{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.mlab as mlab\n",
      "import matplotlib.pyplot as plt\n",
      "import numpy as np\n",
      "import pandas\n",
      "import scipy\n",
      "import scipy.optimize"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The original data has 44 measurements, but 4 are duplicates where the same temperature was recorded at each corner. We remove these duplicate measurements and then vizualize the measurements."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data = pandas.read_table('data/volcanodata.txt', names=('x', 'y', 't'), sep='\\s+')\n",
      "data = data.drop_duplicates()\n",
      "\n",
      "im = np.zeros((11, 11))\n",
      "for _, (x, y, t) in data.iterrows():\n",
      "    im[x / 10, y / 10] = t\n",
      "plt.matshow(im.T, origin='lower')\n",
      "ticks = [''] + [str(10 * i * 2) for i in xrange(6)]\n",
      "plt.gca().set_xticklabels(ticks)\n",
      "plt.gca().set_yticklabels(ticks)\n",
      "plt.colorbar()\n",
      "plt.title('Temperature Measurements')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "<matplotlib.text.Text at 0xa10114c>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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AEKVH0LWp6/jBO0DpESiOAZYBFmCABYDfMcAiYKjSI7gzWW7TMSIq3zwiIjJa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\n/3wEEVdJNTJEAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa0de96c>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Define the model."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def temperature(x, y, b_params):\n",
      "    \"\"\"T(x, y | t0, t1, x0, y0, lamda)\"\"\"\n",
      "    t0, t1, x0, y0, lam = b_params\n",
      "    return t0 + (t1 - t0) * np.exp(-(((x - x0)**2 + (y - y0)**2) / (2 * lam**2)))\n",
      "\n",
      "sigma = 30.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Find the optimal parameters."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def chisquare_temperature(b_params):\n",
      "    return np.sum(((data.t - temperature(data.x, data.y, b_params)) / sigma)**2)\n",
      "\n",
      "b_params_guess = [0, 200, 60, 60, 30]\n",
      "results = scipy.optimize.minimize(chisquare_temperature, b_params_guess)\n",
      "b_params_opt = results.x\n",
      "b_opt_cov = np.linalg.inv(0.5 * np.linalg.inv(results.hess_inv))\n",
      "\n",
      "print \"Optimal parameters:\", zip(['t0', 't1', 'x0', 'y0', 'lambda'], b_params_opt)\n",
      "print \"Min function value:\", results.fun\n",
      "print \"Standard errors of each parameter separately:\", np.sqrt(np.diag(b_opt_cov))\n",
      "print \"Parameter covariance:\"\n",
      "print b_opt_cov"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimal parameters: [('t0', 17.673484842250879), ('t1', 195.30201364892139), ('x0', 56.066149683291705), ('y0', 79.098303551787154), ('lambda', 38.891319341124706)]\n",
        "Min function value: 24.5336951143\n",
        "Standard errors of each parameter separately: [ 20.13250369  39.78433256   3.47624678   9.81651262   7.17102511]\n",
        "Parameter covariance:\n",
        "[[  405.31770485   -37.3178964     18.29330954    71.70887784\n",
        "    -87.35883056]\n",
        " [  -37.3178964   1582.79311745   -71.90147235  -332.81252471\n",
        "   -207.91406254]\n",
        " [   18.29330954   -71.90147235    12.08429165    19.83609197\n",
        "      5.70954513]\n",
        " [   71.70887784  -332.81252471    19.83609197    96.36392008\n",
        "     30.24770931]\n",
        " [  -87.35883056  -207.91406254     5.70954513    30.24770931\n",
        "     51.42360115]]\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot confidence regions around the maximim likelihood estimate of the volcano's position."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Marginalize out everything except x0 and y0.\n",
      "b_cov_x0_yo = b_opt_cov[2:4, 2:4]\n",
      "b_cov_x0_yo"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "array([[ 12.08429165,  19.83609197],\n",
        "       [ 19.83609197,  96.36392008]])"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Extract the parameters of the MVN distribution.\n",
      "_, _, x0, y0, _ = b_params_opt\n",
      "sqrt_b_cov_x0_yo = np.sqrt(b_cov_x0_yo)\n",
      "(sigma_x, sigma_xy), (_, sigma_y) = sqrt_b_cov_x0_yo\n",
      "x0, y0, sigma_x, sigma_y, sigma_xy"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "(56.066149683291705,\n",
        " 79.098303551787154,\n",
        " 3.4762467759197113,\n",
        " 9.8165126231358961,\n",
        " 4.4537727798602615)"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_error_ellipse(mu, sigma, std=2, ax=None, **kwargs):\n",
      "    if ax is None:\n",
      "        ax = plt.gca()\n",
      "    num_points = 1000\n",
      "    L = np.linalg.cholesky(sigma)\n",
      "    points = np.linspace(0, 1, num=num_points)\n",
      "    circle = np.array([np.cos(2 * np.pi * points),\n",
      "                       np.sin(2 * np.pi * points)]) * std\n",
      "    ellipse = L.dot(circle) + np.tile(mu, (num_points, 1)).T\n",
      "    ax.plot(ellipse[0, :], ellipse[1, :], **kwargs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mu = [x0, y0]\n",
      "covar = [[sigma_x, sigma_xy], [sigma_xy, sigma_y]]\n",
      "plt.xlim(0, 100)\n",
      "plt.ylim(0, 100)\n",
      "for std, alpha in zip((1, 2, 3), (1.0, 0.6, 0.3)):\n",
      "    plot_error_ellipse(mu, covar, linewidth=2, std=std, color='red', alpha=alpha)\n",
      "plt.gca().set_aspect('equal')\n",
      "plt.title('Volcano location: $1/2/3-\\sigma$ confidence regions')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "<matplotlib.text.Text at 0xa396a0c>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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98YV9fNFFxet//NGOIoiIYEghUIGQLxOsWLFCgwYN0jXXXCNJatWqla699lr9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\ncYQBAAAc9/8ByRFgdY4UWHsAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa0d3d4c>"
       ]
      }
     ],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}