ks.test.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.stats\n",
    "from astropy.io import fits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# reset defalult plotting values\n",
    "plt.rcParams['figure.figsize'] = (10, 7)\n",
    "plt.rc('font', family='sans-serif')\n",
    "plt.rc('axes', labelsize=14)\n",
    "plt.rc('axes', labelweight='bold')\n",
    "plt.rc('axes', titlesize=16)\n",
    "plt.rc('axes', titleweight='bold')\n",
    "plt.rc('axes', linewidth=2)\n",
    "plt.rc('xtick',labelsize=14)\n",
    "plt.rc('ytick',labelsize=14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Kolmogorov–Smirnov (KS) Testing\n",
    "## Checking for inconsistent distributions of data\n",
    "\n",
    "![]()\n",
    "\n",
    "### Prof. Robert Quimby\n",
    "&copy; 2019 Robert Quimby"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## In this tutorial you will..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Discuss how to determine if two samples are drawn from the same distribution\n",
    "- Plot histograms of the data values\n",
    "- See that plotting the cumulative distributions is more revealing\n",
    "- Learn about the Kolmogorov–Smirnov or \"KS\" Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Weakness of histograms in comparing distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = ???\n",
    "mean = ???\n",
    "std = ???\n",
    "sample = np.random.normal(mean, std, size=n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "scrolled": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "plt.hist(sample);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Plotting the parent distribution on top is not revealing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "low = mean - 5 * std\n",
    "high = mean + 5 * std\n",
    "plt.hist(sample, range=????, bins=????)\n",
    "xs = np.linspace(low, high, 100)\n",
    "ys = n / np.sqrt(2 * np.pi * std**2) * np.exp(-(xs - mean)**2 / 2 / std**2)\n",
    "plt.plot(xs, ys);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Check the sample moments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.mean(sample), np.std(sample), scipy.stats.skew(sample)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cumulative Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "????\n",
    "cfracs = ????\n",
    "plt.plot(sample, cfracs, ls='steps-mid');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "plt.plot(sample, cfracs, ls='steps-mid');\n",
    "\n",
    "# compare to the parent distributon\n",
    "cdf = lambda x: scipy.stats.norm.cdf(x, loc=mean, scale=std)\n",
    "plt.plot(xs, cdf(xs))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Kolmogorov–Smirnov (KS) Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can you confidently reject the **null hypothesis** that the sample is drawn from the parent population?\n",
    "\n",
    "* [Test explanation from NIST](https://itl.nist.gov/div898/handbook/eda/section3/eda35g.htm)\n",
    "* [KS-Test Wikipedia entry](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# in python\n",
    "????"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The \"KS statistic\" is the maximum absolute distance between the two curves"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "# KS statistic\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The p-value is the probability a random test would have a KS statistic as large or larger than the value obtained."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The relation between the KS statistic (D) and the p-value depends on\n",
    "- sample size\n",
    "- the parent population's distribution (e.g. Poisson, Gaussian)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.4"
  },
  "livereveal": {
   "scroll": "True",
   "start_slideshow_at": "selected"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}