{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Practical 7.2: Kolmogorov-Smirnov Test\n", "\n", "* Use the triolbite data above and perform a two-sided Kolmogorov-Smirnov test (two samples) on whether they come from the same distribution, or not.\n", "* Test whether each trilobite data comes from an exponential distribution or not. (You will need to generate an exponential distribution for this question, and normalise all the distributions first)\n", "\n", "**NB** : The necessary function CAN be found in scipy.stats [here](http://docs.scipy.org/doc/scipy-0.15.1/reference/generated/scipy.stats.ks_2samp.html)\n", "\n", "## BONUS (Harder)\n", "\n", "* Plot the cumulative distribution functions (CDF) for the trilobite data in one figure as linegraphs.\n", "* Plot an exponential CDF on top of this as well (with a dashed line)\n", "* Was there ever any suspicion the data might be exponential?\n", "\n", "**HINT** numpy has a function 'cumsum' which should make this easier." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p-value = 0.153861546495\n", "Accept the hypothesis that the samples comes from the same distribution.\n" ] }, { "data": { "image/png": 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QjB4+HGVSInVzNrAA+DXQHvgAKAZuzGFOIg1fKgUCd58HNHH3Le4+DhgcbVoi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\nSUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "H0: Trilobyte dataset 1 is an exponential distribution\n", "p-value = 2.91067367394e-06\n", "Reject H0 the samples do not come from an exponential distribution.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "H0: Trilobyte dataset 2 is an exponential distribution\n", "p-value = 6.18818081532e-10\n", "Reject H0 the samples do not come from an exponential distribution.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "\n", "import numpy as np\n", "from scipy import stats\n", "import matplotlib.pyplot as plt\n", "\n", "# Read in the records\n", "record1 = np.recfromtxt(\"./triloshape1.csv\") \n", "record2 = np.recfromtxt(\"./triloshape2.csv\") \n", "\n", "triloshape1 = np.array(record1, dtype=float)\n", "triloshape2 = np.array(record2, dtype=float)\n", "\n", "D, p = stats.ks_2samp(triloshape1, triloshape2)\n", "\n", "print \"p-value = {}\".format(p)\n", "if p <= 0.05:\n", " print \"Reject\",\n", "else:\n", " print \"Accept\",\n", "print \"the hypothesis that the samples comes from the same distribution.\"\n", "\n", "\n", "\n", "plt.figure()\n", "# Generate a histogram of the data and overlay the normalised exponential \n", "# distribution\n", "n, bins, patches = plt.hist(triloshape1, 10, normed=True)\n", "plt.plot(bins,stats.expon.pdf(bins),color = 'k', lw = 2., \n", " linestyle='--', label = 'Exponential Distribution')\n", "plt.xlim(0,2)\n", "plt.xlabel(\"Samples\")\n", "plt.ylabel(\"Mean length:width ratio\")\n", "plt.legend(loc = 'best')\n", "plt.show()\n", "\n", "D1, p1 = stats.ks_2samp(triloshape1, stats.expon.pdf(bins))\n", "# As before we compare triloshape1 with another dataset. This time\n", "# it is with an exponential distribution.\n", "print \"H0: Trilobyte dataset 1 is an exponential distribution\"\n", "print \"p-value = {}\".format(p1)\n", "if p1 <= 0.05:\n", " print \"Reject H0 the samples do not\",\n", "else:\n", " print \"Accept H0 the samples do\",\n", "print \"come from an exponential distribution.\"\n", "\n", "\n", "# Repeat as just done, for the second set of data\n", "plt.figure()\n", "n, bins, patches = plt.hist(triloshape2, 10, normed=True)\n", "plt.plot(bins,stats.expon.pdf(bins),color = 'k', lw = 2., \n", " linestyle='--', label = 'Exponential Distribution')\n", "plt.xlim(0,2)\n", "plt.xlabel(\"Samples\")\n", "plt.ylabel(\"Mean length:width ratio\")\n", "plt.legend(loc = 'best')\n", "plt.show()\n", "\n", "D2, p2 = stats.ks_2samp(triloshape2, stats.expon.pdf(bins))\n", "print \"H0: Trilobyte dataset 2 is an exponential distribution\"\n", "print \"p-value = {}\".format(p2)\n", "if p2 <= 0.05:\n", " print \"Reject H0 the samples do not\",\n", "else:\n", " print \"Accept H0 the samples do\",\n", "print \"come from an exponential distribution.\"\n", "\n", "\n", "\"\"\" BONUS \"\"\"\n", "\n", "plt.figure()\n", "n1, bins1, patches1 = plt.hist(triloshape2, 10, normed=True, \n", " cumulative=True, histtype = 'step', label = 'Trilobyte CDF 2')\n", "n2, bins2, patches2 = plt.hist(triloshape1, 10, normed=True, \n", " cumulative=True, histtype = 'step', label = 'Trilobyte CDF 1')\n", "plt.plot(bins2, stats.expon.cdf(bins2), color='r', lw=2., \n", " linestyle='--', label = 'Exponential Distribution')\n", "plt.xlabel(\"Samples\")\n", "plt.ylabel(\"Mean length:width ratio\")\n", "plt.legend(loc = 'best')\n", "plt.show()\n", "\n", "# There was not really any reason to suspect exponetiality however this is a useful exercise\n", "# on manipulating distributions other than normal." ] } ], "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": "2.7.11" } }, "nbformat": 4, "nbformat_minor": 0 }