Tsutomu-KKE/正規分布性の検証

## 正規分布性の検証
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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#\u6b63\u898f\u5206\u5e03\u6027\u306e\u691c\u8a3c\n",
      "\u6307\u5b9a\u3055\u308c\u305f\u5217\u306e\u6b63\u898f\u5206\u5e03\u3057\u3066\u3044\u308b\u5ea6\u5408\u3044\u3092$\\chi^2$\u691c\u5b9a\u3067\u8a08\u7b97\u3059\u308b\u3002 \u30ea\u30b9\u30c8\u306e\u500b\u6570\u306f\u300110\u306e\u500d\u6570\u3067\u300150\u4ee5\u4e0a\u3068\u3059\u308b\u3002\n",
      "\n",
      "\u6b63\u898f\u5206\u5e03\u306e\u9762\u7a4d\u304c10(=s)\u7b49\u5206\u306b\u306a\u308b\u3088\u3046\u306b\u5206\u5272\u3059\u308b\n",
      "\n",
      "\\begin{array}{rl}\n",
      "n_s =& \\frac{n}{s} \\\\\n",
      "Q =& \\sum_i{\\frac{(n_s - k_i)^2}{n_s}} \\\\\n",
      "f(x, n) =& \\frac{1}{2^{\\frac{n}{2}}\\Gamma(\\frac{n}{2})}\n",
      "  \\int^x_0(u^{\\frac{n}{2} - 1} exp(-\\frac{u}{2}))du \\\\\n",
      "p =& 1 - f(Q, s - 1)\n",
      "\\end{array}\n",
      "\n",
      "N(0,1) \u306b\u304a\u3044\u3066\n",
      "- P(x>0.253) = 0.4\n",
      "- P(x>0.524) = 0.3\n",
      "- P(x>0.842) = 0.2\n",
      "- P(x>1.282) = 0.1"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#\u30c7\u30fc\u30bf\n",
      "random.seed(1)\n",
      "orig_data = random.randn(10000) * 10 + 50\n",
      "orig_data = [sum(random.random(5)) for i in range(10000)]\n",
      "hist(orig_data)\n",
      "m, s = mean(orig_data), std(orig_data)\n",
      "data = (orig_data - m) / s # \u6b63\u898f\u5316\n",
      "cum = [len(data[data < i]) for i in (-1.282, -0.842, -0.524, -0.253, 0, 0.253, 0.524, 0.842, 1.282)]\n",
      "num = [b - a for a, b in zip([0] + cum, cum + [len(data)])]\n",
      "ns = len(data) / len(num)\n",
      "Q = sum((i - ns)**2 / ns for i in num)\n",
      "from scipy.integrate import quad\n",
      "def f(x, n):\n",
      "    return quad(lambda u:u**(n / 2 - 1) * exp(-u / 2), 0, x)[0] / 2**(n / 2) / math.gamma(n / 2)\n",
      "p = 1 - f(Q, len(num) - 1 - 2)\n",
      "print('p = %.2f  Q = %.2f  %s' % (p, Q, num))\n",
      "print('\u6b63\u898f\u5206\u5e03\u3067\u306a\u3044\u3068\u306f\u3044\u3048\u306a\u3044' if p > 0.05 else\n",
      "      '\u6b63\u898f\u5206\u5e03\u3067\u306a\u3044')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "p = 0.46  Q = 6.68  [1054, 972, 987, 991, 990, 984, 975, 991, 1035, 1021]\n",
        "\u6b63\u898f\u5206\u5e03\u3067\u306a\u3044\u3068\u306f\u3044\u3048\u306a\u3044\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x6ae6588>"
       ]
      }
     ],
     "prompt_number": 2
    }
   ],
   "metadata": {}
  }
 ]
}
	{
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	"worksheets": [
	{
	"cells": [
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"#\u6b63\u898f\u5206\u5e03\u6027\u306e\u691c\u8a3c\n",
	"\u6307\u5b9a\u3055\u308c\u305f\u5217\u306e\u6b63\u898f\u5206\u5e03\u3057\u3066\u3044\u308b\u5ea6\u5408\u3044\u3092$\\chi^2$\u691c\u5b9a\u3067\u8a08\u7b97\u3059\u308b\u3002 \u30ea\u30b9\u30c8\u306e\u500b\u6570\u306f\u300110\u306e\u500d\u6570\u3067\u300150\u4ee5\u4e0a\u3068\u3059\u308b\u3002\n",
	"\n",
	"\u6b63\u898f\u5206\u5e03\u306e\u9762\u7a4d\u304c10(=s)\u7b49\u5206\u306b\u306a\u308b\u3088\u3046\u306b\u5206\u5272\u3059\u308b\n",
	"\n",
	"\\begin{array}{rl}\n",
	"n_s =& \\frac{n}{s} \\\\\n",
	"Q =& \\sum_i{\\frac{(n_s - k_i)^2}{n_s}} \\\\\n",
	"f(x, n) =& \\frac{1}{2^{\\frac{n}{2}}\\Gamma(\\frac{n}{2})}\n",
	" \\int^x_0(u^{\\frac{n}{2} - 1} exp(-\\frac{u}{2}))du \\\\\n",
	"p =& 1 - f(Q, s - 1)\n",
	"\\end{array}\n",
	"\n",
	"N(0,1) \u306b\u304a\u3044\u3066\n",
	"- P(x>0.253) = 0.4\n",
	"- P(x>0.524) = 0.3\n",
	"- P(x>0.842) = 0.2\n",
	"- P(x>1.282) = 0.1"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"#\u30c7\u30fc\u30bf\n",
	"random.seed(1)\n",
	"orig_data = random.randn(10000) * 10 + 50\n",
	"orig_data = [sum(random.random(5)) for i in range(10000)]\n",
	"hist(orig_data)\n",
	"m, s = mean(orig_data), std(orig_data)\n",
	"data = (orig_data - m) / s # \u6b63\u898f\u5316\n",
	"cum = [len(data[data < i]) for i in (-1.282, -0.842, -0.524, -0.253, 0, 0.253, 0.524, 0.842, 1.282)]\n",
	"num = [b - a for a, b in zip([0] + cum, cum + [len(data)])]\n",
	"ns = len(data) / len(num)\n",
	"Q = sum((i - ns)**2 / ns for i in num)\n",
	"from scipy.integrate import quad\n",
	"def f(x, n):\n",
	" return quad(lambda u:u*(n / 2 - 1) exp(-u / 2), 0, x)[0] / 2**(n / 2) / math.gamma(n / 2)\n",
	"p = 1 - f(Q, len(num) - 1 - 2)\n",
	"print('p = %.2f Q = %.2f %s' % (p, Q, num))\n",
	"print('\u6b63\u898f\u5206\u5e03\u3067\u306a\u3044\u3068\u306f\u3044\u3048\u306a\u3044' if p > 0.05 else\n",
	" '\u6b63\u898f\u5206\u5e03\u3067\u306a\u3044')"
	],
	"language": "python",
	"metadata": {},
	"outputs": [
	{
	"output_type": "stream",
	"stream": "stdout",
	"text": [
	"p = 0.46 Q = 6.68 [1054, 972, 987, 991, 990, 984, 975, 991, 1035, 1021]\n",
	"\u6b63\u898f\u5206\u5e03\u3067\u306a\u3044\u3068\u306f\u3044\u3048\u306a\u3044\n"
	]
	},
	{
	"metadata": {},
	"output_type": "display_data",
	"png": 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	"text": [
	"<matplotlib.figure.Figure at 0x6ae6588>"
	]
	}
	],
	"prompt_number": 2
	}
	],
	"metadata": {}
	}
	]
	}