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April 7, 2015 09:36
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PRML 2.5
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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# PRML 2.5 ノンパラメトリック法\n", | |
| "\n", | |
| "社内PRML勉強会 担当@hagino3000" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## ヒストグラム密度推定法\n", | |
| "\n", | |
| "$\\displaystyle\n", | |
| "pi = \\frac{n_{i}}{N\\Delta i}$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 99, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "from __future__ import division\n", | |
| "import scipy.stats" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 131, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "# 図2.24を再現するための確率分布 (混合正規分布)\n", | |
| "GAUSSIAN_1_MEAN = 0.28\n", | |
| "GAUSSIAN_1_STD = 0.08\n", | |
| "GAUSSIAN_2_MEAN = 0.77\n", | |
| "GAUSSIAN_2_STD = 0.10\n", | |
| "\n", | |
| "def multinormal_rand(n):\n", | |
| " ret = []\n", | |
| " for _ in xrange(n):\n", | |
| " if np.random.rand() <= 0.25:\n", | |
| " ret.append(np.random.normal(GAUSSIAN_1_MEAN, GAUSSIAN_1_STD))\n", | |
| " else:\n", | |
| " ret.append(np.random.normal(GAUSSIAN_2_MEAN, GAUSSIAN_2_STD))\n", | |
| " return np.array(ret)\n", | |
| "\n", | |
| "def multinormal_pdf(x):\n", | |
| " return (0.25 * scipy.stats.norm.pdf(x, GAUSSIAN_1_MEAN, GAUSSIAN_1_STD) + \n", | |
| " 0.75 * scipy.stats.norm.pdf(x, GAUSSIAN_2_MEAN, GAUSSIAN_2_STD))\n", | |
| "\n", | |
| "multinormal_pdf = np.vectorize(multinormal_pdf)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 389, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "<matplotlib.text.Text at 0x10f7ef550>" | |
| ] | |
| }, | |
| "execution_count": 389, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
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| "gg==\n" | |
| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114314910>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "def plot_pdf():\n", | |
| " X = np.linspace(0, 1, 1000)\n", | |
| " Y = multinormal_pdf(X)\n", | |
| " plt.plot(X, Y)\n", | |
| " plt.ylim(0, 5)\n", | |
| " \n", | |
| "plot_pdf()\n", | |
| "plt.title(u'図2.24を再現するために用いる混合正規分布')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 335, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "N = 50\n", | |
| "Xn = multinormal_rand(N)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 336, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def plot_histogram_density_estimation_method(X, N, delta=0.04):\n", | |
| " hist, bins = np.histogram(X, bins=np.arange(0, 1+delta, delta))\n", | |
| " # pi = ni/NDeltai\n", | |
| " hist = hist/(N*delta)\n", | |
| " plt.bar(bins[:-1], hist, width=delta)\n", | |
| " plt.title('$\\Delta = %f$' % delta)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 390, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x111eebd10>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_histogram_density_estimation_method(Xn, N, delta=0.04)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 391, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x11426e5d0>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 410 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_histogram_density_estimation_method(Xn, N, delta=0.08)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 392, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x110fcdad0>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_histogram_density_estimation_method(Xn, N, delta=0.25)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "$\\Delta$は大きすぎても小さすぎてもデータの特徴を捉えられない" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "- Good\n", | |
| " - 一旦ヒストグラムを求めてしまえば、元のデータ集合自体は破棄できる\n", | |
| " - ヒストグラムによるアプローチは、データ点が逐次的にあたえられても、容易に適用できる\n", | |
| "- Bad\n", | |
| " - 推定した密度が区間の縁で不連続になる\n", | |
| " - 次元数の増加に共なう計算規模の増大 (次元の呪い)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 2.5.1 カーネル密度推定法\n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "ある$D$次元のユークリッド空間中の未知の確率密度 $p(\\bf{x})$ から、観測値の集合が得られていて、この集合から $p(\\bf{x})$ の値を推定したいとする。ここで $\\bf{x}$ を含むある小さな領域 $R$ を考える。この領域に割りあてられた確率は\n", | |
| "\n", | |
| "$P = \\int_{R}{p(x)dx}$ (2.242)\n", | |
| "\n", | |
| "$p(\\bf{x})$から得られた$N$個の観測値からなるデータ集合をあつめたとする。各データ点が$R$中にある確率は$P$なので、$R$内の点の総数$K$は、二項分布に従う。\n", | |
| "\n", | |
| "$\\displaystyle\n", | |
| "Bin(K|N,P) = \\frac{N!}{K!(N - K)!}P^K(1 - P)^{N-K}$ (2.243)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "----\n", | |
| "[復習] \n", | |
| "二項分布 \n", | |
| "観測値$x \\in [0, 1]$が$N$個あり、それぞれ$x=1$となる確率が$\\mu$の場合に、$x=1$が$m$回出る時の確率分布\n", | |
| "\n", | |
| "$Bin(m|N,\\mu) = \\binom{N}{m}\\mu^m(1 - \\mu)^{N-m}$\n", | |
| "\n", | |
| "$\\displaystyle\n", | |
| "\\binom{N}{m} \\equiv \\frac{N!}{(N-M)!m!}$\n", | |
| "\n", | |
| "---" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "二項分布の平均の式 (2.11) より、点がこの領域内にある平均個数は$NP$、\n", | |
| "\n", | |
| "点がこの領域内にある平均割合は $E[K/N] = P$ \n", | |
| "平均の周りの分散は $var[K/N] = P(1-P)/N$ となる。よって大きなNについて、この分布は平均の周囲で鋭く尖ったものとなり\n", | |
| "\n", | |
| "$K \\simeq NP$ (2.244)\n", | |
| "\n", | |
| "しかし、$R$が、確率密度が $p(x)$ がこの領域内でほぼ一定とみなせるほど十分に小さいとも仮定できるのであれば\n", | |
| "\n", | |
| "$P \\simeq p(x)V$ (2.245)\n", | |
| "\n", | |
| "が成り立つ\n", | |
| "\n", | |
| "(2.244)と(2.245)から $p(x) = \\frac{K}{NV}$ (2.246)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## カーネルとしてParzen窓を使った場合\n", | |
| "\n", | |
| "カーネル関数の一例、単位立方体を表す\n", | |
| "\n", | |
| "$k((x - x_{n})/h)$ は、xを中心とする一辺がhの立方体の内部にデータ点$x_{n}$があれば1、そうでなければ0。この立方体内部の総点数は\n", | |
| "\n", | |
| "$\\displaystyle\n", | |
| "K = \\sum_{n=1}^{N}k(\\frac{x - x_n}{h})$ (2.248)\n", | |
| "\n", | |
| "この$K$を (2.246) に代入すると\n", | |
| "\n", | |
| "$\\displaystyle\n", | |
| "p(x) = \\frac{1}{N}\\sum_{n=1}^{N}\\frac{1}{h^D}k(\\frac{x-x_n}{h})$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 295, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def parzen(x):\n", | |
| " return 1 if np.abs(x) <= 1/2 else 0\n", | |
| "\n", | |
| "parzen = np.vectorize(parzen)\n", | |
| "\n", | |
| "def kernel_density_parzen(X, x, h):\n", | |
| " return 1/len(X) * np.sum(parzen((X - x)/h)/h)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 320, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "N = 50\n", | |
| "Xn = multinormal_rand(N)\n", | |
| "\n", | |
| "def plot_kernel_density_parzen(h=0.005):\n", | |
| " X = np.linspace(0, 1, 200)\n", | |
| " ret = []\n", | |
| " for x in X:\n", | |
| " ret.append(kernel_density_parzen(Xn, x, h))\n", | |
| " plt.plot(X, ret)\n", | |
| " plt.title('h=%f' % h)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 393, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x111b67590>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_kernel_density_parzen(0.005)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 394, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": [ | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x10febf810>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_kernel_density_parzen(0.07)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 402, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": [ | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x10ff57190>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_kernel_density_parzen(0.2)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "より滑らかな密度モデルとしてガウスカーネルを使った場合\n", | |
| "\n", | |
| "$\\displaystyle\n", | |
| "p(x) = \\frac{1}{N}\\sum_{n=1}^{N}\\frac{1}{(2\\pi h^2)^{D/2}}exp\\{-\\frac{||x-x_n||^2}{2h^2}\\}$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 404, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def kernel_density_gaussian(X, x, h):\n", | |
| " # 1次元データに限定した実装 (ユークリッド空間中の1次元データ同士の距離をnp.absを使って出している)\n", | |
| " return 1/len(X) * np.sum(1/np.sqrt((2*np.pi*h**2)) * np.exp(-1 * np.abs(X - x)**2/(2*h**2)))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 397, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def plot_kernel_density_gaussian(h=0.005):\n", | |
| " X = np.linspace(0, 1, 200)\n", | |
| " ret = []\n", | |
| " for x in X:\n", | |
| " ret.append(kernel_density_gaussian(Xn, x, h))\n", | |
| " plt.plot(X, ret)\n", | |
| " plt.title('h=%f' % h)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 398, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x113fea410>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_kernel_density_gaussian()\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 399, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| "rkJggg==\n" | |
| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114ac4ad0>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_kernel_density_gaussian(0.02)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 400, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| "3Ex2djZjx471wCcS3qZ1THUvStbXGGMxNGgdU0J4IqZuuOEGdDodq1evpqGhode+2tpatm3bhk6n\n", | |
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| "PEaPHu2xzyUGhyv9bO61xAbg3nvvJSwsjNWrV/P222/3bA8KCuLhhx8+5+3Shepbh4eH88wzz/Dm\n", | |
| "m2+Sn59Pfn5+z77MzEx+8IMf9Ky+KwYnrWMKoLGxkcbGxn6PkTehg5cnYkoMbZ6Iqdtuuw2j0cgH\n", | |
| "H3zAO++807Ndp9Nx0003cccdd2j3AYTP0Tqmvve972E0Gtm0aRPr1q1j3bp1Pfv0ej0LFy7knnvu\n", | |
| "0fZDCL/ka8/mitq9apMXtbS0UFBQQEtLC7GxsUycOHHAtayLi4spKirC5XKRkZEhBQOGGE/ElBja\n", | |
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| "7hOJjRBCCCGEEEIMhNeKBwghhBBCCCGEViSxEUIIIYQQQvg9SWyEEEIIIYQQfk8SGyGEEEIIIYTf\n", | |
| "k8RGCCGEEEII4fcksRFCCCGEEEL4PUlshBBCCCGEEH5PEhshhBBCCCGE35PERgghhBBCCOH3JLER\n", | |
| "QgghhBBC+D1JbIQQQgghhBB+TxIbIYQQQgghhN+TxEYIIYQQQgjh9ySxEUIIIYQQQvg9SWyEEEII\n", | |
| "IYQQfk8SGyGEEEIIIYTfk8RGCCGEEEII4fcksRFCCCGEEEL4vf8PqA3dDXYpuWEAAAAASUVORK5C\n", | |
| "YII=\n" | |
| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114523050>" | |
| ] | |
| }, | |
| "metadata": { | |
| "image/png": { | |
| "height": 310, | |
| "width": 411 | |
| } | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "plot_kernel_density_gaussian(0.1)\n", | |
| "plot_pdf()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "hは平滑化パラメータの役割を果たすのがわかる。画像処理のガウシアンフィルタもこれ、式を見ると重そうだけど並列処理ができるのでGPUを使えば良い。" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Python 2", | |
| "language": "python", | |
| "name": "python2" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 2 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython2", | |
| "version": "2.7.9" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 0 | |
| } |
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