fig03_04.ipynb

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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h1>Figure 3.4</h1>\n",
      "<h2> Transform a distribution </h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load http://www.astroml.org/_downloads/fig_transform_distribution.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r\"\"\"\n",
      "Transformation of Distribution\n",
      "------------------------------\n",
      "Figure 3.4.\n",
      "\n",
      "An example of transforming a uniform distribution. In the left panel, x\n",
      "is sampled from a uniform distribution of unit width centered on x = 0.5\n",
      "(:math:`\\mu` = 0 and W = 1; see Section 3.3.1). In the right panel,\n",
      "the distribution is transformed via y = exp(x). The form of the resulting\n",
      "pdf is computed from eq. 3.20.\n",
      "\"\"\"\n",
      "# Author: Jake VanderPlas\n",
      "# License: BSD\n",
      "#   The figure produced by this code is published in the textbook\n",
      "#   \"Statistics, Data Mining, and Machine Learning in Astronomy\" (2013)\n",
      "#   For more information, see http://astroML.github.com\n",
      "#   To report a bug or issue, use the following forum:\n",
      "#    https://groups.google.com/forum/#!forum/astroml-general\n",
      "import numpy as np\n",
      "from scipy import stats\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "#----------------------------------------------------------------------\n",
      "# This function adjusts matplotlib settings for a uniform feel in the textbook.\n",
      "# Note that with usetex=True, fonts are rendered with LaTeX.  This may\n",
      "# result in an error if LaTeX is not installed on your system.  In that case,\n",
      "# you can set usetex to False.\n",
      "from astroML.plotting import setup_text_plots\n",
      "setup_text_plots(fontsize=16, usetex=True)\n",
      "\n",
      "#------------------------------------------------------------\n",
      "# Set up the data\n",
      "np.random.seed(0)\n",
      "\n",
      "# create a uniform distribution\n",
      "uniform_dist = stats.uniform(0, 1)\n",
      "x_sample = uniform_dist.rvs(1000)\n",
      "x = np.linspace(-0.5, 1.5, 1000)\n",
      "Px = uniform_dist.pdf(x)\n",
      "\n",
      "# transform the data\n",
      "y_sample = np.exp(x_sample)\n",
      "y = np.exp(x)\n",
      "Py = Px / y\n",
      "\n",
      "#------------------------------------------------------------\n",
      "# Plot the results\n",
      "fig = plt.figure(figsize=(10, 5))\n",
      "fig.subplots_adjust(left=0.11, right=0.95, wspace=0.3, bottom=0.17, top=0.9)\n",
      "\n",
      "ax = fig.add_subplot(121)\n",
      "ax.hist(x_sample, 20, histtype='stepfilled', fc='#CCCCCC', normed=True)\n",
      "ax.plot(x, Px, '-k')\n",
      "ax.set_xlim(-0.2, 1.2)\n",
      "ax.set_ylim(0, 1.4001)\n",
      "ax.xaxis.set_major_locator(plt.MaxNLocator(6))\n",
      "ax.text(0.95, 0.95, r'$p_x(x) = {\\rm Uniform}(x)$',\n",
      "        va='top', ha='right',\n",
      "        transform=ax.transAxes)\n",
      "ax.set_xlabel('$x$')\n",
      "ax.set_ylabel('$p_x(x)$')\n",
      "\n",
      "\n",
      "ax = fig.add_subplot(122)\n",
      "ax.hist(y_sample, 20, histtype='stepfilled', fc='#CCCCCC', normed=True)\n",
      "ax.plot(y, Py, '-k')\n",
      "ax.set_xlim(0.85, 2.9)\n",
      "ax.xaxis.set_major_locator(plt.MaxNLocator(6))\n",
      "ax.text(0.95, 0.95, '$y=\\exp(x)$\\n$p_y(y)=p_x(\\ln y) / y$',\n",
      "        va='top', ha='right',\n",
      "        transform=ax.transAxes)\n",
      "ax.set_xlabel('$y$')\n",
      "ax.set_ylabel('$p_y(y)$')\n",
      "\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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iQy31g9aOMY1bNTDb7fpClGSKSWw2G8cccwyvv/46q1at4pxzzqGk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\nCabGYEZBd3e3xmBKQlEiISKx4tChQ7z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5QLeHUzCx962cizCOml1fMPexfn3H/T1D5OZk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\niIiIiEjM+v9gD0InnWKQOAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10c03eed0>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I have some questions about the concepts behind this figure:\n",
      "+ Is Equation 3.20 the same as $p(y) = p(x)\\frac{dx}{dy}$ ?    --> *yes*\n",
      "+ What is the meaning of the subscripts, is it merely to remind us what the variable is?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "from scipy import stats\n",
      "from matplotlib import pyplot as plt\n",
      "from astroML.plotting import setup_text_plots\n",
      "setup_text_plots(fontsize=16, usetex=True)\n",
      "\n",
      "#------------------------------------------------------------\n",
      "# Set up the data\n",
      "np.random.seed(0)\n",
      "\n",
      "# create a uniform distribution\n",
      "uniform_dist = stats.uniform(0, 1)\n",
      "x_sample = uniform_dist.rvs(1000)\n",
      "x = np.linspace(-0.5, 1.5, 1000)\n",
      "Px = uniform_dist.pdf(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# transform the data\n",
      "y_sample = np.exp(x_sample)\n",
      "y = np.exp(x)\n",
      "Py = Px / y  # This is the same as Py = Px / (dy/dx) \n",
      "\n",
      "\n",
      "#Let's say the transformation is y(x) = x^2\n",
      "#y_sample = x_sample**2\n",
      "#y = x**2\n",
      "#dy_dx= 2.0*x\n",
      "#Py = Px / (dy_dx)  # This is the same as Py = Px / (dy/dx) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#------------------------------------------------------------\n",
      "# Plot the results\n",
      "fig = plt.figure(figsize=(10,5), facecolor = '#f3f3f3')\n",
      "fig.subplots_adjust(left=0.10, right=0.95, wspace=0.3, bottom=0.20, top=0.9)\n",
      "ax = fig.add_subplot(121, axis_bgcolor='#f3f3f3')\n",
      "\n",
      "ax.hist(x_sample, 20, histtype='stepfilled', fc='#00FF00',alpha=0.2, normed=True)\n",
      "ax.plot(x, Px, '-k')\n",
      "ax.set_xlim(-0.2, 1.2)\n",
      "ax.set_ylim(0, 1.4001)\n",
      "ax.xaxis.set_major_locator(plt.MaxNLocator(6))\n",
      "ax.text(0.30, 0.95, r'$p_x(x) = {\\rm Uniform}(x)$',\n",
      "        va='top', ha='center',\n",
      "        transform=ax.transAxes)\n",
      "ax.set_xlabel('$x$')\n",
      "ax.set_ylabel('$p_x(x)$')\n",
      "ax = fig.add_subplot(122, axis_bgcolor='#f3f3f3')\n",
      "ax.hist(y_sample, 20, histtype='stepfilled', fc='#00FF00', alpha=0.2, normed=True)\n",
      "ax.plot(y, Py, '-k')\n",
      "ax.set_xlim(0.85, 2.9)\n",
      "ax.xaxis.set_major_locator(plt.MaxNLocator(6))\n",
      "ax.text(0.95, 0.95, '$y=\\exp(x)$\\n$p(y)dy = p(x)dx $',\n",
      "        va='top', ha='right',\n",
      "        transform=ax.transAxes)\n",
      "ax.set_xlabel('$y$')\n",
      "ax.set_ylabel('$p_y(y)$')\n",
      "\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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JTM+eLm4f+ZqSMUXucCkB5OTkYNOmTbBarTCbzdi+fbvfFEjxxul04oUXXhDvswhmdCEm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\nREQkDZNPIiIiIpKGyScRERERSfMfBodDJwSe8kwAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10f16d890>"
       ]
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h3 align='center'> Recap </h3>\n",
      "\n",
      "So I learned some things.  The first, and most important thing, is that my intuition was correct all along.  Equation 3.20 is overly complicated.  Why do we need to bring $\\phi^{-1}(y)$ into this, when we could easily just say $x$?  Here's a simpler way to say the same thing:\n",
      "$$ p(y) = p(x) (\\frac{dy}{dx})^{-1} $$\n",
      "or:\n",
      "$$ p(y)dy = p(x)dx $$\n",
      "I guess this is the \"easy to show part\", but it took me a little pencil and paper work, and some examples to convince myself that Equation 3.20 is the same, and that there were no hidden Jacobians somewhere.  In fact, Equation 3.20 really is just a Jacobian, I think.  This example just happens to be in one dimension, so the Jacobian is sort of simple.\n",
      "\n",
      "What else did I learn?\n",
      "\n",
      "That's mainly it.  Except that the names for background colors are inconsistent in Matplotlib-  `facecolor`, `axis_bgcolor`, and `fc`, are examples, all of which appeared in this short text figure.\n",
      "\n",
      "*Addendum-* Notice that the **units** work out.  If $p(x)$ is a probability density, then $p(x)dx$ is the probability (dimensionless) of a draw between $x_0$ and $x_0+dx$.  This makes sense, since $p(y)$ will have units of $\\frac{1}{y}$.  So $p(y)dy$ is dimensionless, too. "
     ]
    }
   ],
   "metadata": {}
  }
 ]
}