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": 21,
   "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 0x1138e28d0>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "565uXmYZ9gCAdmU3GwcA0D7CHgAKQNgDQAEIewAoAGEPAAUg7AGgAIQ9ABSAsAeAAvw/GwT/ob4e\n",
       "rPcAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113867b00>"
      ]
     },
     "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": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_masked = np.ma.masked_equal(np.append(y, np.zeros(10)), value=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = pm.MvNormalCov('y', mu, Sigma, value=y_masked, observed=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "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%-----------------] 5000 of 5000 complete in 615.7 sec"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/fonnescj/GitHub/pymc/pymc/StepMethods.py:1272: UserWarning: \n",
      "Covariance was not positive definite and proposal_sd cannot be computed by \n",
      "Cholesky decomposition. The next jumps will be based on the last \n",
      "valid covariance matrix. This situation may have arisen because no \n",
      "jumps were accepted during the last `interval`. One solution is to \n",
      "increase the interval, or specify an initial covariance matrix with \n",
      "a smaller variance. For this simulation, each time a similar error \n",
      "occurs, proposal_sd will be reduced by a factor .9 to reduce the \n",
      "jumps and increase the likelihood of accepted jumps.\n",
      "  warnings.warn(adjustmentwarning)\n"
     ]
    }
   ],
   "source": [
    "M.sample(5000, 4000)"
   ]
  },
  {
   "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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       "JKlZhpgkqVmGmCSpWYaYJKlZhpgkqVn/HzjZQyfaszXKAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11390de80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.summary_plot(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot predictions (blue points)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x113ab2470>]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "Awh7AOgAwh4AOoCwB4AOIOwBoAMIewDogP8HW4NYIP8RanAAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113ab28d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, _y, 'r.')\n",
    "plt.plot(np.linspace(-6, 6, 10), y.value[-10:], 'bo')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting ρ2\n"
     ]
    },
    {
     "data": {
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       "loiIiMiQKNASERERGRIFWiIiIiJDokBLREREZEgUaImIiIgMyf8H+WtDoMPW8akAAAAASUVORK5C\n",
       "YII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113b7aa58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.plot(rho2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting η2\n"
     ]
    },
    {
     "data": {
      "image/png": [
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       "Q5UkSVIFDlWSJEkVOFRJkiRV4FAlSZJUgUOVJElSBQ5VkiRJFXwDjM0Rk6zGDKAAAAAASUVORK5C\n",
       "YII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11a5b1c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.plot(eta2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting σ2\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11a5b1ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pm.Matplot.plot(sigma2)"
   ]
  }
 ],
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