Linear Regression/Linear Regression with Gradient Descent.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Regression with Gradient Descent: A Python Tutorial\n",
    "\n",
    "Ozzie Liu - ozzieliu.com - ozzie@ozzieliu.com\n",
    "\n",
    "*I learn best by doing and teaching*. And while Python has some excellent packages available for linear regression (like Statsmodels or Scikit-learn), I wanted to understand the intuition behind ordinay least squares (OLS) linear regression. How is the best fit found? How do you actually implement gradient descent?\n",
    "\n",
    "This guide is primarily modeled after Andrew Ng's excellent [Machine Learning course](http://www.ml-class.org/) available online for free. The programming exercises there are straightfoward and intuitive, but it's mainly in Matlab/Octave. I wanted to implement the same thing in Python with Numpy arrays.\n",
    "\n",
    "Contents:\n",
    "- Loading and looking at data\n",
    "- Cost function\n",
    "- Gradient Descent\n",
    "\n",
    "Packages:  \n",
    "`Pandas` for data frames and easy to read csv files  \n",
    "`Numpy` for array and matrix mathematics functions  \n",
    "`Matplotlib` for plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Loading the Data\n",
    "\n",
    "In the first exercise, we're performing a linear regression with one variable to predict profits for a food truck. The data contains 2 columns, population of a city (in 10,000s) and the profits of the food truck (in 10,000s). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = pd.read_csv('ex1data1.txt', names = ['population', 'profit'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   population   profit\n",
      "0      6.1101  17.5920\n",
      "1      5.5277   9.1302\n",
      "2      8.5186  13.6620\n",
      "3      7.0032  11.8540\n",
      "4      5.8598   6.8233\n"
     ]
    }
   ],
   "source": [
    "print data.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "## Split population and profit into X and y\n",
    "X_df = pd.DataFrame(data.population)\n",
    "y_df = pd.DataFrame(data.profit)\n",
    "\n",
    "## Length, or number of observations, in our data\n",
    "m = len(y_df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x106c84250>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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lpNYFAEBajGID75+T9Bq1b+D9KXe/by2F9lTcBIawpE6PJLUuAADSYqghzMwukvQGSbtU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\n2qUlRkLKFwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106b24250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,8))\n",
    "plt.plot(X_df, y_df, 'kx')\n",
    "plt.xlabel('Population of City in 10,000s')\n",
    "plt.ylabel('Profit in $10,000s')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The idea of linear regression is to find a relationship between our target or dependent variable (y) and a set of explanatory variables ($x_1, x_2...$). This relatonship can then be used to predict other values.\n",
    "\n",
    "In our case with one variable, this relationship is a line defined by parameters $\\beta$ and the following form: $y = \\beta_0 + \\beta_1x$, where $\\beta_0$ is our intercept.\n",
    "\n",
    "This can be extended to multivariable regression by extending the equation in vector form: $y=X\\beta$\n",
    "\n",
    "So how do I make the best line? In this figure, there are many possible lines. Which one is the best?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x106ef6410>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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RPwYuA0qNMV8YY74ASoAE4N4WVymenY26UF3x8fEUFBREZwOmTLCYYK2loLyA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\nAZMwSZKkDpiESZIkdeD/AcxhE4gHWB5vAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106c793d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,8))\n",
    "plt.plot(X_df, y_df, 'k.')\n",
    "plt.plot([5, 22], [6,6], '-')\n",
    "plt.plot([5, 22], [0,20], '-')\n",
    "plt.plot([5, 15], [-5,25], '-')\n",
    "plt.xlabel('Population of City in 10,000s')\n",
    "plt.ylabel('Profit in $10,000s')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cost Function\n",
    "It turns out that to make the best line to model the data, we want to pick parameters $\\beta$ that allows our predicted value to be as close to the actual value as possible. In other words, we want the distance or residual between our hypothesis $h(x)$ and y to be minimized.\n",
    "\n",
    "So we formally define a cost function using ordinary least squares that is simply the sum of the squared distances. To find the liner regression line, we minimize:\n",
    "$$J(\\beta) = \\frac{1}{2m}\\sum_{i=1}^m(h_\\beta(x^{(i)})-y^{(i)})^2$$\n",
    "\n",
    "Again the hypothesis that we're trying to find is given by the linear model:\n",
    "$$h_\\beta(x) = \\beta^{T}x = \\beta_0 + \\beta_1x_1$$\n",
    "\n",
    "The parameters of the model are the beta values. We adjust $\\beta_j$ to minimze the cost function $J(\\beta)$. \n",
    "\n",
    "And we can use batch gradient descent where each iteration performs the update\n",
    "$$\\beta_j := \\beta_j - \\alpha\\frac{1}{m}\\sum_{i=1}^m (h_\\beta(x^{(i)})-y^{(i)})x_{j}^{(i)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Whoa, what's gradient descent? And why are we updating that?\n",
    "\n",
    "Gradient descent simply is an algorithm that makes small steps along a function to find a local minimum. We can look at a simply quadratic equation such as this one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x_quad = [n/10 for n in range(0, 100)]\n",
    "y_quad = [(n-4)**2+5 for n in x_quad]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x107236e90>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8PC3MyPCUCAAi35QpCw4oX5KUl/eQMjIWekoE1A0KWEDqFVX+qU0JhYUBJwGA\n6FFUVPlETWFhQsBJgLpFAQtISWJi5ceTkgJOAgDRIzGx8qvHk5JKA04C1C0KWEAuTk/X2A4dDjh2\nXYMG2n788Z4SAUDkS0+/WB06jD3gWIcOYzRixEWeEgF1g20oArQ0M1MLMzKUUFio0qQknXXNNbr9\n3ns1fvx4DRkyxHc8AIhImZlLlZGxUIWFCUpKKtWIERexAB8Rpc53wvclVgtYZdasWaN+/fpp9erV\nOp7RMADQzp07VVhYqOOOO853FKBKKGBRqqCgQM2bN/cdAwC8Kykp0aBBg3TWWWfpwQcf9B0HqBIK\nGAAgajnndNtttyk3N1eZmZmqX7++70hAlbATPgAgak2ePFlLlixRTk4O5QsxjwIGAPBu1qxZmjBh\ngt5++20lJyf7jgOEHdtQRKCcnBw9+uijvmMAQGA+/PBDvf766zr55JN9RwECwRqwCLR582add955\nbE8BAEAUYBF+DFmzZo369u2rN954Qz179vQdBwAAHEJNChhTkBEqJSVF06dP1xVXXKG8gz7EGwAA\nRDcKWATr37+/7r33Xv3sZz9TaSmfewYgNjjntH37dt8xAK+YgowCubm56tixo+8YAFAnJk+erLlz\n52r+/Pm+owB1gjVgAICINnv2bP3617/W22+/zRWPiBlsxAoAiFgrVqzQjTfeqMzMTMoX4h5rwAAA\nYbdp0yYNGjRIf/3rX3XOOef4jgN4xxRkFJozZ45atGjB9hQAosbUqVNVUFCg0aNH+44C1DnWgMWJ\nuXPn6sYbb1R2drY6dOjgOw4AAHGNfcDixIABA3Tvvfdq4MCBXMoNAEAUYgQsit1xxx1atWqV5s2b\npwYNGviOAwBAXGIKMs6UlpZq8ODBOvHEE/WXv/zFdxwA2K+4uFj169f3HQMIBFOQcSYhIUEvvfSS\nfvnLX/qOAgD7rVixQj/+8Y9VVFTkOwoQsRgBAwDUmU2bNum8887TlClTNHjwYN9xgEAwAgYA8Gbn\nzp269NJLdfvtt1O+gCNgBAwAUGslJSUaNGiQWrdurSeffFJm1RoMAKIaI2CQJL3xxhtsTwEgUDk5\nOXLO6bHHHqN8AVXACFgMuvPOO7Vy5Uq2pwAQqLKyMh11FP+uR/xhGwpI+s/2FM2bN9fUqVP51ygA\nAGHEFCQk/Wd7ijVr1ujhhx/2HQcAAByknu8ACI9GjRppzpw56tGjhzp06KAhQ4b4jgQghjjnGF0H\naoERsBitMaYxAAAUJklEQVR24okn6s0331Tnzp19RwEQQ3bu3Km0tDRt3LjRdxQgajECFuNSUlJ8\nRwAQQ0pKSjRkyBCdeuqpOvnkk33HAaIWI2AAgCpxzmnkyJEqKSnR448/zhQkUAuMgAEAqmTy5Mla\nsmSJcnJy+KBtoJYYAYtDixcvFtt8AKiOzZs3a/LkyXrzzTeVnJzsOw4Q9dgHLM4UFxerR48euuKK\nKzRmzBjfcQBEkT179ujoo4/2HQOIODXZB4wpyDhTv359tqcAUCOUL6DuUMDi0L7tKfr27as2bdqo\nZ8+eviMBABBXWAMWp1JSUvTcc8/piiuuUF5enu84AADEFQpYHBswYIAmTJig0tJS31EARBDnnEaP\nHq1ly5b5jgLELBbhAwAO8Oc//1nPPPOMcnJyuOIRqAIW4QMAamXWrFmaMGGC3n77bcoXEEYUMACA\nJGnFihW66aablJmZyccMAWHGGjAcwDmnTz75xHcMAAErLi7W0KFD9de//lXnnHOO7zhAzGMNGA6Q\nn5+vs846S7NmzdJ5553nOw6AAH399dc6/vjjfccAok5N1oCFdQTMzKaa2VYz+6DCseZmttDMPjWz\nBWbWNJwZUD2tW7fW9OnTNXjwYK1fv953HAABonwBwQn3FOQ0SZccdOxuSQudc50k/SP0NSJI//79\ndd9992nAgAHavn277zgAAMScsE9BmllbSXOcc2eEvv5YUm/n3FYzaykpyznX+aDXMAUZAe644w6t\nWrVK8+bNU4MGDXzHAQAgIkXcFOQhnOCc2xq6v1XSCR4yoAomTJigs846SwUFBb6jAKhjc+bM0Suv\nvOI7BhC3vF4FGRrmYqgrQiUkJGjixIlq2bKl7ygA6tCKFSv03//932w1AXjkYx+wrWbW0jm3xcxa\nSfq6sieNGzdu//20tDSlpaUFkw4AYtimTZs0aNAgtpsAaiErK0tZWVm1OoePNWDjJX3rnPujmd0t\nqalz7u6DXsMaMACoYzt37lRqaqquu+46jR492nccIGbUZA1YWAuYmc2Q1FvSsSpf73WfpFmSXpb0\nI0kbJV3tnPvuoNdRwCKUc07ffvutjj32WN9RAFTTkCFD1LRpUz355JMyq9Z/KwAcRsQVsJqigEWu\nBQsWKD09Xe+8846aNWvmOw6Aavj000/Vrl071a9f33cUIKZQwBCIUaNGadWqVZo/fz7bUwAA4h4F\nDIEoLS3V4MGD1bx5c02dOpWpDABAXIuWfcAQ5RISEvTSSy9pzZo1evjhh33HAQAg6lDAUCONGjXS\nnDlztGLFCu3du9d3HAAH2bRpk6ZNm+Y7BoBDoIChxk488US99tprrAMDIszOnTt16aWXatu2bb6j\nADgE1oABQAwpKSnRZZddpjZt2rDdBBCQmqwB87ETPmLU0sxMLZgyRfWKilSSmKiL09N1wcCBvmMB\nccM5p9tvv12lpaV67LHHKF9ABKOAoU4szczU/JEj9VBe3v5jY0P3KWFAMP7yl78oKytLOTk57PUF\nRDimIFEnft+vn/5nwYIfHL+3Xz89OG+eh0RA/Pn000+VmJjIh2wDAWMKEt7UKyqq9HhCYWHASYD4\n1alTJ98RAFQRV0GiTpQkJlZ6fM369dq9e3fAaQAAiGwUMNSJi9PTNbZDhwOO3dWunUo6dNADDzzg\nKRUQmwoKCvTYY4+JpRpA9GIKEnVi30L7ezMylFBYqNKkJA0cMUJ/HDhQJSUlntMBseOtt97S8OHD\nNXjwYJWVlSkhIcF3JAA1wCJ8AIgChYWFuueee/TKK69o6tSpuvjii31HAhDCZ0EiahQUFDB9AlTR\nxo0bdfbZZys/P1+rV6+mfAExgBEwePGLX/xC33//vZ555hkdd9xxvuMAEW337t3KzMzUlVdeyeaq\nQARiBAxRY+rUqercubO6deumzMxM33GAiNawYUNdddVVlC8ghjACBq+WLFmiYcOGqX///po4caIa\nNWrkOxIAANXCCBiiTu/evbV69Wrt2rVLc+bM8R0H8KqgoEC33367vv/+e99RAIQZBQzeJScn6/nn\nn9eQIUN8RwG8eeutt9StWzc559haAogD7AOGiMH6FsSjPXv26J577tGrr76qadOm6aKLLvIdCUAA\nGAFDRPvkk0/YrgIxa+fOnerevbs2b96sNWvWUL6AOMIifEQs55xSU1N13HHH6emnn2a7CsSknJwc\n9ezZkxFgIIqxCB8xxcy0aNEiderUSd26ddPcuXN9RwLqXK9evShfQBxiBAxRge0qAACRihEwxKx9\n21UUFRXpiy++8B0HqJaCggJde+21+uCDD3xHARAhKGCIGsnJyZo6daq6dOniOwpQZfu2lzj22GN1\nyimn+I4DIEKwDQUAhAHbSwA4HEbAEBMWLVrEdhWIGM459enTR19++SXbSwCoFIvwEfV27Nih3r17\nq127dnrqqafYrgIRYcOGDWrbti1XOAJxgEX4iEvJyclavnw521UgorRr147yBeCQGAFDTNm3XcWA\nAQM0ceJENWzY0HckxLh9f1dRtoD4xQgY4t6+7SqaNGniOwriQEFBgX7+85/rhRde8B0FQJShgCHm\nJCcn65FHHmH0C2G1b3uJli1b6sorr/QdB0CUYRsKAKiGwsJC3XPPPXrllVfYXgJAjTEChrjx/fff\n68UXX2S7CtTKTTfdpPz8fK1evZryBaDGWISPuPH5559r0KBBOvnkk/XMM8+wXQVqZMeOHWrSpAmL\n7gHsxyJ84DBOPvlkLV++XJ07d1a3bt2UmZnpOxKiUHJyMuULQK0xAoa4tGTJEl1//fUaMGCAHnvs\nMSUkJPiOhAjjnNPevXuVmJjoOwqACMcIGFBFvXv31po1a/STn/yE8oUf2L59u4YOHaqxY8f6jgIg\nRlHAELeSk5N10003+Y6BCPOPf/xDKSkpatmypR588EHfcQDEKLahAACVby8xZswYvfzyy2wvASDs\nGAEDDrJ69Wo988wzbFcRZ/70pz8pPz9fa9asoXwBCDsW4QMH+eijjzR06FC1bdtWTz/9NNtVxImS\nkhIlJCRwhSOAamMRPlAHunTpouXLl+vUU09Vt27dNHfuXN+REIB69epRvgAEhhEw4DCWLFmiYcOG\n6eqrr9alvXtrwZQpqldUpJLERF2cnq4LBg70HRFVkJm5VFOmLFBRUT0lJhZr2LAeuuaa//IdC0CM\nqMkIGIvwgcPo3bu3Vq9erReffFLzR47UQ3l5+x8bG7pPCYtsmZlLNXLkfOXlPbT/WHb2cCUnJ2vg\nwAs8JgMQz5iCBI4gOTlZmxctOqB8SdJDeXlaMGWKp1SoqkceeeOA8iVJu3dPU0bGQk+JAIARMKBK\n6hUVVXr83cWL1bt3b/Xq1Uupqanq2bOnmjZtGnA6HM4nn+RXerywkA14AfjDCBhQBSWH+Diasy64\nQGPGjFG9evU0ceJEtW7dWj179mQLi4Ds2LFD8+bN07333qulS5dW+pwzz+xU6fGkpNJwRgOAw2IE\nDKiCi9PTNTYv74BpyDEdOmjAqFG6oF8/9evXT5JUXFysvLy8Sq+m27FjhzZs2KAzzjiDjz+qhXff\nfVfPP/+8srOz9dlnn+nss89Wr169dOyxx1b6/PT0i5WXN/aAacgOHcZoxIhLgooMAD/AVZBAFS3N\nzNTCjAwlFBaqNClJF40YUa0F+O+//76uu+46bd68WT169FBqaqpSU1N1zjnnqFGjRmFMHlsWL16s\nFStWKDU1VWeddZYaNGhwxNdkZi5VRsZCFRYmKCmpVCNGXMQCfAB1piZXQVLAgIB98803evvtt5Wd\nna2cnBx1795dkydP9h3Luz179ujdd9/d/3Np3ry5XnjhBd+xAOCIKGBAFHLOVTpl+eabb2rLli1K\nTU3VqaeeGrObhH7xxRcaMmSI1qxZo9NPP33/yGDPnj11wgkn+I4HAEdEAQNiyNy5c/XSSy8pOztb\nu3btUq9evdSrVy8NHTpUrVu39h2vWpxz+vzzz9W2bdsfPFZUVKR33nlH55xzjho2bBh8OACoJQoY\nEKM2bdqknJwc5eTk6Je//KVSUlJ8Rzqs4uJirVy5Ujk5OcrOzlZ2drYSExP18ccfU7IAxJyoKmBm\ntlHSvyWVSip2zp1T4TEKGFBNl156qdq0abN/Cu9HP/qRt2nLM888U2VlZUpNTd2/R9qPfvQjL1kA\nINyirYBtkPQT51xBJY9RwIBqevfddw8Ycapfv77OP/98/e1vf1O9enW740x+fr6ys7PVvXt3dejQ\n4QeP7927t0pXJwJALIjGAna2c+7bSh6jgAG14JxTXl6eVq1apSuvvPIHj5eUlKioqOiA7S8yMzM1\nZcoUFRUVKTExUenp6RoY2mYjLy9PCxYs2F/wdu3apdTUVI0ZM0bdu3cP7PsCgEgUbR/G7SS9ZWal\nkv7qnHvaYxYgppiZTjnlFJ1yyimVPv7hhx+qV69e6tq1q1JTU5WUlKQZM2Zo48aN+5+TF9p0duDA\ngVq8eLHee+89XXjhhbr33nvVqVOnmL0qEwCC4HMErJVz7iszO07SQkkjnHPLQo8xAgaE2Z49e/Te\ne+8pJydHjz76qLZt2/aD5/Tr10/z5s3zkA4AokdUjYA5574K/fmNmb0u6RxJy/Y9Pm7cuP3PTUtL\nU1paWsAJgdh29NFH64ILLtAFF1yg+fPna8mSJT94TmFhoYdkABDZsrKylJWVVatzeBkBM7OGkhKc\nczvNrJGkBZIecM4tCD3OCBgQoH79+mnBggWVHmcEDAAOryYjYEeFK8wRnCBpmZmtkrRc0pv7yheA\n4KWnp//gasYOHTpoxIgRnhIBQGxjI1YAksqvgszIyFBhYaGSkpI0YsSI/VdBAgAOLaq2oTgcChgA\nAIgW0TQFCQAAELcoYAAAAAGjgAEAAASMAgYAABAwChgAAEDAKGAAAAABo4ABAAAEjAIGAAAQMAoY\nAABAwChgAAAAAaOAAQAABIwCBgAAEDAKGAAAQMAoYAAAAAGjgAEAAASMAgYAABAwChgAAEDAKGAA\nAAABo4ABAAAEjAIGAAAQMAoYAABAwChgAAAAAaOAAQAABIwCBgAAEDAKGAAAQMAoYAAAAAGjgAEA\nAASMAgYAABAwChgAAEDAKGAAAAABo4ABAAAEjAIGAAAQMAoYAABAwChgAAAAAaOAAQAABIwCBgAA\nEDAKGAAAQMAoYAAAAAGjgAEAAASMAgYAABAwChgAAEDAKGAAAAABo4ABAAAEjAIGAAAQMAoYAABA\nwChgAAAAAaOAAQAABIwCBgAAEDAKGAAAQMAoYAAAAAGjgAEAAASMAgYAABAwChgAAEDAKGAAAAAB\no4ABAAAEzEsBM7NLzOxjM8s1s7t8ZAAAAPAl8AJmZgmSHpN0iaTTJA01sy5B50D4ZGVl+Y6AWuD3\nF7343UU3fn/xxccI2DmSPnPObXTOFUuaKWmQhxwIE/4SiW78/qIXv7voxu8vvvgoYCdJ2lTh6/zQ\nMQAAgLjgo4A5D+8JAAAQMcy5YPuQmfWQNM45d0no63sklTnn/ljhOZQ0AAAQNZxzVp3n+yhg9SR9\nIqmPpM2S3pU01Dn3UaBBAAAAPKkX9Bs650rM7DZJ8yUlSHqW8gUAAOJJ4CNgAAAA8S7idsJnk9bo\nZWZtzGyxma01sw/NLN13JlSPmSWY2Uozm+M7C6rHzJqa2atm9pGZrQutt0UUMLNRob8zPzCzl8ws\n0XcmHJqZTTWzrWb2QYVjzc1soZl9amYLzKzpkc4TUQWMTVqjXrGkUc65rpJ6SPoNv7+oM1LSOnG1\ncjSaLGmuc66LpBRJLO2IAmZ2kqQRkn7inDtD5Utzfu43FY5gmsp7SkV3S1ronOsk6R+hrw8rogqY\n2KQ1qjnntjjnVoXuf6/y/wCc6DcVqsrMWksaIOkZSdW6mgd+mVmypPOdc1Ol8rW2zrkdnmOh6upJ\nahi6SK2hpC8958FhOOeWSdp+0OHLJE0P3Z8u6fIjnSfSChibtMYIM2sr6UxJy/0mQTX8SdJvJZX5\nDoJqayfpGzObZmb/MrOnzayh71A4Mufcl5ImSfpC5TsDfOece8tvKtTACc65raH7WyWdcKQXRFoB\nY9ojBpjZMZJelTQyNBKGCGdml0r62jm3Uox+RaN6ks6S9IRz7ixJu1SFKRD4Z2bNVD560lblMwbH\nmNkvvIZCrbjyqxuP2GcirYB9KalNha/bqHwUDFHCzOpLek3SC865N3znQZX1lHSZmW2QNEPShWb2\nvOdMqLp8SfnOufdCX7+q8kKGyNdX0gbn3LfOuRJJf1f5/x8RXbaaWUtJMrNWkr4+0gsirYC9L6mj\nmbU1swaShkia7TkTqsjMTNKzktY55/7sOw+qzjk3xjnXxjnXTuULgBc55673nQtV45zbImmTmXUK\nHeoraa3HSKi6zyX1MLOjQ3+H9lX5hTCILrMlDQvdHybpiAMQgW/Eejhs0hr1ekm6VtIaM1sZOnaP\nc26ex0yoGZYDRJ8Rkl4M/eM1T9Jwz3lQBc65d83sVUn/klQS+vMpv6lwOGY2Q1JvScea2SZJ90l6\nRNLLZnajpI2Srj7iediIFQAAIFiRNgUJAAAQ8yhgAAAAAaOAAQAABIwCBgAAEDAKGAAAQMAoYAAA\nAAGjgAEAAASMAgYAABAwChiAmGdm3c1stZklmlkjM/vQzE7znQtA/GInfABxwcwelJQk6WhJm5xz\nf/QcCUAco4ABiAtmVl/S+5L2SDrP8ZcfAI+YggQQL46V1EjSMSofBQMAbxgBAxAXzGy2pJcktZfU\nyjk3wnMkAHGsnu8AABBuZna9pCLn3EwzO0rS22aW5pzL8hwNQJxiBAwAACBgrAEDAAAIGAUMAAAg\nYBQwAACAgFHAAAAAAkYBAwAACBgFDAAAIGAUMAAAgIBRwAAAAAL2/wHjcvnOnOgiXgAAAABJRU5E\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106c84dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (10,7))\n",
    "plt.plot(x_quad, y_quad, 'k--')\n",
    "plt.axis([0,10,0,30])\n",
    "plt.plot([1, 2, 3], [14, 9, 6], 'ro')\n",
    "plt.plot([5, 7, 8],[6, 14, 21], 'bo')\n",
    "plt.plot(4, 5, 'ko')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x)')\n",
    "plt.title('Quadratic Equation')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We're trying to find the local minimum on this function. If we start at the first red dot at x = 2, we find the gradient and we move against it. In this case, the gradient is the slope. And since the slope is negative, our next attempt is further to the right. Thus bringing us closer to the minimum. \n",
    "\n",
    "Indeed, we keep updating our parameter beta to get us closer and closer to the minumum.\n",
    "$$\\beta_j := \\beta_j - \\alpha\\frac{\\partial}{\\partial \\beta_j} J(\\beta)$$\n",
    "Where $\\alpha$ is our learning rate and $J(\\beta)$ is our cost function. By adjusting alpha, we can change how quickly we converge on the minimum (at the risk of overshooting it entirely and does not converge on our local minimum)\n",
    "\n",
    "Derive into the final formula?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, let's try to implement this in Python. First I declare some parameters. Alpha is my learning rate, and iterations defines how many times I want to perform the update.\n",
    "\n",
    "Then I transform the data frame holding my data into an array for simpler matrix math. And then write a helper function to calculate the cost function as defined above. Using np.dot for inner matrix multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "iterations = 1500\n",
    "alpha = 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "## Add a columns of 1s as intercept to X\n",
    "X_df['intercept'] = 1\n",
    "\n",
    "## Transform to Numpy arrays for easier matrix math and start theta at 0\n",
    "X = np.array(X_df)\n",
    "y = np.array(y_df).flatten()\n",
    "theta = np.array([0, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def cost_function(X, y, theta):\n",
    "    \"\"\"\n",
    "    cost_function(X, y, theta) computes the cost of using theta as the\n",
    "    parameter for linear regression to fit the data points in X and y\n",
    "    \"\"\"\n",
    "    ## number of training examples\n",
    "    m = len(y) \n",
    "    \n",
    "    ## Calculate the cost with the given parameters\n",
    "    J = np.sum((X.dot(theta)-y)**2)/2/m\n",
    "    \n",
    "    return J"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32.072733877455676"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cost_function(X, y, theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, I split the gradient descent algorithm into 4 parts so that I can see what's going on. Plus, I like to check my matrix dimensions to make sure that I'm doing the math in the right order.\n",
    "1. calculate hypothesis[97x1] = x[97x2] * theta[2x1]\n",
    "2. calculte loss [97x1] with element-wise subtraction\n",
    "3. calculate gradient [2x1] = X' [2x97] * loss [97*1]\n",
    "4. update parameter theta [2x1] after element-wise subtraction multiplied by a scalar\n",
    "5. find the cost by using cost_function()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def gradient_descent(X, y, theta, alpha, iterations):\n",
    "    \"\"\"\n",
    "    gradient_descent Performs gradient descent to learn theta\n",
    "    theta = GRADIENTDESENT(X, y, theta, alpha, num_iters) updates theta by \n",
    "    taking num_iters gradient steps with learning rate alpha\n",
    "    \"\"\"\n",
    "    cost_history = [0] * iterations\n",
    "    \n",
    "    for iteration in range(iterations):\n",
    "        hypothesis = X.dot(theta)\n",
    "        loss = hypothesis-y\n",
    "        gradient = X.T.dot(loss)/m\n",
    "        theta = theta - alpha*gradient\n",
    "        cost = cost_function(X, y, theta)\n",
    "        cost_history[iteration] = cost\n",
    "\n",
    "    return theta, cost_history\n",
    "     "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "(t, c) = gradient_descent(X,y,theta,alpha, iterations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.16636235 -3.63029144]\n"
     ]
    }
   ],
   "source": [
    "## Print theta parameters\n",
    "print t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.45197678677\n",
      "4.53424501294\n"
     ]
    }
   ],
   "source": [
    "## Prediction\n",
    "print np.array([3.5, 1]).dot(t)\n",
    "print np.array([7, 1]).dot(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "## Plotting the best fit line\n",
    "best_fit_x = np.linspace(0, 25, 20)\n",
    "best_fit_y = [t[1] + t[0]*xx for xx in best_fit_x]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x107537510>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HMl5yIlnqjhQREZGUGTYMunWDkhJo3jwzx1y0YhFXjryS3/3zdxy/8/FMvnRy1idgoDph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\nM7MKfAFsKzyYr+7I5AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106c9af50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.plot(X_df.population, y_df, '.')\n",
    "plt.plot(best_fit_x, best_fit_y, '-')\n",
    "plt.axis([0,25,-5,25])\n",
    "plt.xlabel('Population of City in 10,000s')\n",
    "plt.ylabel('Profit in $10,000s')\n",
    "plt.title('Profit vs. Population with Linear Regression Line')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}