{ "metadata": { "name": "", "signature": "sha256:7e6c22c200b7983bb4fd26bf8baf94d18d5b373dd09d5d84310d2c242ba5f3a7" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "#Rejection Sampling" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Using Uniform Distribution as an Envelop Distribution" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import random\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import matplotlib.mlab as mlab\n", "from scipy.stats import truncnorm\n", "\n", "#Domain of X\n", "xdomain = [-3, 3]\n", "\n", "#Multiplier Constant\n", "M = 2.0\n", "\n", "def pdf(x):\n", " \"\"\"\n", " Probability distribution function for Random Variable X\n", " from which we want to sample points. Here we assume \n", " we have truncated standard normal distribution in the domain of -3 to 3\n", " \"\"\"\n", " return truncnorm.pdf(x, xdomain[0], xdomain[1])\n", "\n", "def random_point_within_enveloping_region():\n", " \"\"\"\n", " Return random point within the enveloping region. \n", " For x we will randomly sample point between -3 and 3\n", " Since we are assuming uniform distribution, the height of the enveloping region\n", " at any x is 1/6. So for Y we randomly sample point between 0 and 1/6\n", " \"\"\"\n", " #Randomly sample x from -3 to 3\n", " x = random.uniform(xdomain[0], xdomain[1])\n", " \n", " # probability of obtain any x is equal to 1/6. i.e. height of enveloping region\n", " # for any X is 1/6. \n", " y = random.uniform(0, height_of_enveloping_region(x) ) \n", " \n", " return (x,y)\n", "\n", "def height_of_enveloping_region(x):\n", " \"\"\"Return height of enveloping region at x.\"\"\"\n", " return M * 1.0/6.0\n", " \n", "#Number of sample points to sample \n", "n = 100\n", "\n", "#Creating two arrays to capture accepted and rejected points\n", "accepted = []\n", "rejected = []\n", "M = 2.0\n", "\n", "#Run this loop until we got required number of valid points\n", "while len(accepted) < n:\n", " \n", " #Get random point\n", " x, y = random_point_within_enveloping_region()\n", " \n", " #If for any x if envelping region is below the distribution from which we want to sample points\n", " #increment the multipler constant and resample all the points. \n", " if height_of_enveloping_region(x) < pdf(x):\n", " print \"Increasing M from {0} to {1}\".format(M, M+1)\n", " accepted = []\n", " rejected = []\n", " M += 1.0 \n", " continue\n", " \n", " #If y is below blue curve then accept it\n", " if y < pdf(x):\n", " accepted.append((x, y))\n", " #otherwise reject it. \n", " else:\n", " rejected.append((x, y))\n", " \n", "x = np.linspace(a, b, 100)\n", "plt.plot(x, [pdf(i) for i in x], color='blue')\n", "plt.plot(x, [height_of_enveloping_region(x)/M for i in x], color='black', ls='dashed', lw=1)\n", "plt.plot(x, [height_of_enveloping_region(x) for i in x], color='black', ls='dashed', lw=2)\n", "plt.plot([x[0] for x in accepted], [x[1] for x in accepted] , 'ro', color='g')\n", "plt.plot([x[0] for x in rejected], [x[1] for x in rejected] , 'ro', color='r')\n", "plt.show() \n", "\n", "#Calculate expected value for the truncated standard normal distribution\n", "approxMean = sum([x[0] for x in accepted])/len(accepted)\n", "print \"Expected Mean = \", 0, pdf(0)\n", "print \"Approximated Mean = \", approxMean, pdf(approxMean)\n", "print \"Approximated Variance = \", sum([(x[0] - approxMean)**2 for x in accepted])/(len(accepted)-1)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Increasing M from 2.0 to 3.0\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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GDEn/HSE000q/fnDWWXDHHfDyy9oxZs+Gu+6C885Lb3qxi1vPVfw87ZJ2tmyhbPp0wH+a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\nCSTXs0MPO0wrqnXFFVBb63zxNi9Np36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"text": [ "" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "Expected Mean = 0 0.400022258921\n", "Approximated Mean = -0.00944774946307 0.400004406332\n", "Approximated Variance = 0.822905055332\n" ] } ], "prompt_number": 111 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using Gaussian Distribution as an Envelop Distribution" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import random\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import matplotlib.mlab as mlab\n", "from scipy.stats import truncnorm\n", "import matplotlib.mlab as mlab\n", "\n", "#Domain of X\n", "xdomain = [-3, 3]\n", "\n", "#Multiplier Constant\n", "M = 2.0\n", "\n", "\n", "\n", "def pdf(x):\n", " \"\"\"\n", " Probability distribution function for Random Variable X\n", " from which we want to sample points. Here we assume \n", " we have truncated standard normal distribution in the domain of -3 to 3\n", " \"\"\"\n", " return truncnorm.pdf(x, xdomain[0], xdomain[1])\n", "\n", "def random_point_within_enveloping_region():\n", " \"\"\"\n", " Assuming gaussian distribution as an enveloping distribution \n", " \"\"\"\n", " #Randomly sample x from gaussian distribution\n", " x = random.gauss(0, 3)\n", " y = random.uniform(0, height_of_enveloping_region(x)) \n", " return (x,y)\n", "\n", "def height_of_enveloping_region(x):\n", " \"\"\"Return height of enveloping region at x.\"\"\"\n", " return M * mlab.normpdf(x,0,3)\n", " \n", "#Number of sample points to sample \n", "n = 100\n", "\n", "#Creating two arrays to capture accepted and rejected points\n", "accepted = []\n", "rejected = []\n", "\n", "\n", "#Run this loop until we got required number of valid points\n", "while len(accepted) < n:\n", " \n", " #Get random point\n", " x, y = random_point_within_enveloping_region()\n", " if x < -3 or x > 3: continue\n", " \n", " #If for any x if envelping region is below the distribution from which we want to sample points\n", " #increment the multipler constant and resample all the points. \n", " if height_of_enveloping_region(x) < pdf(x):\n", " print \"Increasing M from {0} to {1}\".format(M, M+1)\n", " accepted = []\n", " rejected = []\n", " M += 1.0 \n", " continue\n", " \n", " #If y is below blue curve then accept it\n", " if y < pdf(x):\n", " accepted.append((x, y))\n", " #otherwise reject it. \n", " else:\n", " rejected.append((x, y))\n", " \n", "x = np.linspace(a, b, 100)\n", "plt.plot(x, [pdf(i) for i in x], color='blue', lw=2)\n", "plt.plot(x, [height_of_enveloping_region(i)/M for i in x], color='black', ls='dashed', lw=2)\n", "plt.plot(x, [height_of_enveloping_region(i) for i in x], color='black', ls='dashed', lw=2)\n", "plt.plot([x[0] for x in accepted], [x[1] for x in accepted] , 'ro', color='g')\n", "plt.plot([x[0] for x in rejected], [x[1] for x in rejected] , 'ro', color='r')\n", "plt.show() \n", "\n", "#Calculate expected value for the truncated standard normal distribution\n", "approxMean = sum([x[0] for x in accepted])/len(accepted)\n", "print \"Expected Mean = \", 0, pdf(0)\n", "print \"Approximated Mean = \", approxMean, pdf(approxMean)\n", "print \"Approximated Variance = \", sum([(x[0] - approxMean)**2 for x in accepted])/(len(accepted)-1)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Increasing M from 2.0 to 3.0\n", "Increasing M from 3.0 to 4.0\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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U7o+vWnSgzInBbLbUMhyaYNXIhRC89K9/8UYFTW8eJauGWIvnmgqlmaZjwsOZ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\ngsOHD7Nr1y5at26tmDNj27ZtvPeePGNOkiRiYmL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"text": [ "" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "Expected Mean = 0 0.400022258921\n", "Approximated Mean = 0.0957004025778 0.398194631387\n", "Approximated Variance = 1.10797491153\n" ] } ], "prompt_number": 113 } ], "metadata": {} } ] }