statistics/Coefficient of Determination (R2).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Coefficient of Determination (R2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn\n",
    "seaborn.set_style('whitegrid')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def r2(actual, predicted):\n",
    "    if isinstance(actual, list):\n",
    "        actual = np.array(actual)\n",
    "    if isinstance(predicted, list):\n",
    "        predicted = np.array(predicted)\n",
    "        \n",
    "    plt.scatter(actual, predicted);\n",
    "    plt.plot(actual, actual, 'r', alpha=0.5)\n",
    "    plt.scatter(actual, actual, facecolors='none', edgecolor='r', linestyle='-')\n",
    "    plt.axhline(y=actual.mean(), ls='dashed')\n",
    "        \n",
    "    mean = actual.mean()\n",
    "    ss_total = ((actual - mean) ** 2).sum()\n",
    "    ss_residual = ((actual - predicted) ** 2).sum()\n",
    "    return 1 - ss_residual / ss_total"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll get a perfect `R2=1` if `actual` and `predicted` values are the same"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "actual = list(range(10))\n",
    "actual"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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Gj9HIJQ/quZV32z0tolif/nnHyU4AQoLHI334oXTokBQTI82bJy1YIMXFyTFqlJZMn6Z8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\nAEFCyAIAECT/H7dEYB8YX2o1AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112b30550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "predicted = list(range(10))\n",
    "r2(actual, predicted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now if we're a bit off on our predictions R2 will be still pretty high but not a perfect 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5, 6, 7, 9, 10]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predicted = list(range(8)) + [9, 10]\n",
    "predicted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.97575757575757571"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KTIt5yBAzQahPH7srExEJKgWv2K+62gymv3wZ0tJg6VIzK1dtZRGJQQpesU9L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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1126d4e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, predicted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Intuitively it makes sense that we get a worse `R2` if the predictions are farther off."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5, 6, 7, 13, 15]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predicted = list(range(8)) + [13, 15]\n",
    "predicted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.26060606060606062"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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kqirT5ba0mLH8fBO406aZW4YAYIAifBE9ti0dPWq63L17zWpUSUnSjBkmdEeO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\nAOAwwhcAAIcRvgAAOIzwBQDAYf8fiGNEieoSMYoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1122c87b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, predicted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now here's the interesting situation: `R2` can be negative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5, 6, 7, 100, 150]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predicted = list(range(8)) + [100, 150]\n",
    "predicted"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is because `R2` is calculated relative to a hypothetical model that always predicts the mean of the actual values. Obviously we don't have access to the mean of actual values at the time of prediction but we do at evaluation time.\n",
    "\n",
    "So looking at lines `12-13` in `In[2]` you can see that:\n",
    "```\n",
    "mean = actual.mean()\n",
    "ss_total = ((actual - mean) ** 2).sum()\n",
    "```\n",
    "\n",
    "and then `R2` is:\n",
    "```\n",
    "ss_residual = ((actual - predicted) ** 2).sum()\n",
    "R2 = 1 - ss_residual / ss_total\n",
    "```\n",
    "\n",
    "So obviously if we're way off on our predictions we'll be worse than this hypothetical mean model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-342.57575757575756"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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2uzVixAhJ0oABA7R169ZoDn9WXptDH22tUrKvUbe635YjGGjalrP4E/k9O8Kf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\nKrvI1ESzreYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10769ac50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, predicted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Special cases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "actual = [1] * 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/amir.ziai/anaconda/lib/python3.5/site-packages/ipykernel/__main__.py:15: RuntimeWarning: invalid value encountered in true_divide\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11212e2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, [1] * 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/amir.ziai/anaconda/lib/python3.5/site-packages/ipykernel/__main__.py:15: RuntimeWarning: divide by zero encountered in true_divide\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-inf"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CDUCStCtcr7Vb/0e7wvXdvRUAl+hSqLdt26b58+e33a6trVUoFFJ+fr5Wr14tSWptbVVJ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\nAIYRagAADCPUAAAY9n9LQqw5iQxRcwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11249d588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, [-100] * 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "actual = list(range(10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.4848484848484849"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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1e3evJwMARAAh65Vz56y9Hj8udekizZ0r3Xcf7RUA4gghG22OI23fbu21vV26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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111e707f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, [1] * 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Predicting exactly the same as mean everytime gives us an `R2=0`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x112c8fda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r2(actual, [4.5] * 10)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}