Notes 02 - Linear Regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "from scipy.optimize import minimize\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>area</th>\n",
       "      <th>value</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>57.02</td>\n",
       "      <td>202970</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>96.07</td>\n",
       "      <td>312626</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>93.18</td>\n",
       "      <td>264921</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>91.05</td>\n",
       "      <td>272091</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>69.76</td>\n",
       "      <td>209846</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    area   value\n",
       "0  57.02  202970\n",
       "1  96.07  312626\n",
       "2  93.18  264921\n",
       "3  91.05  272091\n",
       "4  69.76  209846"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = pd.read_csv(\"data/houses.csv\")\n",
    "d.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10ea5b790>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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KM6tZLYFkVcnxM8AjwDum8yGSfo+sd/KR1BMqz3vOMw+6obTAZhtfr2xi6nKj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\nMyvE/wcecL/n2DEkewAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e79c410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(d.area, d.value)\n",
    "plt.xlabel(\"house area in $m^2$\")\n",
    "plt.ylabel(\"house value in USD\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ASUMIMOS** que la relación entre el área ($y$) y el valor ($x_1$) es lineal, para la casa $i$:\n",
    "\n",
    "$$y^{(i)} = \\theta_0 + \\theta_1 x_1^{(i)}$$\n",
    "\n",
    "si establecemos que $x_0^{(i)}=1$ para todas las casas, entonces podemos escribir la expresión anterior de forma vectorial, con $\\theta = [\\theta_0, \\theta_1]$ y $\\mathbf{x}=[x_0, x_1]$\n",
    "\n",
    "$$y^{(i)} = \\theta\\mathbf{x}^{(i)}$$\n",
    "\n",
    "queremos encontrar los valores $\\theta_{min}$ que parametrizan una recta de modo que\n",
    "\n",
    "$$ J(\\theta) =  \\sum_{i=0}^{m-1} (\\mathbf{x}\\theta^{(i)}-y^{(i)})^2$$\n",
    "\n",
    "$$\\theta_{min} = \\underset{\\theta}{arg\\,min} \\,\\,\\,J(\\theta)$$ \n",
    "\n",
    "en modo matricial\n",
    "\n",
    "$$J(\\theta) = sum(\\mathbf{X}\\theta-\\mathbf{y})^2$$\n",
    "\n",
    "P.ej., los parámetros $\\theta_0=0$, $\\theta_1=3800$ definen una recta que es mejorable respecto a los datos que tenemos. Observa como creamos las matrices $X$ e $y$, añadiendo una columna de unos a la matrix $X$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100, 2) (100,)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10ed7ebd0>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KfDXtz4tTHKO3Ck3KNREzKzGzl8xsdvh7DzOba2ZLzKzWzLolnDvWzOrNbLGZVSakDzSz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a52c990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.hstack((np.ones((len(d),1)), d[[\"area\"]].as_matrix()))\n",
    "y = d.value.as_matrix()\n",
    "print X.shape, y.shape\n",
    "\n",
    "t = np.array([0,3800])\n",
    "\n",
    "xg = np.vstack((np.ones(100), np.linspace(np.min(d.area), np.max(d.area),100))).T\n",
    "plt.scatter(d.area, d.value)\n",
    "plt.plot(xg[:,1], xg.dot(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Una primera aproximación para obtener la recta con los parámetros $\\theta_{min}$ según definido anteriormente es usando optimización _caja negra_. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cost(t):\n",
    "    return np.sum((X.dot(t)-y)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      fun: 87173814841.43971\n",
      " hess_inv: array([[  6.22486109e-02,  -5.63373937e-04],\n",
      "       [ -5.63373937e-04,   5.54403305e-06]])\n",
      "      jac: array([    0., -4096.])\n",
      "  message: 'Desired error not necessarily achieved due to precision loss.'\n",
      "     nfev: 135\n",
      "      nit: 4\n",
      "     njev: 31\n",
      "   status: 2\n",
      "  success: False\n",
      "        x: array([ 24848.96670977,   2936.34079734])\n"
     ]
    }
   ],
   "source": [
    "r = minimize(cost, [0,0], method=\"BFGS\")\n",
    "print r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10ed8e110>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8eJHHNZHW/MJLese95moodXXuf/2r+8iR7u3bf+jwd7WHi+xlSKAmUqDzhwtH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\nNWokl9cAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ed8e710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = r.x\n",
    "xg = np.vstack((np.ones(100), np.linspace(np.min(d.area), np.max(d.area),100))).T\n",
    "plt.scatter(d.area, d.value)\n",
    "plt.plot(xg[:,1], xg.dot(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El error de predicción medio que tenemos es de:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAASCAYAAAB4p22AAAAABHNCSVQICAgIfAhkiAAABPpJREFU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\nuqKKdn3oQx/60Ic+dId/ASsR0ihq81WOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$22825.4393205$$"
      ],
      "text/plain": [
       "22825.4393205"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(np.abs(X.dot(t)-y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El gradiente de $J$ es\n",
    "\n",
    "$$\\nabla J= 2X^{T}\\cdot(X\\cdot\\theta-Y)$$\n",
    "\n",
    "que podemos usar en los procesos de minimización"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def grad(t):\n",
    "    return 2*X.T.dot(X.dot(t)-y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      fun: 87173791346.97466\n",
      " hess_inv: array([[  6.23113465e-02,  -5.64148580e-04],\n",
      "       [ -5.64148580e-04,   5.55323927e-06]])\n",
      "      jac: array([  6.40284270e-10,  -1.00815415e-08])\n",
      "  message: 'Optimization terminated successfully.'\n",
      "     nfev: 6\n",
      "      nit: 3\n",
      "     njev: 6\n",
      "   status: 0\n",
      "  success: True\n",
      "        x: array([ 24794.85968752,   2936.83230103])\n"
     ]
    }
   ],
   "source": [
    "r = minimize(cost, [0,0], jac=grad, method=\"BFGS\")\n",
    "print r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "también podemos obtener el gradiente usando sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAWCAYAAABqgnq6AAAABHNCSVQICAgIfAhkiAAAAsFJREFU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\nXLMouoRnyK5miQsssMC8cg+b6ZKTxmECUwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( \\theta_{0}, \\quad \\theta_{1}\\right )$$"
      ],
      "text/plain": [
       "(θ₀, θ₁)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t0,t1 = sympy.symbols(\"theta_0 theta_1\")\n",
    "t0,t1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "fíjate en la expresión que obtenemos con un número reducido de datos y cómo la simplifica sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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k3Y+lEhJNbh5GuDOzD/9JeLTMmOnTH+UuUYfclZ++/pK7hu8uSOcvucuG3JUf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\nCCGEEEIIIQbD/wN7oF6XBqfHHgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{100} \\left(1.0 \\theta_{0} + 57.02 \\theta_{1} - 202970\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 57.62 \\theta_{1} - 161472\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 69.76 \\theta_{1} - 209846\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 81.67 \\theta_{1} - 241927\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 91.05 \\theta_{1} - 272091\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 93.18 \\theta_{1} - 264921\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 96.07 \\theta_{1} - 312626\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 99.59 \\theta_{1} - 317444\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 111.62 \\theta_{1} - 350402\\right)^{2} + \\frac{1}{100} \\left(1.0 \\theta_{0} + 149.57 \\theta_{1} - 478948\\right)^{2}$$"
      ],
      "text/plain": [
       "                            2                               2                 \n",
       "(1.0⋅θ₀ + 57.02⋅θ₁ - 202970)    (1.0⋅θ₀ + 57.62⋅θ₁ - 161472)    (1.0⋅θ₀ + 69.7\n",
       "───────────────────────────── + ───────────────────────────── + ──────────────\n",
       "             100                             100                             1\n",
       "\n",
       "              2                               2                               \n",
       "6⋅θ₁ - 209846)    (1.0⋅θ₀ + 81.67⋅θ₁ - 241927)    (1.0⋅θ₀ + 91.05⋅θ₁ - 272091)\n",
       "─────────────── + ───────────────────────────── + ────────────────────────────\n",
       "00                             100                             100            \n",
       "\n",
       "2                               2                               2             \n",
       "    (1.0⋅θ₀ + 93.18⋅θ₁ - 264921)    (1.0⋅θ₀ + 96.07⋅θ₁ - 312626)    (1.0⋅θ₀ + \n",
       "─ + ───────────────────────────── + ───────────────────────────── + ──────────\n",
       "                 100                             100                          \n",
       "\n",
       "                  2                                2                          \n",
       "99.59⋅θ₁ - 317444)    (1.0⋅θ₀ + 111.62⋅θ₁ - 350402)    (1.0⋅θ₀ + 149.57⋅θ₁ - 4\n",
       "─────────────────── + ────────────────────────────── + ───────────────────────\n",
       "   100                             100                              100       \n",
       "\n",
       "      2\n",
       "78948) \n",
       "───────\n",
       "       "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = 0\n",
    "for i in range(10):\n",
    "    expr += (X[i,0]*t0+X[i,1]*t1-y[i])**2\n",
    "expr = expr/len(X)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[202970 312626 264921 272091 209846 317444 161472 478948 241927 350402]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([[   1.  ,   57.02],\n",
       "       [   1.  ,   96.07],\n",
       "       [   1.  ,   93.18],\n",
       "       [   1.  ,   91.05],\n",
       "       [   1.  ,   69.76],\n",
       "       [   1.  ,   99.59],\n",
       "       [   1.  ,   57.62],\n",
       "       [   1.  ,  149.57],\n",
       "       [   1.  ,   81.67],\n",
       "       [   1.  ,  111.62]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print y[:10]\n",
    "X[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8Xh1PSn0LlPmefPspui9gXFTtqVrxOW9R9aJ4r5qeqmVobzvfc7QxScwVsYKMwZr4eOBt\ndH9pYCI2yGdVR29l0BNrIi4x9ZGRtPFyInlBol8M6iQ7kRizxFwRK9gBe2s0Ec5E7Aq6eEFpD+x7\nmaGkiXV8EVMfaby+vEhekOgXvWoreVFi3BF7xTQRjz2Bb2AvB1xI9e2+Ng4CzsfMbzFwVLTSJQZN\nDH0kXn6kk+xEbNI8k0jUEHvFNJFIJBJjl3SSnUgkxg3lP0M51liBffvwSOxzPftgbyY/MchCJRKJ\nRCIqB2GPB85m+A9vLBtoiRKJRCKRSCQSiUQikUgkEolEIpFIJBKJRCKRSCQSiUQikUgkEolEIpFI\nJBID4/+ct1D9WXRDUwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.1 \\theta_{0}^{2} + 18.143 \\theta_{0} \\theta_{1} - 56252.94 \\theta_{0} + 890.582285 \\theta_{1}^{2} - 5542599.536 \\theta_{1} + 8647291505.71$$"
      ],
      "text/plain": [
       "      2                                             2                         \n",
       "0.1⋅θ₀  + 18.143⋅θ₀⋅θ₁ - 56252.94⋅θ₀ + 890.582285⋅θ₁  - 5542599.536⋅θ₁ + 86472\n",
       "\n",
       "        \n",
       "91505.71"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = expr.simplify()\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "obtenemos la expresion del coste para todos los datos y las derivadas parciales para el gradiente"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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39W72qxHta6W2HYfWGV9NLbsObUxpx3MJZVqgrT/ffKF7Y1xvX2wTJ/hAnPkg4uBbgeNS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\ngg==\n",
      "text/latex": [
       "$$100.0 \\theta_{1}^{2} + 20317.82 \\theta_{1} \\theta_{2} - 64629002.0 \\theta_{1} + 1122072.0633 \\theta_{2}^{2} - 7094452455.22 \\theta_{2} + 11306015873817.0$$"
      ],
      "text/plain": [
       "        2                                                   2                 \n",
       "100.0⋅θ₁  + 20317.82⋅θ₁⋅θ₂ - 64629002.0⋅θ₁ + 1122072.0633⋅θ₂  - 7094452455.22⋅\n",
       "\n",
       "                     \n",
       "θ₂ + 11306015873817.0"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_cost = 0\n",
    "for i in range(len(X)):\n",
    "    expr_cost += (X[i,0]*t0+X[i,1]*t1-y[i])**2\n",
    "expr_cost = expr_cost.simplify()\n",
    "expr_cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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r4kQikUgkEolEIpFIJBKJRCKRSCQSiUQikUgkEolEIiH8H8o9VTKh4PA/AAAAAElFTkSu\nQmCC\n",
      "text/latex": [
       "$$\\left ( 200.0 \\theta_{1} + 20317.82 \\theta_{2} - 64629002.0, \\quad 20317.82 \\theta_{1} + 2244144.1266 \\theta_{2} - 7094452455.22\\right )$$"
      ],
      "text/plain": [
       "(200.0⋅θ₁ + 20317.82⋅θ₂ - 64629002.0, 20317.82⋅θ₁ + 2244144.1266⋅θ₂ - 70944524\n",
       "55.22)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_dt0 = expr_cost.diff(t0)\n",
    "expr_dt1 = expr_cost.diff(t1)\n",
    "expr_dt0, expr_dt1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y la función de coste y el jacobiano evaluable en Python"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "s_cost = sympy.lambdify([[t0,t1]], expr_cost, \"numpy\")\n",
    "\n",
    "d0 = sympy.lambdify([[t0,t1]], expr_dt0, \"numpy\")\n",
    "d1 = sympy.lambdify([[t0,t1]], expr_dt1, \"numpy\")\n",
    "s_grad = lambda x: np.array([d0(x), d1(x)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y ahora los usamos en el minimizador"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      fun: 87173791346.97119\n",
      " hess_inv: array([[  6.23113465e-02,  -5.64150650e-04],\n",
      "       [ -5.64150650e-04,   5.55327713e-06]])\n",
      "      jac: array([ -2.23517418e-08,   0.00000000e+00])\n",
      "  message: 'Optimization terminated successfully.'\n",
      "     nfev: 38\n",
      "      nit: 4\n",
      "     njev: 37\n",
      "   status: 0\n",
      "  success: True\n",
      "        x: array([ 24794.85968752,   2936.83230103])\n"
     ]
    }
   ],
   "source": [
    "r = minimize(s_cost, [0,0], jac=s_grad, method=\"BFGS\")\n",
    "print r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "fíjate como las funciones son las mismas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "theta: [ 24359.01378228  20662.31165051]\n",
      "cost analytic: 3.52476985235e+14\n",
      "cost symbolic: 3.52476985235e+14\n",
      "gradient analytic: [  3.60055930e+08   3.97696749e+10]\n",
      "gradient symbolic: [  3.60055930e+08   3.97696749e+10]\n"
     ]
    }
   ],
   "source": [
    "t = np.array([np.random.random()*5000+20000, np.random.random()*1000+20000])\n",
    "print \"theta:\",t\n",
    "print \"cost analytic:\", cost(t)\n",
    "print \"cost symbolic:\", s_cost(t)\n",
    "\n",
    "print \"gradient analytic:\", grad(t)\n",
    "print \"gradient symbolic:\", s_grad(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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