CovarianceCorrelation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Covariance and Correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Covariance measures how two variables vary in tandem from their means.\n",
    "\n",
    "For example, let's say we work for an e-commerce company, and they are interested in finding a correlation between page speed (how fast each web page renders for a customer) and how much a customer spends.\n",
    "\n",
    "numpy offers covariance methods, but we'll do it the \"hard way\" to show what happens under the hood. Basically we treat each variable as a vector of deviations from the mean, and compute the \"dot product\" of both vectors. Geometrically this can be thought of as the angle between the two vectors in a high-dimensional space, but you can just think of it as a measure of similarity between the two variables.\n",
    "\n",
    "First, let's just make page speed and purchase amount totally random and independent of each other; a very small covariance will result as there is no real correlation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10703983911657081"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DbK2u1IDGRoYChTNWrDJuv2ZtrOGPBOCOa4fbomNsjTeoEC4RlNv3JBxQM3N1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\ngiB0CSLQBUEQugQR6IIgCF3C/wer3cdfWPGGygAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x28959ed50b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "from pylab import *\n",
    "\n",
    "def de_mean(x):\n",
    "    xmean = mean(x)\n",
    "    return [xi - xmean for xi in x]\n",
    "\n",
    "def covariance(x, y):\n",
    "    n = len(x)\n",
    "    return dot(de_mean(x), de_mean(y)) / (n-1)\n",
    "\n",
    "pageSpeeds = np.random.normal(3.0, 1.0, 1000)\n",
    "purchaseAmount = np.random.normal(50.0, 10.0, 1000)\n",
    "\n",
    "scatter(pageSpeeds, purchaseAmount)\n",
    "\n",
    "covariance (pageSpeeds, purchaseAmount)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Now we'll make our fabricated purchase amounts an actual function of page speed, making a very real correlation. The negative value indicates an inverse relationship; pages that render in less time result in more money spent:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-12.101174437411698"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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dkoluXVxuWaFVvP3utTGPI3zqxdf9UHEzazqTrdEvjIiL2fQlYGE23Qm8WLLehaztBpI2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\n6QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2895ae12b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "purchaseAmount = np.random.normal(50.0, 10.0, 1000) / pageSpeeds\n",
    "\n",
    "scatter(pageSpeeds, purchaseAmount)\n",
    "\n",
    "covariance (pageSpeeds, purchaseAmount)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "But, what does this value mean? Covariance is sensitive to the units used in the variables, which makes it difficult to interpret. Correlation normalizes everything by their standard deviations, giving you an easier to understand value that ranges from -1 (for a perfect inverse correlation) to 1 (for a perfect positive correlation):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.40512114003756994"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def correlation(x, y):\n",
    "    stddevx = x.std()\n",
    "    stddevy = y.std()\n",
    "    return covariance(x,y) / stddevx / stddevy  #In real life you'd check for divide by zero here\n",
    "\n",
    "correlation(pageSpeeds, purchaseAmount)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "numpy can do all this for you with numpy.corrcoef. It returns a matrix of the correlation coefficients between every combination of the arrays passed in:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.        , -0.40471602],\n",
       "       [-0.40471602,  1.        ]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.corrcoef(pageSpeeds, purchaseAmount)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "(It doesn't match exactly just due to the math precision available on a computer.)\n",
    "\n",
    "We can force a perfect correlation by fabricating a totally linear relationship (again, it's not exactly -1 just due to precision errors, but it's close enough to tell us there's a really good correlation here):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0010010010010011"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hpSbpLGAO8J1yx2I5kv4EuBK4DyAiDpU6GUDtJoRG4OVBz/dSpf/w1CpJU4Fm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\nWqzuAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2895ae4e128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "purchaseAmount = 100 - pageSpeeds * 3\n",
    "\n",
    "scatter(pageSpeeds, purchaseAmount)\n",
    "\n",
    "correlation (pageSpeeds, purchaseAmount)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Remember, correlation does not imply causality!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Activity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "numpy also has a numpy.cov function that can compute Covariance for you. Try using it for the pageSpeeds and purchaseAmounts data above. Interpret its results, and compare it to the results from our own covariance function above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.96858    -2.90573999]\n",
      " [-2.90573999  8.71721996]] \n",
      " [[ 0.96858    -2.90573999]\n",
      " [-2.90573999  8.71721996]]\n",
      "Same result\n"
     ]
    }
   ],
   "source": [
    "covariance2 = np.cov(pageSpeeds, purchaseAmount)\n",
    "print(covariance, \"\\n\", covariance2)\n",
    "if all(covariance == covariance2):\n",
    "    print(\"Same result\")\n",
    "else:\n",
    "    print(\"Not same\")"
   ]
  }
 ],
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