Linear_Regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.optimize\n",
    "import time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load and preprocess the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data_original = np.loadtxt('stanford_dl_ex/ex1/housing.data')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = np.insert(data_original, 0, 1, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np.random.shuffle(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create train & test sets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "train_X = data[:400, :-1]\n",
    "train_y = data[:400, -1]\n",
    "\n",
    "test_X = data[400:, :-1]\n",
    "test_y = data[400:, -1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "m, n = train_X.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the cost function and how to compute the gradient.<br>\n",
    "Both are needed for the subsequent optimization procedure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def cost_function(theta, X, y): \n",
    "    squared_errors = (X.dot(theta) - y) ** 2\n",
    "    J = 0.5 * squared_errors.sum()\n",
    "    return J"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def gradient(theta, X, y):\n",
    "    errors = X.dot(theta) - y\n",
    "    return errors.dot(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run a timed optimization and store the iteration values of the cost function (for latter investigation)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 3909.307674\n",
      "         Iterations: 25\n",
      "         Function evaluations: 34\n",
      "         Gradient evaluations: 34\n",
      "Optimization took 0.004139900207519531 seconds\n"
     ]
    }
   ],
   "source": [
    "J_history = []\n",
    "\n",
    "t0 = time.time()\n",
    "res = scipy.optimize.minimize(\n",
    "    fun=cost_function,\n",
    "    x0=np.random.rand(n),\n",
    "    args=(train_X, train_y),\n",
    "    method='bfgs',\n",
    "    jac=gradient,\n",
    "    options={'maxiter': 200, 'disp': True},\n",
    "    callback=lambda x: J_history.append(cost_function(x, train_X, train_y)),\n",
    ")\n",
    "t1 = time.time()\n",
    "\n",
    "print('Optimization took {s} seconds'.format(s=t1 - t0))\n",
    "optimal_theta = res.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's always interesting to take a more detailed look at the optimization results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1097fca90>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097ffac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(J_history, marker='o')\n",
    "plt.xlabel('Iterations')\n",
    "plt.ylabel('J(theta)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now compute the Root Mean Square Error on both the train and the test set and hopefully they are similar to each other. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RMS train error: 4.421146725707398\n",
      "RMS test error: 5.638168304221702\n"
     ]
    }
   ],
   "source": [
    "for dataset, (X, y) in (\n",
    "    ('train', (train_X, train_y)),\n",
    "    ('test', (test_X, test_y)),\n",
    "):\n",
    "    actual_prices = y\n",
    "    predicted_prices = X.dot(optimal_theta)\n",
    "    print(\n",
    "        'RMS {dataset} error: {error}'.format(\n",
    "            dataset=dataset,\n",
    "            error=np.sqrt(np.mean((predicted_prices - actual_prices) ** 2))\n",
    "        )\n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, let's have a more intuitive look at the predictions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1098966a0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "rkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097ffda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 8))\n",
    "plt.scatter(np.arange(test_y.size), sorted(test_y), c='b', edgecolor='None', alpha=0.5, label='actual')\n",
    "plt.scatter(np.arange(test_y.size), sorted(test_X.dot(optimal_theta)), c='g', edgecolor='None', alpha=0.5, label='predicted')\n",
    "plt.legend(loc='upper left')\n",
    "plt.ylabel('House price ($1000s)')\n",
    "plt.xlabel('House #')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}