Regression project/linear_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linear Regression\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Step 1 : load the data using numpy\n",
    "import numpy as np\n",
    "in_data = np.loadtxt('data100m.csv',delimiter=',')\n",
    "\n",
    "# In CSV file first column contains year and second column winning time\n",
    "# Extract data accordingly\n",
    "x = in_data[:,0] # year\n",
    "t = in_data[:,1] # winning time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Step 2: plot the data using Matplot library\n",
    "import pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'Winning time')"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1b4b2b201d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,t,'ro')\n",
    "#label the axes to ease the identification of data points\n",
    "plt.xlabel('Olympic year')\n",
    "plt.ylabel('Winning time')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Step 3: compute the necessary values for  w0 and  w1\n",
    "xbar = x.mean()\n",
    "tbar = t.mean()\n",
    "xxbar = (x*x).mean()\n",
    "xtbar = (x*t).mean()\n",
    "\n",
    "w_1 = (xtbar - xbar*tbar)/(xxbar - xbar*xbar)\n",
    "w_0 = (t - w_1*x).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10832af90>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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BwGHAL81s/wztT6RRW2wB55wTzhiOOAL+/d/DLauvvpq9fcbRkU8kkzq2cLvlZvZ9d19u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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10833fc50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Step 4: make the new plot\n",
    "plt.figure()\n",
    "plt.plot(x,t,'ro')\n",
    "plt.plot([x[0],x[-1]],[w_0 + w_1*x[0],w_0 + w_1*x[-1]],'b')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Step 5: quadratic!\n",
    "Derivation is theoretically easy but a pain as it involves solving a system of three simultaneous equations (we'll see how to make it easy next time). Probably the easiest way to do it is through gaussian elimination of the following matrix system (see e.g. [https://www.math.hmc.edu/calculus/tutorials/linearsystems/]()). The matrix system is obtained by differentiating with respect to $w_0$, $w_1$ and $w_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ Aw = b $$\n",
    "\n",
    "where\n",
    "\n",
    "$$ A = \\left[ \\begin{array}{ccc} 1 & \\bar{x} & \\bar{x^2} \\\\\n",
    "                                \\bar{x} & \\bar{x^2} & \\bar{x^3} \\\\\n",
    "                                \\bar{x^2} & \\bar{x^3} & \\bar{x^4}\n",
    "               \\end{array}\\right] $$\n",
    "            \n",
    "$$ w = \\left[ \\begin{array}{c} w_0 \\\\ w_1 \\\\ w_2 \\end{array}\\right] $$\n",
    "\n",
    "and \n",
    "\n",
    "$$ b = \\left[ \\begin{array}{c} \\bar{t} \\\\ \\bar{xt} \\\\ \\bar{x^2t} \\end{array}\\right] $$\n",
    "\n",
    "and \n",
    "\n",
    "$$ \\bar{f} = \\frac{1}{N}\\sum_{n=1}^N f_n $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Add more parameters to see how it will affect the curve/ prediction of future value \n",
    "tbar = t.mean()\n",
    "xbar = x.mean()\n",
    "x2bar = (x*x).mean()\n",
    "x3bar = (x*x*x).mean()\n",
    "x4bar = (x*x*x*x).mean()\n",
    "x2tbar = (x*x*t).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000110151552656\n",
      "-0.443160488183\n",
      "455.597858393\n"
     ]
    }
   ],
   "source": [
    "#Computed w2, w1 and w0 using derived formula - (Using calculus) - (check the link mentioned above)\n",
    "num = x2tbar - x2bar*tbar - (x3bar-x2bar*xbar)*(xtbar-xbar*tbar)/(x2bar-xbar*xbar)\n",
    "den = x4bar - x2bar*x2bar - (x3bar-x2bar*xbar)*(x3bar-x2bar*xbar)/(x2bar-xbar*xbar)\n",
    "w_2 = num/den\n",
    "print w_2\n",
    "w_1 = (xtbar-xbar*tbar)/(x2bar-xbar*xbar) - w_3*(x3bar-xbar*x2bar)/(x2bar-xbar*xbar)\n",
    "print w_1\n",
    "w_0 = tbar - w_1*xbar - w_2*x2bar\n",
    "print w_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x108dc5890>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Cxizh3d7fMOJHI9loI2ODDaBvX1h//QRLlsylZ88G9thjMOutV7LG5My4KZmk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\nkjElExERydj/ARM8gsn60rmkAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108afcf10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Make the new plot\n",
    "plt.figure()\n",
    "xvals = np.linspace(x[0],x[-1],100)\n",
    "tvals = w_0 + w_1*xvals + w_2*xvals**2\n",
    "plt.plot(x,t,'ro')\n",
    "plt.plot(xvals,tvals,'b')\n",
    "plt.xlabel('Olympic year')\n",
    "plt.ylabel('Winning time')"
   ]
  }
 ],
 "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}