u304b\u3063\u3066\u3044\u305f\u3089\u3001\u305d\u306e\u671f\u5f85\u5024\u3084\u5206\u6563\u306a\u3069\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u3060\u3051\u63a8\u5b9a\u3059\u308c\u3070\u3044\u3044(\u3053\u308c\u3092\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u3068\u3044\u3046)\u306e\u3067\u3059\u304c\u3001
\u305d\u308c\u3055\u3048\u3082\u308f\u304b\u3089\u306a\u3044\u3068\u304d\u306b\u4f7f\u308f\u308c\u308b\u624b\u6cd5\u3067\u3042\u308a\u3001\u30ce\u30f3\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>\n<\/blockquote>\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u6cd5\u306e\u7d39\u4ecb | \u5206\u6790\u30ce\u30fc\u30c8<\/a><\/p>\n\n\n\n

\u901a\u5e38\u306f\u3001\u6b63\u898f\u5206\u5e03\u3068\u304b\u3001\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u3068\u304b\u3001\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u304b\u3001\u6bcd\u96c6\u56e3\u306e\u5206\u5e03\u306e\u5f62\u72b6\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u591a\u3044\u304c\u3001\u307e\u3063\u305f\u304f\u308f\u304b\u3089\u306a\u3044\u5834\u5408\u306b\u4f7f\u3046\u3053\u3068\u304c\u60f3\u5b9a\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u5206\u5e03\u306e\u5f62\u72b6\u304c\u308f\u304b\u308b\u5834\u5408\u306f\u3001\u305d\u306e\u5206\u5e03\u306e\u7279\u5fb4\u3092\u793a\u3059\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u3067\u6bcd\u96c6\u56e3\u304c\u63a8\u5b9a\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

\u307e\u3063\u305f\u304f\u308f\u304b\u3089\u306a\u3044\u5834\u5408\u306e\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u306f\u3001\u7d50\u679c\u306f\u30b0\u30e9\u30d5\u3067\u8868\u73fe\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u6cd5\u306f\u3069\u3093\u306a\u63a8\u5b9a\u65b9\u6cd5\u306a\u306e\u304b\uff1f<\/h2>\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u6cd5\u306f\u3001\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 K \u3092\u7528\u3044\u3066\u3001\u63a8\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n

$$ \\hat{p} (x) = \\frac{1}{nh} \\sum_{i=1}^{n} {K \\left( \\frac{x – X_i}{h} \\right)} $$<\/p>\n\n\n\n\n\n\n\n

n\u306f\u30c7\u30fc\u30bf\u306e\u500b\u6570\u3001Xi \u304c\u5b9f\u969b\u306e\u30c7\u30fc\u30bf\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u4efb\u610f\u306b\u6c7a\u3081\u305f\uff58\u306e\u3068\u304d\u306e\u78ba\u7387\u5bc6\u5ea6\u3092\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306b\u3088\u3063\u3066\u63a8\u5b9a\u3059\u308b\u3068\u3044\u3046\u4ed5\u7d44\u307f\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u306e\u30d0\u30f3\u30c9\u5e45\u3068\u306f\uff1f<\/h3>\n\n\n\n

h\u306f\u30d0\u30f3\u30c9\u5e45\u3068\u547c\u3070\u308c\u3066\u9069\u5207\u306b\u6c7a\u3081\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

h \u306fSilvermann\u306e\u65b9\u6cd5\u304c\u6709\u540d\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

$$ h = \\frac{0.9 \\times \\min \\left( s, \\frac{IQR}{1.34} \\right)}{n^{1\/5}} $$<\/p>\n\n\n\n\n\n\n\n

s \u306f\u30b5\u30f3\u30d7\u30eb\u306e\u6a19\u6e96\u504f\u5dee\u3001IQR\u306f\u56db\u5206\u4f4d\u7bc4\u56f2\u3067\u3001\u6a19\u6e96\u504f\u5dee\u3068IQR\/1.34\u306e\u3069\u3061\u3089\u304b\u5c0f\u3055\u3044\u307b\u3046\u30920.9\u500d\u3057\u305f\u3082\u306e\u304c\u5206\u5b50\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u5206\u6bcd\u306f\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\u306e1\/5\u4e57\u3002<\/p>\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306b\u306f\u3044\u304f\u3064\u304b\u7a2e\u985e\u304c\u3042\u308b\u304c\u3001\u3082\u3063\u3068\u3082\u3088\u304f\u7528\u3044\u3089\u308c\u308b\u306e\u306f Gaussian \u30ac\u30a6\u30b9\u95a2\u6570\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u306f\u3044\u308f\u3086\u308b\u6b63\u898f\u5206\u5e03\u306e\u95a2\u6570\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

$$ K(x) = \\frac{1}{\\sqrt{2 \\pi}} \\exp{\\left( – \\frac{x^2}{2} \\right)} $$<\/p>\n\n\n\n\n\n\n\n

\uff1e\uff1e\u3082\u3046\u7d71\u8a08\u3067\u60a9\u3080\u306e\u306f\u7d42\u308f\u308a\u306b\u3057\u307e\u305b\u3093\u304b\uff1f\u00a0<\/a><\/strong><\/span><\/p>\r\n\"\"<\/a>\r\n

\u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div>

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u308b<\/h2>\n\n\n\n

\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u30b0\u30e9\u30d5\u306e\u63cf\u753b with Excel<\/a><\/p>\n\n\n\n

\u3053\u3061\u3089\u306e\u30b5\u30f3\u30d7\u30eb\u30c7\u30fc\u30bf\u3092\u7528\u3044\u3066\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u308b\u3002<\/p>\n\n\n\n

\u7d71\u8a08\u30bd\u30d5\u30c8R\u3067\u5b9f\u884c\u3057\u3066\u307f\u308b\u3002<\/p>\n\n\n\n

\u30af\u30ea\u30c3\u30d7\u30dc\u30fc\u30c9\u7d4c\u7531\u3067\u8aad\u307f\u8fbc\u3080\u3002<\/p>\n\n\n\n

dat <-<\/span> read.table<\/span>(<\/span>\"clipboard\"<\/span>,<\/span> header=<\/span>T<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u30d2\u30b9\u30c8\u30b0\u30e9\u30e0\u3092\u63cf\u304f\u3068\u3053\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u7d71\u8a08\u30bd\u30d5\u30c8R\u306b\u306fdensity()\u3068\u3044\u3046\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u95a2\u6570\u304c\u3042\u308a\u3001\u305d\u308c\u3092\u4f7f\u3046\u3068\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
density<\/span>(<\/span>dat$<\/span>x)<\/span>\n<\/code><\/pre>\n\n\n\n
\u30d0\u30f3\u30c9\u5e45\u306e\u8a2d\u5b9a\u306f\u3069\u3046\u3059\u308b\uff1f<\/h3>\n\n\n\n
\u30d0\u30f3\u30c9\u5e45 h \u306f\u3001Silvermann\u306e\u65b9\u6cd5\uff08\u30c7\u30d5\u30a9\u30eb\u30c8\u8a2d\u5b9a\uff09\u3067\u3001\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306f\u30ac\u30a6\u30b9\u95a2\u6570\uff08\u30c7\u30d5\u30a9\u30eb\u30c8\u8a2d\u5b9a\uff09\u3092\u5229\u7528\u3059\u308b\u3002<\/p>\n\n\n\n
\u8a08\u7b97\u7d50\u679c\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
h\u306f Bandwidth ‘bw’\u3067\u8868\u793a\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
7.876\u3068\u8a08\u7b97\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
plot<\/span>(<\/span>desity<\/span>(<\/span>dat$<\/span>x))<\/span>\n<\/code><\/pre>\n\n\n\n
\u3068plot()\u95a2\u6570\u3092\u5b9f\u884c\u3059\u308b\u3068\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u30b0\u30e9\u30d5\u304c\u63cf\u753b\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u3053\u306e\u7d50\u679c\u3067\u63cf\u304b\u308c\u305f\u30b0\u30e9\u30d5\u304c\u6bcd\u96c6\u56e3\u306e\u63a8\u5b9a\u30b0\u30e9\u30d5\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u5de6\u53f3\u5bfe\u79f0\u306e\u91e3\u308a\u9418\u578b\u306e\u6b63\u898f\u5206\u5e03\u3092\u3057\u3066\u3044\u308c\u3070\u3001\u6bcd\u96c6\u56e3\u304c\u6b63\u898f\u5206\u5e03\u3067\u3042\u308b\u3068\u63a8\u5b9a\u3055\u308c\u305f\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u4f8b\u3048\u3070\u3001\u3082\u3068\u3082\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6b63\u898f\u5206\u5e03\u3092\u601d\u308f\u305b\u308b\u30d2\u30b9\u30c8\u30b0\u30e9\u30e0\u3060\u3063\u305f\u5834\u5408\u3001<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u307b\u307c\u6b63\u898f\u5206\u5e03\u306b\u8fd1\u3044\u30b0\u30e9\u30d5\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u307e\u3068\u3081<\/h2>\n\n\n\n
\u3042\u308b\u30b5\u30f3\u30d7\u30eb\u304c\u5f97\u3089\u308c\u3066\u3044\u308b\u6642\u306b\u3001\u30ce\u30f3\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u306b\u6bcd\u96c6\u56e3\u306e\u5206\u5e03\u3092\u63a8\u5b9a\u3059\u308b\u65b9\u6cd5\u3068\u3057\u3066\u3001\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u304c\u3042\u308b\u3002<\/p>\n\n\n\n
\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u306e\u65b9\u6cd5\u3092\u7c21\u5358\u306b\u7d39\u4ecb\u3057\u3001\u7d71\u8a08\u30bd\u30d5\u30c8R\u306edensity()\u95a2\u6570\u3092\u4f7f\u3048\u3070\u3001\u7c21\u5358\u306b\u63a8\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3001\u30b0\u30e9\u30d5\u63cf\u753b\u3067\u304d\u308b\u3053\u3068\u3092\u7d39\u4ecb\u3057\u305f\u3002<\/p>\n\n\n\n
\u95a2\u9023\u8a18\u4e8b<\/h2>\n\n\n
\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\u3042\u308f\u305b\u3066\u8aad\u307f\u305f\u3044<\/span>\n\t\t\t\t\t
\"\"<\/figure><\/div>\t\t\t\t\t
\n\t\t\t\t\t\tEZR \u306e\u6563\u5e03\u56f3\u884c\u5217\u3067\u8868\u793a\u3055\u308c\u308b\u66f2\u7dda\u30b0\u30e9\u30d5\u306e\u610f\u7fa9<\/a>\n\t\t\t\t\t\tEZR\u306e\u6563\u5e03\u56f3\u884c\u5217\u306b\u304a\u3044\u3066\u5404\u5909\u6570\u306e\u6587\u5b57\u304c\u8868\u793a\u3055\u308c\u3066\u3044\u308b\u30b9\u30e0\u30fc\u30ba\u306a\u30b0\u30e9\u30d5\u306e\u89e3\u8aac\u3002 EZR \u306e\u6563\u5e03\u56f3\u884c\u5217\u3067\u8868\u793a\u3055\u308c\u308b\u66f2\u7dda\u30b0\u30e9\u30d5 \u4ee5\u4e0b\u306e\u8d64\u77e2\u5370\u3067\u793a\u3057\u305f\u66f2\u7dda\u30b0\u30e9\u30d5\u306e\u3053\u3068\u3092\u8a00\u3063…<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\n\n\n\n\n
\u53c2\u8003\u8cc7\u6599\u30fb\u30b5\u30a4\u30c8\u3000<\/h2>\n\n\n\n
\u30ce\u30f3\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u5bc6\u5ea6\u63a8\u5b9a\u6cd5<\/p>\n\n\n\n
http:\/\/ibis.t.u-tokyo.ac.jp\/suzuki\/lecture\/2015\/dataanalysis\/L9.pdf<\/a><\/p>\n\n\n\n
\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a<\/a><\/p>\n\n\n\n
\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u6cd5\u306e\u7d39\u4ecb | \u5206\u6790\u30ce\u30fc\u30c8<\/a><\/p>\n\n\n\n
\u30ce\u30f3\u30d1\u30e9\u30e1\u30c8\u30ea\u30c3\u30af\u5bc6\u5ea6\u63a8\u5b9a\uff08\u30d2\u30b9\u30c8\u30b0\u30e9\u30e0\u5bc6\u5ea6\u63a8\u5b9a\u3001\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\uff09 – \u6a5f\u68b0\u3068\u5b66\u7fd2\u3059\u308b<\/a><\/p>\n\n\n\n
\u30ab\u30fc\u30cd\u30eb\u5bc6\u5ea6\u63a8\u5b9a\u30b0\u30e9\u30d5\u306e\u63cf\u753b with Excel<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
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