machine-learning/gp-regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Import the required modules\n",
    "import numpy as np\n",
    "import matplotlib.pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Kernel Function\n",
    "# Currently only RBF kernel is implemented.\n",
    "def kernel_fn(x1, x2, kernel_type=\"RBF\", sigma_f=1.27, sigma_n=.3, l=1):\n",
    "    if kernel_type == \"RBF\":\n",
    "        '''\n",
    "        Sanity Check - kernel_fn(2, 2) = 1.27*1.27 + .3*.3 ~ 1.7\n",
    "        '''\n",
    "        return sigma_f**2 * np.exp(-(x1 - x2)**2/(2*l**2)) + (x1==x2)*sigma_n**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f54866c1c50>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f54866d16d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Training set \n",
    "x = np.array([-1.50, -1.00, -0.75, -0.40, -0.25, 0.00], dtype=np.double)\n",
    "y = np.array([[-1.75, -1.25, -.30, .20, .50, 1]], dtype=np.double)\n",
    "plt.plot(x.ravel(), y.ravel(), \"r*\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Prepare the K Matrix\n",
    "K = np.zeros(shape=(x.shape[0],x.shape[0]), dtype=np.double)\n",
    "\n",
    "# Fill in the K Matrix\n",
    "for i in range(K.shape[0]):\n",
    "    for j in range(K.shape[1]):\n",
    "        K[i][j] = kernel_fn(x[i], x[j])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Test Data\n",
    "# x_star = np.array([0.20, -1.2, -.6], dtype=np.double)\n",
    "x_star = np.linspace(-2.5, 1, 100)\n",
    "\n",
    "# Prepare the K Matrix\n",
    "K_star = np.zeros(shape=(x_star.shape[0],x.shape[0]), dtype=np.double)\n",
    "K_star_star = np.zeros(shape=(x_star.shape[0],x_star.shape[0]), dtype=np.double)\n",
    "\n",
    "# Fill in the K_star matrix\n",
    "for i in range(K_star.shape[0]):\n",
    "    for j in range(K_star.shape[1]):\n",
    "        K_star[i][j] = kernel_fn(x_star[i], x[j])\n",
    "\n",
    "# Fill in the K_star_star matrix\n",
    "for i in range(K_star_star.shape[0]):\n",
    "    for j in range(K_star_star.shape[1]):\n",
    "        K_star_star[i][j] = kernel_fn(x_star[i], x_star[j])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Compute the predictions with their variances.\n",
    "exp_y_pred = np.dot(np.dot(K_star, np.linalg.inv(K)), y.T)\n",
    "var_y_pred = K_star_star - np.dot(np.dot(K_star, np.linalg.inv(K)), K_star.T)\n",
    "#print(\"Expected Value of y-star = {} with co-variance = {}\".format(exp_y_pred, var_y_pred))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f5483bd0350>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Dk8nNfcyS5+B2dcFvfgOrVsErr8BUkx6n+m3fen3938nLe1IeEemF+vu7qapa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\nEH5Iwl0IIfzQ/wNdggyJmnD5mAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f54841b4890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Show the predictions, along with the variances.\n",
    "plt.plot(x.ravel(), y.ravel(), \"r*\")\n",
    "plt.plot(x_star.ravel(), exp_y_pred.ravel(), \"b\")\n",
    "plt.plot(x_star.ravel(), exp_y_pred.ravel() - 2*np.sqrt(np.diag(var_y_pred)), \"y\")\n",
    "plt.plot(x_star.ravel(), exp_y_pred.ravel() + 2*np.sqrt(np.diag(var_y_pred)), \"y\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}