linear_regression_example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook is for unsderstanding Linear Regression of statsmodels\n",
    "\n",
    "Reference : \n",
    "https://www.statsmodels.org/stable/regression.html\n",
    "\n",
    "versions :  \n",
    "- python 3.6.2  \n",
    "- statsmodels 0.8.0  \n",
    "- scikit-learn 0.19.1  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.simplefilter(action='ignore', category=FutureWarning)\n",
    "\n",
    "import matplotlib.pylab as plt\n",
    "import pandas as pd\n",
    "from sklearn import datasets\n",
    "from sklearn.model_selection import train_test_split\n",
    "import seaborn as sns\n",
    "import statsmodels.api as sm\n",
    "from statsmodels.stats.outliers_influence import variance_inflation_factor\n",
    "\n",
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# load data to use\n",
    "boston_data = datasets.load_boston()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prepare Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X :  (506, 13)\n",
      "Y :  (506,)\n",
      "Feature names ['CRIM' 'ZN' 'INDUS' 'CHAS' 'NOX' 'RM' 'AGE' 'DIS' 'RAD' 'TAX' 'PTRATIO'\n",
      " 'B' 'LSTAT']\n"
     ]
    }
   ],
   "source": [
    "# check features and target\n",
    "print('X : ', boston_data.data.shape)\n",
    "print('Y : ', boston_data.target.shape)\n",
    "\n",
    "print('Feature names', boston_data.feature_names)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00632</td>\n",
       "      <td>18.0</td>\n",
       "      <td>2.31</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.538</td>\n",
       "      <td>6.575</td>\n",
       "      <td>65.2</td>\n",
       "      <td>4.0900</td>\n",
       "      <td>1.0</td>\n",
       "      <td>296.0</td>\n",
       "      <td>15.3</td>\n",
       "      <td>396.90</td>\n",
       "      <td>4.98</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.02731</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>6.421</td>\n",
       "      <td>78.9</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>396.90</td>\n",
       "      <td>9.14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.02729</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>7.185</td>\n",
       "      <td>61.1</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>392.83</td>\n",
       "      <td>4.03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.03237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>6.998</td>\n",
       "      <td>45.8</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>394.63</td>\n",
       "      <td>2.94</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.06905</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>7.147</td>\n",
       "      <td>54.2</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>396.90</td>\n",
       "      <td>5.33</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      CRIM    ZN  INDUS  CHAS    NOX     RM   AGE     DIS  RAD    TAX  \\\n",
       "0  0.00632  18.0   2.31   0.0  0.538  6.575  65.2  4.0900  1.0  296.0   \n",
       "1  0.02731   0.0   7.07   0.0  0.469  6.421  78.9  4.9671  2.0  242.0   \n",
       "2  0.02729   0.0   7.07   0.0  0.469  7.185  61.1  4.9671  2.0  242.0   \n",
       "3  0.03237   0.0   2.18   0.0  0.458  6.998  45.8  6.0622  3.0  222.0   \n",
       "4  0.06905   0.0   2.18   0.0  0.458  7.147  54.2  6.0622  3.0  222.0   \n",
       "\n",
       "   PTRATIO       B  LSTAT  \n",
       "0     15.3  396.90   4.98  \n",
       "1     17.8  396.90   9.14  \n",
       "2     17.8  392.83   4.03  \n",
       "3     18.7  394.63   2.94  \n",
       "4     18.7  396.90   5.33  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = pd.DataFrame(boston_data.data, columns=boston_data.feature_names)\n",
    "X.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# target\n",
    "Y = pd.DataFrame(boston_data.target, columns=['target'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>target</th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>target</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.385832</td>\n",
       "      <td>0.360445</td>\n",
       "      <td>-0.483725</td>\n",
       "      <td>0.175260</td>\n",
       "      <td>-0.427321</td>\n",
       "      <td>0.695360</td>\n",
       "      <td>-0.376955</td>\n",
       "      <td>0.249929</td>\n",
       "      <td>-0.381626</td>\n",
       "      <td>-0.468536</td>\n",
       "      <td>-0.507787</td>\n",
       "      <td>0.333461</td>\n",
       "      <td>-0.737663</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CRIM</th>\n",
       "      <td>-0.385832</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.199458</td>\n",
       "      <td>0.404471</td>\n",
       "      <td>-0.055295</td>\n",
       "      <td>0.417521</td>\n",
       "      <td>-0.219940</td>\n",
       "      <td>0.350784</td>\n",
       "      <td>-0.377904</td>\n",
       "      <td>0.622029</td>\n",
       "      <td>0.579564</td>\n",
       "      <td>0.288250</td>\n",
       "      <td>-0.377365</td>\n",
       "      <td>0.452220</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ZN</th>\n",
       "      <td>0.360445</td>\n",
       "      <td>-0.199458</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.533828</td>\n",
       "      <td>-0.042697</td>\n",
       "      <td>-0.516604</td>\n",
       "      <td>0.311991</td>\n",
       "      <td>-0.569537</td>\n",
       "      <td>0.664408</td>\n",
       "      <td>-0.311948</td>\n",
       "      <td>-0.314563</td>\n",
       "      <td>-0.391679</td>\n",
       "      <td>0.175520</td>\n",
       "      <td>-0.412995</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>INDUS</th>\n",
       "      <td>-0.483725</td>\n",
       "      <td>0.404471</td>\n",
       "      <td>-0.533828</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.062938</td>\n",
       "      <td>0.763651</td>\n",
       "      <td>-0.391676</td>\n",
       "      <td>0.644779</td>\n",
       "      <td>-0.708027</td>\n",
       "      <td>0.595129</td>\n",
       "      <td>0.720760</td>\n",
       "      <td>0.383248</td>\n",
       "      <td>-0.356977</td>\n",
       "      <td>0.603800</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CHAS</th>\n",
       "      <td>0.175260</td>\n",
       "      <td>-0.055295</td>\n",
       "      <td>-0.042697</td>\n",
       "      <td>0.062938</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.091203</td>\n",
       "      <td>0.091251</td>\n",
       "      <td>0.086518</td>\n",
       "      <td>-0.099176</td>\n",
       "      <td>-0.007368</td>\n",
       "      <td>-0.035587</td>\n",
       "      <td>-0.121515</td>\n",
       "      <td>0.048788</td>\n",
       "      <td>-0.053929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NOX</th>\n",
       "      <td>-0.427321</td>\n",
       "      <td>0.417521</td>\n",
       "      <td>-0.516604</td>\n",
       "      <td>0.763651</td>\n",
       "      <td>0.091203</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.302188</td>\n",
       "      <td>0.731470</td>\n",
       "      <td>-0.769230</td>\n",
       "      <td>0.611441</td>\n",
       "      <td>0.668023</td>\n",
       "      <td>0.188933</td>\n",
       "      <td>-0.380051</td>\n",
       "      <td>0.590879</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RM</th>\n",
       "      <td>0.695360</td>\n",
       "      <td>-0.219940</td>\n",
       "      <td>0.311991</td>\n",
       "      <td>-0.391676</td>\n",
       "      <td>0.091251</td>\n",
       "      <td>-0.302188</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.240265</td>\n",
       "      <td>0.205246</td>\n",
       "      <td>-0.209847</td>\n",
       "      <td>-0.292048</td>\n",
       "      <td>-0.355501</td>\n",
       "      <td>0.128069</td>\n",
       "      <td>-0.613808</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AGE</th>\n",
       "      <td>-0.376955</td>\n",
       "      <td>0.350784</td>\n",
       "      <td>-0.569537</td>\n",
       "      <td>0.644779</td>\n",
       "      <td>0.086518</td>\n",
       "      <td>0.731470</td>\n",
       "      <td>-0.240265</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.747881</td>\n",
       "      <td>0.456022</td>\n",
       "      <td>0.506456</td>\n",
       "      <td>0.261515</td>\n",
       "      <td>-0.273534</td>\n",
       "      <td>0.602339</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>DIS</th>\n",
       "      <td>0.249929</td>\n",
       "      <td>-0.377904</td>\n",
       "      <td>0.664408</td>\n",
       "      <td>-0.708027</td>\n",
       "      <td>-0.099176</td>\n",
       "      <td>-0.769230</td>\n",
       "      <td>0.205246</td>\n",
       "      <td>-0.747881</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.494588</td>\n",
       "      <td>-0.534432</td>\n",
       "      <td>-0.232471</td>\n",
       "      <td>0.291512</td>\n",
       "      <td>-0.496996</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RAD</th>\n",
       "      <td>-0.381626</td>\n",
       "      <td>0.622029</td>\n",
       "      <td>-0.311948</td>\n",
       "      <td>0.595129</td>\n",
       "      <td>-0.007368</td>\n",
       "      <td>0.611441</td>\n",
       "      <td>-0.209847</td>\n",
       "      <td>0.456022</td>\n",
       "      <td>-0.494588</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.910228</td>\n",
       "      <td>0.464741</td>\n",
       "      <td>-0.444413</td>\n",
       "      <td>0.488676</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>TAX</th>\n",
       "      <td>-0.468536</td>\n",
       "      <td>0.579564</td>\n",
       "      <td>-0.314563</td>\n",
       "      <td>0.720760</td>\n",
       "      <td>-0.035587</td>\n",
       "      <td>0.668023</td>\n",
       "      <td>-0.292048</td>\n",
       "      <td>0.506456</td>\n",
       "      <td>-0.534432</td>\n",
       "      <td>0.910228</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.460853</td>\n",
       "      <td>-0.441808</td>\n",
       "      <td>0.543993</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>PTRATIO</th>\n",
       "      <td>-0.507787</td>\n",
       "      <td>0.288250</td>\n",
       "      <td>-0.391679</td>\n",
       "      <td>0.383248</td>\n",
       "      <td>-0.121515</td>\n",
       "      <td>0.188933</td>\n",
       "      <td>-0.355501</td>\n",
       "      <td>0.261515</td>\n",
       "      <td>-0.232471</td>\n",
       "      <td>0.464741</td>\n",
       "      <td>0.460853</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.177383</td>\n",
       "      <td>0.374044</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>B</th>\n",
       "      <td>0.333461</td>\n",
       "      <td>-0.377365</td>\n",
       "      <td>0.175520</td>\n",
       "      <td>-0.356977</td>\n",
       "      <td>0.048788</td>\n",
       "      <td>-0.380051</td>\n",
       "      <td>0.128069</td>\n",
       "      <td>-0.273534</td>\n",
       "      <td>0.291512</td>\n",
       "      <td>-0.444413</td>\n",
       "      <td>-0.441808</td>\n",
       "      <td>-0.177383</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.366087</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>LSTAT</th>\n",
       "      <td>-0.737663</td>\n",
       "      <td>0.452220</td>\n",
       "      <td>-0.412995</td>\n",
       "      <td>0.603800</td>\n",
       "      <td>-0.053929</td>\n",
       "      <td>0.590879</td>\n",
       "      <td>-0.613808</td>\n",
       "      <td>0.602339</td>\n",
       "      <td>-0.496996</td>\n",
       "      <td>0.488676</td>\n",
       "      <td>0.543993</td>\n",
       "      <td>0.374044</td>\n",
       "      <td>-0.366087</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           target      CRIM        ZN     INDUS      CHAS       NOX        RM  \\\n",
       "target   1.000000 -0.385832  0.360445 -0.483725  0.175260 -0.427321  0.695360   \n",
       "CRIM    -0.385832  1.000000 -0.199458  0.404471 -0.055295  0.417521 -0.219940   \n",
       "ZN       0.360445 -0.199458  1.000000 -0.533828 -0.042697 -0.516604  0.311991   \n",
       "INDUS   -0.483725  0.404471 -0.533828  1.000000  0.062938  0.763651 -0.391676   \n",
       "CHAS     0.175260 -0.055295 -0.042697  0.062938  1.000000  0.091203  0.091251   \n",
       "NOX     -0.427321  0.417521 -0.516604  0.763651  0.091203  1.000000 -0.302188   \n",
       "RM       0.695360 -0.219940  0.311991 -0.391676  0.091251 -0.302188  1.000000   \n",
       "AGE     -0.376955  0.350784 -0.569537  0.644779  0.086518  0.731470 -0.240265   \n",
       "DIS      0.249929 -0.377904  0.664408 -0.708027 -0.099176 -0.769230  0.205246   \n",
       "RAD     -0.381626  0.622029 -0.311948  0.595129 -0.007368  0.611441 -0.209847   \n",
       "TAX     -0.468536  0.579564 -0.314563  0.720760 -0.035587  0.668023 -0.292048   \n",
       "PTRATIO -0.507787  0.288250 -0.391679  0.383248 -0.121515  0.188933 -0.355501   \n",
       "B        0.333461 -0.377365  0.175520 -0.356977  0.048788 -0.380051  0.128069   \n",
       "LSTAT   -0.737663  0.452220 -0.412995  0.603800 -0.053929  0.590879 -0.613808   \n",
       "\n",
       "              AGE       DIS       RAD       TAX   PTRATIO         B     LSTAT  \n",
       "target  -0.376955  0.249929 -0.381626 -0.468536 -0.507787  0.333461 -0.737663  \n",
       "CRIM     0.350784 -0.377904  0.622029  0.579564  0.288250 -0.377365  0.452220  \n",
       "ZN      -0.569537  0.664408 -0.311948 -0.314563 -0.391679  0.175520 -0.412995  \n",
       "INDUS    0.644779 -0.708027  0.595129  0.720760  0.383248 -0.356977  0.603800  \n",
       "CHAS     0.086518 -0.099176 -0.007368 -0.035587 -0.121515  0.048788 -0.053929  \n",
       "NOX      0.731470 -0.769230  0.611441  0.668023  0.188933 -0.380051  0.590879  \n",
       "RM      -0.240265  0.205246 -0.209847 -0.292048 -0.355501  0.128069 -0.613808  \n",
       "AGE      1.000000 -0.747881  0.456022  0.506456  0.261515 -0.273534  0.602339  \n",
       "DIS     -0.747881  1.000000 -0.494588 -0.534432 -0.232471  0.291512 -0.496996  \n",
       "RAD      0.456022 -0.494588  1.000000  0.910228  0.464741 -0.444413  0.488676  \n",
       "TAX      0.506456 -0.534432  0.910228  1.000000  0.460853 -0.441808  0.543993  \n",
       "PTRATIO  0.261515 -0.232471  0.464741  0.460853  1.000000 -0.177383  0.374044  \n",
       "B       -0.273534  0.291512 -0.444413 -0.441808 -0.177383  1.000000 -0.366087  \n",
       "LSTAT    0.602339 -0.496996  0.488676  0.543993  0.374044 -0.366087  1.000000  "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Correlation coefficient\n",
    "Z = pd.concat([Y, X], axis=1)\n",
    "Z.corr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Modeling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>         <td>target</td>      <th>  R-squared:         </th> <td>   0.751</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.743</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   90.43</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Tue, 12 Feb 2019</td> <th>  Prob (F-statistic):</th> <td>6.22e-109</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:36:50</td>     <th>  Log-Likelihood:    </th> <td> -1194.3</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   404</td>      <th>  AIC:               </th> <td>   2417.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   390</td>      <th>  BIC:               </th> <td>   2473.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>    13</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "     <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>   <td>   30.2889</td> <td>    5.678</td> <td>    5.335</td> <td> 0.000</td> <td>   19.126</td> <td>   41.452</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>CRIM</th>    <td>   -0.1125</td> <td>    0.035</td> <td>   -3.250</td> <td> 0.001</td> <td>   -0.181</td> <td>   -0.044</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ZN</th>      <td>    0.0301</td> <td>    0.016</td> <td>    1.881</td> <td> 0.061</td> <td>   -0.001</td> <td>    0.062</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>INDUS</th>   <td>    0.0407</td> <td>    0.067</td> <td>    0.605</td> <td> 0.546</td> <td>   -0.092</td> <td>    0.173</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>CHAS</th>    <td>    2.7868</td> <td>    0.956</td> <td>    2.916</td> <td> 0.004</td> <td>    0.908</td> <td>    4.666</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>NOX</th>     <td>  -17.2406</td> <td>    4.237</td> <td>   -4.069</td> <td> 0.000</td> <td>  -25.571</td> <td>   -8.911</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>RM</th>      <td>    4.4325</td> <td>    0.464</td> <td>    9.551</td> <td> 0.000</td> <td>    3.520</td> <td>    5.345</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AGE</th>     <td>   -0.0062</td> <td>    0.015</td> <td>   -0.428</td> <td> 0.669</td> <td>   -0.035</td> <td>    0.022</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>DIS</th>     <td>   -1.4485</td> <td>    0.226</td> <td>   -6.410</td> <td> 0.000</td> <td>   -1.893</td> <td>   -1.004</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>RAD</th>     <td>    0.2621</td> <td>    0.076</td> <td>    3.451</td> <td> 0.001</td> <td>    0.113</td> <td>    0.411</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>TAX</th>     <td>   -0.0106</td> <td>    0.004</td> <td>   -2.506</td> <td> 0.013</td> <td>   -0.019</td> <td>   -0.002</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>PTRATIO</th> <td>   -0.9164</td> <td>    0.144</td> <td>   -6.364</td> <td> 0.000</td> <td>   -1.200</td> <td>   -0.633</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>B</th>       <td>    0.0125</td> <td>    0.003</td> <td>    4.203</td> <td> 0.000</td> <td>    0.007</td> <td>    0.018</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>LSTAT</th>   <td>   -0.5093</td> <td>    0.056</td> <td>   -9.151</td> <td> 0.000</td> <td>   -0.619</td> <td>   -0.400</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>133.030</td> <th>  Durbin-Watson:     </th> <td>   2.113</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td> 578.504</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 1.379</td>  <th>  Prob(JB):          </th> <td>2.40e-126</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 8.173</td>  <th>  Cond. No.          </th> <td>1.49e+04</td> \n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                 target   R-squared:                       0.751\n",
       "Model:                            OLS   Adj. R-squared:                  0.743\n",
       "Method:                 Least Squares   F-statistic:                     90.43\n",
       "Date:                Tue, 12 Feb 2019   Prob (F-statistic):          6.22e-109\n",
       "Time:                        23:36:50   Log-Likelihood:                -1194.3\n",
       "No. Observations:                 404   AIC:                             2417.\n",
       "Df Residuals:                     390   BIC:                             2473.\n",
       "Df Model:                          13                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const         30.2889      5.678      5.335      0.000      19.126      41.452\n",
       "CRIM          -0.1125      0.035     -3.250      0.001      -0.181      -0.044\n",
       "ZN             0.0301      0.016      1.881      0.061      -0.001       0.062\n",
       "INDUS          0.0407      0.067      0.605      0.546      -0.092       0.173\n",
       "CHAS           2.7868      0.956      2.916      0.004       0.908       4.666\n",
       "NOX          -17.2406      4.237     -4.069      0.000     -25.571      -8.911\n",
       "RM             4.4325      0.464      9.551      0.000       3.520       5.345\n",
       "AGE           -0.0062      0.015     -0.428      0.669      -0.035       0.022\n",
       "DIS           -1.4485      0.226     -6.410      0.000      -1.893      -1.004\n",
       "RAD            0.2621      0.076      3.451      0.001       0.113       0.411\n",
       "TAX           -0.0106      0.004     -2.506      0.013      -0.019      -0.002\n",
       "PTRATIO       -0.9164      0.144     -6.364      0.000      -1.200      -0.633\n",
       "B              0.0125      0.003      4.203      0.000       0.007       0.018\n",
       "LSTAT         -0.5093      0.056     -9.151      0.000      -0.619      -0.400\n",
       "==============================================================================\n",
       "Omnibus:                      133.030   Durbin-Watson:                   2.113\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              578.504\n",
       "Skew:                           1.379   Prob(JB):                    2.40e-126\n",
       "Kurtosis:                       8.173   Cond. No.                     1.49e+04\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 1.49e+04. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# fit data to OLS(Ordinary Least Square)\n",
    "X = sm.add_constant(X) # add constant term to features\n",
    "\n",
    "# split training data and validation data\n",
    "X_train, X_val, Y_train, Y_val = train_test_split(X, Y, test_size=0.2, random_state=42)\n",
    "\n",
    "model = sm.OLS(Y_train, X_train)\n",
    "result = model.fit()\n",
    "\n",
    "# dispalay result\n",
    "result.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "const      30.288948\n",
       "CRIM       -0.112463\n",
       "ZN          0.030081\n",
       "INDUS       0.040731\n",
       "CHAS        2.786767\n",
       "NOX       -17.240635\n",
       "RM          4.432488\n",
       "AGE        -0.006240\n",
       "DIS        -1.448485\n",
       "RAD         0.262114\n",
       "TAX        -0.010639\n",
       "PTRATIO    -0.916399\n",
       "B           0.012452\n",
       "LSTAT      -0.509349\n",
       "dtype: float64"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# dispalay estimated params\n",
    "result.params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prediction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# predict\n",
    "preds_train = result.predict(X_train) \n",
    "preds_val = result.predict(X_val)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x122763be0>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAEKCAYAAAAfGVI8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1227056a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# visualize\n",
    "plt.xlabel('prediction')\n",
    "plt.ylabel('target')\n",
    "plt.grid(True)\n",
    "\n",
    "plt.plot(preds_train, Y_train, 'o', color='blue')\n",
    "plt.plot(preds_val, Y_val, 'o', color='orange')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1227959e8>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAD8CAYAAABn919SAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMS4wLCBo\ndHRwOi8vbWF0cGxvdGxpYi5vcmcvpW3flQAADVpJREFUeJzt3X+o3fV9x/Hne/5Yxcv8MfUuJG5X\nWJC6hU28iND9cWJbzJox3ahgkRKZI/90w7LAlrawMVgh/cO2MLY/wnTmj9Fb6bop6igu88wWhl3S\nukUXJM5lnRoSSo3rFem4+N4f96veJPd4ftzz832fD5B7vp/zOee8zyffvPzk++NzIzORJM2+n5p0\nAZKk4TDQJakIA12SijDQJakIA12SijDQJakIA12SijDQJakIA12Sirh4nB92zTXX5MLCwjg/cqq8\n9dZbXH755ZMuY6o4JudyPC7kmMDRo0d/mJnXdus31kBfWFjgyJEj4/zIqdJut2m1WpMuY6o4Judy\nPC7kmEBE/Hcv/TzkIklFGOiSVISBLklFGOiSVISBLklFGOiSVISBLklFGOiSVISBLklFjPVOUc22\nhf1Pdnzu5IHdfb2mU39Jg3OGLklFGOiSVISBLklFGOiSVISBLklFGOiSVISBLklFGOiSVISBLklF\nGOiSVISBLklFGOiSVISBLklFuNqihuKDVmKUNB7O0CWpCANdkoow0CWpCANdkoroOdAj4qKI+H5E\nPNFs3xARz0XEiYj4ekRcOroyJUnd9DNDfwA4vmb7S8BXMnM78AZw/zALkyT1p6dAj4htwG7gr5rt\nAG4HvtF0OQTcNYoCJUm96XWG/lXgD4F3mu2fBc5m5kqz/Sqwdci1SZL60PXGooj4DeBMZh6NiNa7\nzet0zQ6v3wvsBZifn6fdbg9WaQHLy8sz/f337Vjp3qlH747DrI/JsDkeF3JMetfLnaIfAX4zIj4B\nfAj4GVZn7FdGxMXNLH0b8Pp6L87Mg8BBgMXFxWy1WsOoeya1221m+fvfN8S7QU/e2wJmf0yGzfG4\nkGPSu66HXDLzc5m5LTMXgHuAf8rMe4FngE823fYAj42sSklSVxu5Dv2PgD+IiJdZPab+0HBKkiQN\noq/FuTKzDbSbx68Atw6/JEnSILxTVJKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgD\nXZKKMNAlqQgDXZKKMNAlqQgDXZKK6Gu1RWlYFppflrFvx8o5vzjj5IHdkypJmnnO0CWpCANdkoow\n0CWpCANdkorwpKimysKaE6RrebJU6s4ZuiQVYaBLUhEGuiQVYaBLUhEGuiQVYaBLUhEGuiQVYaBL\nUhEGuiQVYaBLUhEGuiQVYaBLUhEGuiQVYaBLUhEGuiQVYaBLUhEGuiQVYaBLUhFdAz0iPhQR342I\nf4uIFyPiT5v2GyLiuYg4ERFfj4hLR1+uJKmTXmboPwFuz8xfAX4V2BURtwFfAr6SmduBN4D7R1em\nJKmbroGeq5abzUua/xK4HfhG034IuGskFUqSetLTMfSIuCgingfOAE8D/wmczcyVpsurwNbRlChJ\n6kVkZu+dI64E/g74Y+CvM/MXm/brgacyc8c6r9kL7AWYn5+/ZWlpaRh1z6Tl5WXm5uYmXcZ7jr32\n5rrtO7Ze0Vf/jZi/DE6/3b1fp5qqmbZ9ZBo4JrBz586jmbnYrd/F/bxpZp6NiDZwG3BlRFzczNK3\nAa93eM1B4CDA4uJitlqtfj6ylHa7zTR9//v2P7lu+8l7W33134h9O1Z48Fj33bBTTdVM2z4yDRyT\n3vVylcu1zcyciLgM+BhwHHgG+GTTbQ/w2KiKlCR118sMfQtwKCIuYvV/AI9m5hMR8R/AUkT8GfB9\n4KER1ilJ6qJroGfmvwM3r9P+CnDrKIqSJPXPO0UlqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAl\nqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgD\nXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqQgDXZKKMNAlqYiLJ12A1IuF/U+u237ywO4xVyJNL2fo\nklSEgS5JRRjoklSEgS5JRRjoklSEgS5JRRjoklSEgS5JRRjoklRE10CPiOsj4pmIOB4RL0bEA037\n1RHxdEScaH5eNfpyJUmd9DJDXwH2ZeaHgduAz0TETcB+4HBmbgcON9uSpAnpGuiZeSozv9c8/jFw\nHNgK3AkcarodAu4aVZGSpO76OoYeEQvAzcBzwHxmnoLV0AeuG3ZxkqTeRWb21jFiDvhn4IuZ+c2I\nOJuZV655/o3MvOA4ekTsBfYCzM/P37K0tDScymfQ8vIyc3Nzky7jPcdee3Pd9h1br+ir/0bMXwan\n3x789Z1qnVXTto9MA8cEdu7ceTQzF7v16ynQI+IS4AngW5n55abtJaCVmaciYgvQzswbP+h9FhcX\n88iRIz19gYra7TatVmvSZbyn3yVpO/XfiH07Vnjw2OCrOFdbPnfa9pFp4JhARPQU6L1c5RLAQ8Dx\nd8O88Tiwp3m8B3hskEIlScPRy9ToI8CngWMR8XzT9nngAPBoRNwP/AC4ezQlSpJ60TXQM/M7QHR4\n+qPDLUeSNCjvFJWkIgx0SSrCQJekIgx0SSpi8AuANTP6vX58FNebSxo9Z+iSVISBLklFGOiSVISB\nLklFGOiSVISBLklFGOiSVISBLklFGOiSVISBLklFGOiSVISBLklFGOiSVISrLUp0XmHy5IHdY65E\nGpwzdEkqwkCXpCIMdEkqwmPommke+5be5wxdkoow0CWpCANdkoow0CWpCE+KalPpdBJVqsAZuiQV\nYaBLUhEGuiQVYaBLUhEGuiQVYaBLUhEGuiQVYaBLUhHeWFSIN828z7HQZtR1hh4RD0fEmYh4YU3b\n1RHxdEScaH5eNdoyJUnd9HLI5RFg13lt+4HDmbkdONxsS5ImqGugZ+azwI/Oa74TONQ8PgTcNeS6\nJEl9GvSk6HxmngJofl43vJIkSYOIzOzeKWIBeCIzf7nZPpuZV655/o3MXPc4ekTsBfYCzM/P37K0\ntDSEsmfT8vIyc3NzI3v/Y6+9ObL3HpX5y+D025OuorMdW68Y6+eNeh+ZRY4J7Ny582hmLnbrN+hV\nLqcjYktmnoqILcCZTh0z8yBwEGBxcTFbrdaAHzn72u02o/z+983glR37dqzw4LHpvdjq5L2tsX7e\nqPeRWeSY9G7QQy6PA3uax3uAx4ZTjiRpUL1ctvg14F+AGyPi1Yi4HzgAfDwiTgAfb7YlSRPU9d+6\nmfmpDk99dMi1SJI2wFv/JakIA12SijDQJakIA12SijDQJakIA12SijDQJakIA12SijDQJamI6V0V\nSR3569Wm1wf92Zw8sHuMlWgzcoYuSUUY6JJUhIEuSUV4DF36AJ6v0Cxxhi5JRRjoklSEgS5JRRjo\nklSEgS5JRRjoklSEgS5JRRjoklSEgS5JRRjoklSEgS5JRRjoklSEgS5JRbjaojQmvazcuG/HCvc1\n/fwNR+qXM3RJKsJAl6QiDHRJKsJAl6QiPCk6Ap1Ofj2y6/KhvI82t373i04nVzu9jydjZ5czdEkq\nwkCXpCIMdEkqYmaOoXu8TxpMv8fc+/27VuHv5rC+w6THYkMz9IjYFREvRcTLEbF/WEVJkvo3cKBH\nxEXAXwC/DtwEfCoibhpWYZKk/mxkhn4r8HJmvpKZ/wcsAXcOpyxJUr82Euhbgf9Zs/1q0yZJmoDI\nzMFeGHE3cEdm/m6z/Wng1sz8/fP67QX2Nps3Ai8NXu7Muwb44aSLmDKOybkcjws5JvALmXltt04b\nucrlVeD6NdvbgNfP75SZB4GDG/icMiLiSGYuTrqOaeKYnMvxuJBj0ruNHHL5V2B7RNwQEZcC9wCP\nD6csSVK/Bp6hZ+ZKRPwe8C3gIuDhzHxxaJVJkvqyoRuLMvMp4Kkh1bIZeOjpQo7JuRyPCzkmPRr4\npKgkabq4loskFWGgj0FE3B0RL0bEOxGxeN5zn2uWTngpIu6YVI3j5rIREBEPR8SZiHhhTdvVEfF0\nRJxofl41yRrHKSKuj4hnIuJ48/flgaZ9045Jvwz08XgB+G3g2bWNzVIJ9wC/BOwC/rJZUqE0l414\nzyOs/rmvtR84nJnbgcPN9maxAuzLzA8DtwGfafaLzTwmfTHQxyAzj2fmejdU3QksZeZPMvO/gJdZ\nXVKhOpeNADLzWeBH5zXfCRxqHh8C7hprUROUmacy83vN4x8Dx1m9+3zTjkm/DPTJ2qzLJ2zW792L\n+cw8BasBB1w34XomIiIWgJuB53BMejYz66FPu4j4R+Dn1nnqC5n5WKeXrdO2GS472qzfWz2IiDng\nb4HPZub/Rqy3u2g9BvqQZObHBnhZT8snFLRZv3cvTkfElsw8FRFbgDOTLmicIuISVsP8bzLzm03z\nph6TfnjIZbIeB+6JiJ+OiBuA7cB3J1zTOLhsRGePA3uax3uATv+6KydWp+IPAccz88trntq0Y9Iv\nbywag4j4LeDPgWuBs8DzmXlH89wXgN9h9Qz/ZzPzHyZW6BhFxCeAr/L+shFfnHBJYxcRXwNarK4m\neBr4E+DvgUeBnwd+ANydmeefOC0pIn4N+DZwDHinaf48q8fRN+WY9MtAl6QiPOQiSUUY6JJUhIEu\nSUUY6JJUhIEuSUUY6JJUhIEuSUUY6JJUxP8D2Clo6zJ25lkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x12068e828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# histogram of residual\n",
    "result.resid.hist(bins=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>VIF value</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>CRIM</th>\n",
       "      <td>1.693529</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ZN</th>\n",
       "      <td>2.465409</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>INDUS</th>\n",
       "      <td>3.877614</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CHAS</th>\n",
       "      <td>1.096576</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NOX</th>\n",
       "      <td>4.470809</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RM</th>\n",
       "      <td>1.948486</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AGE</th>\n",
       "      <td>2.990166</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>DIS</th>\n",
       "      <td>4.169339</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RAD</th>\n",
       "      <td>7.653303</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>TAX</th>\n",
       "      <td>8.943203</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>PTRATIO</th>\n",
       "      <td>1.851625</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>B</th>\n",
       "      <td>1.322576</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>LSTAT</th>\n",
       "      <td>2.815755</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         VIF value\n",
       "CRIM      1.693529\n",
       "ZN        2.465409\n",
       "INDUS     3.877614\n",
       "CHAS      1.096576\n",
       "NOX       4.470809\n",
       "RM        1.948486\n",
       "AGE       2.990166\n",
       "DIS       4.169339\n",
       "RAD       7.653303\n",
       "TAX       8.943203\n",
       "PTRATIO   1.851625\n",
       "B         1.322576\n",
       "LSTAT     2.815755"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# VIF\n",
    "# https://www.statsmodels.org/dev/generated/statsmodels.stats.outliers_influence.variance_inflation_factor.html\n",
    "vifs = [variance_inflation_factor(model.exog, i)for i in range(0, model.exog.shape[1])]\n",
    "pd.DataFrame(vifs, index=model.exog_names, columns=[\"VIF value\"])[1:]"
   ]
  }
 ],
 "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": "3.6.2"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}