fonnesbeck/Gaussian Processes.ipynb

## Gaussian Processes.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pymc as pm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sample data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = (-5, -4.9, -4.8, -4.7, -4.6, -4.5, -4.4, -4.3, -4.2, -4.1, -4, \n",
    "-3.9, -3.8, -3.7, -3.6, -3.5, -3.4, -3.3, -3.2, -3.1, -3, -2.9, \n",
    "-2.8, -2.7, -2.6, -2.5, -2.4, -2.3, -2.2, -2.1, -2, -1.9, -1.8, \n",
    "-1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1, -0.9, -0.8, -0.7, \n",
    "-0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0, 0.1, 0.2, 0.3, 0.4, 0.5, \n",
    "0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, \n",
    "1.9, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3, 3.1, \n",
    "3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4, 4.1, 4.2, 4.3, 4.4, \n",
    "4.5, 4.6, 4.7, 4.8, 4.9, 5)\n",
    "\n",
    "y = (1.04442478194401, 0.948306088493654, 0.357037759697332, 0.492336514646604, \n",
    "0.520651364364746, 0.112629866592809, 0.470995468454158, -0.168442254267804, \n",
    "0.0720344402575861, -0.188108980535916, -0.0160163306512027, \n",
    "-0.0388792158617705, -0.0600673630622568, 0.113568725264636, \n",
    "0.447160403837629, 0.664421188556779, -0.139510743820276, 0.458823971660986, \n",
    "0.141214654640904, -0.286957663528091, -0.466537724021695, -0.308185884317105, \n",
    "-1.57664872694079, -1.44463024170082, -1.51206214603847, -1.49393593601901, \n",
    "-2.02292464164487, -1.57047488853653, -1.22973445533419, -1.51502367058357, \n",
    "-1.41493587255224, -1.10140254663611, -0.591866485375275, -1.08781838696462, \n",
    "-0.800375653733931, -1.00764767602679, -0.0471028950122742, -0.536820626879737, \n",
    "-0.151688056391446, -0.176771681318393, -0.240094952335518, -1.16827876746502, \n",
    "-0.493597351974992, -0.831683011472805, -0.152347043914137, 0.0190364158178343, \n",
    "-1.09355955218051, -0.328157917911376, -0.585575679802941, -0.472837120425201, \n",
    "-0.503633622750049, -0.0124446353828312, -0.465529814250314, \n",
    "-0.101621725887347, -0.26988462590405, 0.398726664193302, 0.113805181040188, \n",
    "0.331353802465398, 0.383592361618461, 0.431647298655434, 0.580036473774238, \n",
    "0.830404669466897, 1.17919105883462, 0.871037583886711, 1.12290553424174, \n",
    "0.752564860804382, 0.76897960270623, 1.14738839410786, 0.773151715269892, \n",
    "0.700611498974798, 0.0412951045437818, 0.303526087747629, -0.139399513324585, \n",
    "-0.862987735433697, -1.23399179134008, -1.58924289116396, -1.35105117911049, \n",
    "-0.990144529089174, -1.91175364127672, -1.31836236129543, -1.65955735224704, \n",
    "-1.83516148300526, -2.03817062501248, -1.66764011409214, -0.552154350554687, \n",
    "-0.547807883952654, -0.905389222477036, -0.737156477425302, -0.40211249920415, \n",
    "0.129669958952991, 0.271142753510592, 0.176311762529962, 0.283580281859344, \n",
    "0.635808289696458, 1.69976647982837, 1.10748978734239, 0.365412229181044, \n",
    "0.788821368082444, 0.879731888124867, 1.02180766619069, 0.551526067300283\n",
    ")\n",
    "\n",
    "N = len(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x118442588>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "bNxgP3eL5GWuA7S/oepOmvsknYuIwZ1ajr7A5VeGNFAkzb6NRp+N2lZ/qqpddWp0ZjMIQ33PJhni\n",
       "565uXmYZ9gCAdmU3GwcA0D7CHgAKQNgDQAEIewAoAGEPAAUg7AGgAIQ9ABSAsAeAAvw/GwT/ob4e\n",
       "rPcAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1183c4a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'r.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Kernel function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "squared_exponential = lambda x, y, η2, ρ2: η2 * np.exp(-ρ2 * (x - y)**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Parameters of kernel function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "eta2 = pm.HalfCauchy('eta2', 0, 5, value=1)\n",
    "rho2 = pm.HalfCauchy('rho2', 0, 5, value=1)\n",
    "sigma2 = pm.HalfCauchy('sigma2', 0, 5, value=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Points on which to evaluate GP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x_pred = np.append(x, np.linspace(-6, 6, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Covariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@pm.deterministic\n",
    "def Sigma(eta2=eta2, rho2=rho2, sigma2=sigma2):\n",
    "    S = np.array([[squared_exponential(xi, yi, eta2, rho2) for xi in x_pred] for yi in x_pred])\n",
    "    # Add variance to \n",
    "    S[np.diag_indices_from(S)] += sigma2\n",
    "    return S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Default mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mu = np.zeros(len(x_pred))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use masked array to make predictions on linear mesh along with observed values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_masked = np.ma.masked_equal(np.append(y, np.zeros(10)), value=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "masked_array(data = [1.04442478194401 0.948306088493654 0.357037759697332 0.492336514646604\n",
       " 0.520651364364746 0.112629866592809 0.470995468454158 -0.168442254267804\n",
       " 0.0720344402575861 -0.188108980535916 -0.0160163306512027\n",
       " -0.0388792158617705 -0.0600673630622568 0.113568725264636\n",
       " 0.447160403837629 0.664421188556779 -0.139510743820276 0.458823971660986\n",
       " 0.141214654640904 -0.286957663528091 -0.466537724021695 -0.308185884317105\n",
       " -1.57664872694079 -1.44463024170082 -1.51206214603847 -1.49393593601901\n",
       " -2.02292464164487 -1.57047488853653 -1.22973445533419 -1.51502367058357\n",
       " -1.41493587255224 -1.10140254663611 -0.591866485375275 -1.08781838696462\n",
       " -0.800375653733931 -1.00764767602679 -0.0471028950122742\n",
       " -0.536820626879737 -0.151688056391446 -0.176771681318393\n",
       " -0.240094952335518 -1.16827876746502 -0.493597351974992 -0.831683011472805\n",
       " -0.152347043914137 0.0190364158178343 -1.09355955218051 -0.328157917911376\n",
       " -0.585575679802941 -0.472837120425201 -0.503633622750049\n",
       " -0.0124446353828312 -0.465529814250314 -0.101621725887347\n",
       " -0.26988462590405 0.398726664193302 0.113805181040188 0.331353802465398\n",
       " 0.383592361618461 0.431647298655434 0.580036473774238 0.830404669466897\n",
       " 1.17919105883462 0.871037583886711 1.12290553424174 0.752564860804382\n",
       " 0.76897960270623 1.14738839410786 0.773151715269892 0.700611498974798\n",
       " 0.0412951045437818 0.303526087747629 -0.139399513324585 -0.862987735433697\n",
       " -1.23399179134008 -1.58924289116396 -1.35105117911049 -0.990144529089174\n",
       " -1.91175364127672 -1.31836236129543 -1.65955735224704 -1.83516148300526\n",
       " -2.03817062501248 -1.66764011409214 -0.552154350554687 -0.547807883952654\n",
       " -0.905389222477036 -0.737156477425302 -0.40211249920415 0.129669958952991\n",
       " 0.271142753510592 0.176311762529962 0.283580281859344 0.635808289696458\n",
       " 1.69976647982837 1.10748978734239 0.365412229181044 0.788821368082444\n",
       " 0.879731888124867 1.02180766619069 0.551526067300283 -- -- -- -- -- -- --\n",
       " -- -- --],\n",
       "             mask = [False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False False False False False False False False\n",
       " False False False False False  True  True  True  True  True  True  True\n",
       "  True  True  True],\n",
       "       fill_value = 0.0)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_masked"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "gp = pm.MvNormalCov('gp', mu, Sigma, value=y_masked, observed=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = pm.MCMC(locals())\n",
    "#M.use_step_method(pm.AdaptiveMetropolis, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [-----------------100%-----------------] 2000 of 2000 complete in 255.6 sec"
     ]
    }
   ],
   "source": [
    "M.sample(2000, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Could not calculate Gelman-Rubin statistics. Requires multiple chains of equal length.\n"
     ]
    },
    {
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       "AElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1183c49e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.summary_plot(gp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot predictions (blue points)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11ee49898>]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ee49860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'r.')\n",
    "plt.plot(np.linspace(-6, 6, 10), gp.value[-10:], 'bo')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting rho2\n"
     ]
    },
    {
     "data": {
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       "/CvAzMzMLC0PW2ZmZmYd8rBlZmZm1iEPW2ZmZmYd8rBlZmZm1iEPW2ZmZmYd8rBlZmZm1iEPW2Zm\n",
       "ZmYd+gu7smaKkVpY/gAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ee53a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.plot(rho2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting eta2\n"
     ]
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ee53278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.plot(eta2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting sigma2\n"
     ]
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ec36a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.plot(sigma2)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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}