Housing+Case+Study+using+RFE.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Selection using RFE (Housing Case Study)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Importing and Understanding Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Supress Warnings\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing Housing.csv\n",
    "housing = pd.read_csv('Housing.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>price</th>\n",
       "      <th>area</th>\n",
       "      <th>bedrooms</th>\n",
       "      <th>bathrooms</th>\n",
       "      <th>stories</th>\n",
       "      <th>mainroad</th>\n",
       "      <th>guestroom</th>\n",
       "      <th>basement</th>\n",
       "      <th>hotwaterheating</th>\n",
       "      <th>airconditioning</th>\n",
       "      <th>parking</th>\n",
       "      <th>prefarea</th>\n",
       "      <th>furnishingstatus</th>\n",
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       "      <td>yes</td>\n",
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       "      <td>no</td>\n",
       "      <td>no</td>\n",
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       "      <td>yes</td>\n",
       "      <td>furnished</td>\n",
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       "      <td>4</td>\n",
       "      <td>yes</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>yes</td>\n",
       "      <td>3</td>\n",
       "      <td>no</td>\n",
       "      <td>furnished</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>12250000</td>\n",
       "      <td>9960</td>\n",
       "      <td>3</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>yes</td>\n",
       "      <td>no</td>\n",
       "      <td>yes</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>2</td>\n",
       "      <td>yes</td>\n",
       "      <td>semi-furnished</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>12215000</td>\n",
       "      <td>7500</td>\n",
       "      <td>4</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>yes</td>\n",
       "      <td>no</td>\n",
       "      <td>yes</td>\n",
       "      <td>no</td>\n",
       "      <td>yes</td>\n",
       "      <td>3</td>\n",
       "      <td>yes</td>\n",
       "      <td>furnished</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>11410000</td>\n",
       "      <td>7420</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>yes</td>\n",
       "      <td>yes</td>\n",
       "      <td>yes</td>\n",
       "      <td>no</td>\n",
       "      <td>yes</td>\n",
       "      <td>2</td>\n",
       "      <td>no</td>\n",
       "      <td>furnished</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      price  area  bedrooms  bathrooms  stories mainroad guestroom basement  \\\n",
       "0  13300000  7420         4          2        3      yes        no       no   \n",
       "1  12250000  8960         4          4        4      yes        no       no   \n",
       "2  12250000  9960         3          2        2      yes        no      yes   \n",
       "3  12215000  7500         4          2        2      yes        no      yes   \n",
       "4  11410000  7420         4          1        2      yes       yes      yes   \n",
       "\n",
       "  hotwaterheating airconditioning  parking prefarea furnishingstatus  \n",
       "0              no             yes        2      yes        furnished  \n",
       "1              no             yes        3       no        furnished  \n",
       "2              no              no        2      yes   semi-furnished  \n",
       "3              no             yes        3      yes        furnished  \n",
       "4              no             yes        2       no        furnished  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Looking at the first five rows\n",
    "housing.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data Preparation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# List of variables to map\n",
    "\n",
    "varlist =  ['mainroad', 'guestroom', 'basement', 'hotwaterheating', 'airconditioning', 'prefarea']\n",
    "\n",
    "# Defining the map function\n",
    "def binary_map(x):\n",
    "    return x.map({'yes': 1, \"no\": 0})\n",
    "\n",
    "# Applying the function to the housing list\n",
    "housing[varlist] = housing[varlist].apply(binary_map)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "      <th>parking</th>\n",
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       "      <th>2</th>\n",
       "      <td>12250000</td>\n",
       "      <td>9960</td>\n",
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       "      <td>2</td>\n",
       "      <td>1</td>\n",
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       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>semi-furnished</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>12215000</td>\n",
       "      <td>7500</td>\n",
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       "      <td>furnished</td>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>furnished</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      ],
      "text/plain": [
       "      price  area  bedrooms  bathrooms  stories  mainroad  guestroom  \\\n",
       "0  13300000  7420         4          2        3         1          0   \n",
       "1  12250000  8960         4          4        4         1          0   \n",
       "2  12250000  9960         3          2        2         1          0   \n",
       "3  12215000  7500         4          2        2         1          0   \n",
       "4  11410000  7420         4          1        2         1          1   \n",
       "\n",
       "   basement  hotwaterheating  airconditioning  parking  prefarea  \\\n",
       "0         0                0                1        2         1   \n",
       "1         0                0                1        3         0   \n",
       "2         1                0                0        2         1   \n",
       "3         1                0                1        3         1   \n",
       "4         1                0                1        2         0   \n",
       "\n",
       "  furnishingstatus  \n",
       "0        furnished  \n",
       "1        furnished  \n",
       "2   semi-furnished  \n",
       "3        furnished  \n",
       "4        furnished  "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the housing dataframe now\n",
    "\n",
    "housing.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dummy Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The variable `furnishingstatus` has three levels. We need to convert these levels into integer as well. For this, we will use something called `dummy variables`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
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       "      <th></th>\n",
       "      <th>furnished</th>\n",
       "      <th>semi-furnished</th>\n",
       "      <th>unfurnished</th>\n",
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      "text/plain": [
       "   furnished  semi-furnished  unfurnished\n",
       "0          1               0            0\n",
       "1          1               0            0\n",
       "2          0               1            0\n",
       "3          1               0            0\n",
       "4          1               0            0"
      ]
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   ],
   "source": [
    "# Get the dummy variables for the feature 'furnishingstatus' and store it in a new variable - 'status'\n",
    "\n",
    "status = pd.get_dummies(housing['furnishingstatus'])\n",
    "\n",
    "# Check what the dataset 'status' looks like\n",
    "status.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "      <td>2</td>\n",
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      "text/plain": [
       "      price  area  bedrooms  bathrooms  stories  mainroad  guestroom  \\\n",
       "0  13300000  7420         4          2        3         1          0   \n",
       "1  12250000  8960         4          4        4         1          0   \n",
       "2  12250000  9960         3          2        2         1          0   \n",
       "3  12215000  7500         4          2        2         1          0   \n",
       "4  11410000  7420         4          1        2         1          1   \n",
       "\n",
       "   basement  hotwaterheating  airconditioning  parking  prefarea  \\\n",
       "0         0                0                1        2         1   \n",
       "1         0                0                1        3         0   \n",
       "2         1                0                0        2         1   \n",
       "3         1                0                1        3         1   \n",
       "4         1                0                1        2         0   \n",
       "\n",
       "  furnishingstatus  semi-furnished  unfurnished  \n",
       "0        furnished               0            0  \n",
       "1        furnished               0            0  \n",
       "2   semi-furnished               1            0  \n",
       "3        furnished               0            0  \n",
       "4        furnished               0            0  "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let's drop the first column from status df using 'drop_first = True'\n",
    "status = pd.get_dummies(housing['furnishingstatus'], drop_first = True)\n",
    "\n",
    "# Add the results to the original housing dataframe\n",
    "housing = pd.concat([housing, status], axis = 1)\n",
    "\n",
    "# Now let's see the head of our dataframe.\n",
    "housing.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>price</th>\n",
       "      <th>area</th>\n",
       "      <th>bedrooms</th>\n",
       "      <th>bathrooms</th>\n",
       "      <th>stories</th>\n",
       "      <th>mainroad</th>\n",
       "      <th>guestroom</th>\n",
       "      <th>basement</th>\n",
       "      <th>hotwaterheating</th>\n",
       "      <th>airconditioning</th>\n",
       "      <th>parking</th>\n",
       "      <th>prefarea</th>\n",
       "      <th>semi-furnished</th>\n",
       "      <th>unfurnished</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>13300000</td>\n",
       "      <td>7420</td>\n",
       "      <td>4</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
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       "      <th>1</th>\n",
       "      <td>12250000</td>\n",
       "      <td>8960</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>12250000</td>\n",
       "      <td>9960</td>\n",
       "      <td>3</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>12215000</td>\n",
       "      <td>7500</td>\n",
       "      <td>4</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>11410000</td>\n",
       "      <td>7420</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      price  area  bedrooms  bathrooms  stories  mainroad  guestroom  \\\n",
       "0  13300000  7420         4          2        3         1          0   \n",
       "1  12250000  8960         4          4        4         1          0   \n",
       "2  12250000  9960         3          2        2         1          0   \n",
       "3  12215000  7500         4          2        2         1          0   \n",
       "4  11410000  7420         4          1        2         1          1   \n",
       "\n",
       "   basement  hotwaterheating  airconditioning  parking  prefarea  \\\n",
       "0         0                0                1        2         1   \n",
       "1         0                0                1        3         0   \n",
       "2         1                0                0        2         1   \n",
       "3         1                0                1        3         1   \n",
       "4         1                0                1        2         0   \n",
       "\n",
       "   semi-furnished  unfurnished  \n",
       "0               0            0  \n",
       "1               0            0  \n",
       "2               1            0  \n",
       "3               0            0  \n",
       "4               0            0  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Drop 'furnishingstatus' as we have created the dummies for it\n",
    "housing.drop(['furnishingstatus'], axis = 1, inplace = True)\n",
    "\n",
    "housing.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Splitting the Data into Training and Testing Sets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "# We specify this so that the train and test data set always have the same rows, respectively\n",
    "\n",
    "df_train, df_test = train_test_split(housing, train_size = 0.7, test_size = 0.3, random_state = 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rescaling the Features \n",
    "\n",
    "We will use MinMax scaling."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import MinMaxScaler\n",
    "scaler = MinMaxScaler()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>price</th>\n",
       "      <th>area</th>\n",
       "      <th>bedrooms</th>\n",
       "      <th>bathrooms</th>\n",
       "      <th>stories</th>\n",
       "      <th>mainroad</th>\n",
       "      <th>guestroom</th>\n",
       "      <th>basement</th>\n",
       "      <th>hotwaterheating</th>\n",
       "      <th>airconditioning</th>\n",
       "      <th>parking</th>\n",
       "      <th>prefarea</th>\n",
       "      <th>semi-furnished</th>\n",
       "      <th>unfurnished</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>359</th>\n",
       "      <td>0.169697</td>\n",
       "      <td>0.155227</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0.333333</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.615152</td>\n",
       "      <td>0.403379</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.333333</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.333333</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>159</th>\n",
       "      <td>0.321212</td>\n",
       "      <td>0.115628</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>0.548133</td>\n",
       "      <td>0.454417</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.666667</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>0.575758</td>\n",
       "      <td>0.538015</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.333333</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0.666667</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        price      area  bedrooms  bathrooms   stories  mainroad  guestroom  \\\n",
       "359  0.169697  0.155227       0.4        0.0  0.000000         1          0   \n",
       "19   0.615152  0.403379       0.4        0.5  0.333333         1          0   \n",
       "159  0.321212  0.115628       0.4        0.5  0.000000         1          1   \n",
       "35   0.548133  0.454417       0.4        0.5  1.000000         1          0   \n",
       "28   0.575758  0.538015       0.8        0.5  0.333333         1          0   \n",
       "\n",
       "     basement  hotwaterheating  airconditioning   parking  prefarea  \\\n",
       "359         0                0                0  0.333333         0   \n",
       "19          0                0                1  0.333333         1   \n",
       "159         1                0                1  0.000000         0   \n",
       "35          0                0                1  0.666667         0   \n",
       "28          1                1                0  0.666667         0   \n",
       "\n",
       "     semi-furnished  unfurnished  \n",
       "359               0            1  \n",
       "19                1            0  \n",
       "159               0            0  \n",
       "35                0            0  \n",
       "28                0            1  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Apply scaler() to all the columns except the 'yes-no' and 'dummy' variables\n",
    "num_vars = ['area', 'bedrooms', 'bathrooms', 'stories', 'parking','price']\n",
    "\n",
    "df_train[num_vars] = scaler.fit_transform(df_train[num_vars])\n",
    "\n",
    "df_train.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dividing into X and Y sets for the model building"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_train = df_train.pop('price')\n",
    "X_train = df_train"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Building our model\n",
    "\n",
    "This time, we will be using the **LinearRegression function from SciKit Learn** for its compatibility with RFE (which is a utility from sklearn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### RFE\n",
    "Recursive feature elimination"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing RFE and LinearRegression\n",
    "from sklearn.feature_selection import RFE\n",
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Running RFE with the output number of the variable equal to 10\n",
    "lm = LinearRegression()\n",
    "lm.fit(X_train, y_train)\n",
    "\n",
    "rfe = RFE(lm, 10)             # running RFE\n",
    "rfe = rfe.fit(X_train, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('area', True, 1),\n",
       " ('bedrooms', True, 1),\n",
       " ('bathrooms', True, 1),\n",
       " ('stories', True, 1),\n",
       " ('mainroad', True, 1),\n",
       " ('guestroom', True, 1),\n",
       " ('basement', False, 3),\n",
       " ('hotwaterheating', True, 1),\n",
       " ('airconditioning', True, 1),\n",
       " ('parking', True, 1),\n",
       " ('prefarea', True, 1),\n",
       " ('semi-furnished', False, 4),\n",
       " ('unfurnished', False, 2)]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(zip(X_train.columns,rfe.support_,rfe.ranking_))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['area', 'bedrooms', 'bathrooms', 'stories', 'mainroad', 'guestroom',\n",
       "       'hotwaterheating', 'airconditioning', 'parking', 'prefarea'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col = X_train.columns[rfe.support_]\n",
    "col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['basement', 'semi-furnished', 'unfurnished'], dtype='object')"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_train.columns[~rfe.support_]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Building model using statsmodel, for the detailed statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating X_test dataframe with RFE selected variables\n",
    "X_train_rfe = X_train[col]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Adding a constant variable \n",
    "import statsmodels.api as sm  \n",
    "X_train_rfe = sm.add_constant(X_train_rfe)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "lm = sm.OLS(y_train,X_train_rfe).fit()   # Running the linear model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                  price   R-squared:                       0.669\n",
      "Model:                            OLS   Adj. R-squared:                  0.660\n",
      "Method:                 Least Squares   F-statistic:                     74.89\n",
      "Date:                Tue, 09 Oct 2018   Prob (F-statistic):           1.28e-82\n",
      "Time:                        13:15:31   Log-Likelihood:                 374.65\n",
      "No. Observations:                 381   AIC:                            -727.3\n",
      "Df Residuals:                     370   BIC:                            -683.9\n",
      "Df Model:                          10                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "===================================================================================\n",
      "                      coef    std err          t      P>|t|      [0.025      0.975]\n",
      "-----------------------------------------------------------------------------------\n",
      "const               0.0027      0.018      0.151      0.880      -0.033       0.038\n",
      "area                0.2363      0.030      7.787      0.000       0.177       0.296\n",
      "bedrooms            0.0661      0.037      1.794      0.074      -0.006       0.139\n",
      "bathrooms           0.1982      0.022      8.927      0.000       0.155       0.242\n",
      "stories             0.0977      0.019      5.251      0.000       0.061       0.134\n",
      "mainroad            0.0556      0.014      3.848      0.000       0.027       0.084\n",
      "guestroom           0.0381      0.013      2.934      0.004       0.013       0.064\n",
      "hotwaterheating     0.0897      0.022      4.104      0.000       0.047       0.133\n",
      "airconditioning     0.0711      0.011      6.235      0.000       0.049       0.093\n",
      "parking             0.0637      0.018      3.488      0.001       0.028       0.100\n",
      "prefarea            0.0643      0.012      5.445      0.000       0.041       0.088\n",
      "==============================================================================\n",
      "Omnibus:                       86.105   Durbin-Watson:                   2.098\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              286.069\n",
      "Skew:                           0.992   Prob(JB):                     7.60e-63\n",
      "Kurtosis:                       6.753   Cond. No.                         13.2\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "#Let's see the summary of our linear model\n",
    "print(lm.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Bedrooms` is insignificant in presence of other variables; can be dropped"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_train_new = X_train_rfe.drop([\"bedrooms\"], axis = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rebuilding the model without `bedrooms`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Adding a constant variable \n",
    "import statsmodels.api as sm  \n",
    "X_train_lm = sm.add_constant(X_train_new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "lm = sm.OLS(y_train,X_train_lm).fit()   # Running the linear model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                  price   R-squared:                       0.666\n",
      "Model:                            OLS   Adj. R-squared:                  0.658\n",
      "Method:                 Least Squares   F-statistic:                     82.37\n",
      "Date:                Tue, 09 Oct 2018   Prob (F-statistic):           6.67e-83\n",
      "Time:                        13:15:31   Log-Likelihood:                 373.00\n",
      "No. Observations:                 381   AIC:                            -726.0\n",
      "Df Residuals:                     371   BIC:                            -686.6\n",
      "Df Model:                           9                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "===================================================================================\n",
      "                      coef    std err          t      P>|t|      [0.025      0.975]\n",
      "-----------------------------------------------------------------------------------\n",
      "const               0.0242      0.013      1.794      0.074      -0.002       0.051\n",
      "area                0.2367      0.030      7.779      0.000       0.177       0.297\n",
      "bathrooms           0.2070      0.022      9.537      0.000       0.164       0.250\n",
      "stories             0.1096      0.017      6.280      0.000       0.075       0.144\n",
      "mainroad            0.0536      0.014      3.710      0.000       0.025       0.082\n",
      "guestroom           0.0390      0.013      2.991      0.003       0.013       0.065\n",
      "hotwaterheating     0.0921      0.022      4.213      0.000       0.049       0.135\n",
      "airconditioning     0.0710      0.011      6.212      0.000       0.049       0.094\n",
      "parking             0.0669      0.018      3.665      0.000       0.031       0.103\n",
      "prefarea            0.0653      0.012      5.513      0.000       0.042       0.089\n",
      "==============================================================================\n",
      "Omnibus:                       91.542   Durbin-Watson:                   2.107\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              315.402\n",
      "Skew:                           1.044   Prob(JB):                     3.25e-69\n",
      "Kurtosis:                       6.938   Cond. No.                         10.0\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "#Let's see the summary of our linear model\n",
    "print(lm.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['const', 'area', 'bathrooms', 'stories', 'mainroad', 'guestroom',\n",
       "       'hotwaterheating', 'airconditioning', 'parking', 'prefarea'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_train_new.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_train_new = X_train_new.drop(['const'], axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Features</th>\n",
       "      <th>VIF</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>area</td>\n",
       "      <td>4.52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>mainroad</td>\n",
       "      <td>4.26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>stories</td>\n",
       "      <td>2.12</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>parking</td>\n",
       "      <td>2.10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>airconditioning</td>\n",
       "      <td>1.75</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>bathrooms</td>\n",
       "      <td>1.58</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>prefarea</td>\n",
       "      <td>1.47</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>guestroom</td>\n",
       "      <td>1.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>hotwaterheating</td>\n",
       "      <td>1.12</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          Features   VIF\n",
       "0             area  4.52\n",
       "3         mainroad  4.26\n",
       "2          stories  2.12\n",
       "7          parking  2.10\n",
       "6  airconditioning  1.75\n",
       "1        bathrooms  1.58\n",
       "8         prefarea  1.47\n",
       "4        guestroom  1.30\n",
       "5  hotwaterheating  1.12"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate the VIFs for the new model\n",
    "from statsmodels.stats.outliers_influence import variance_inflation_factor\n",
    "\n",
    "vif = pd.DataFrame()\n",
    "X = X_train_new\n",
    "vif['Features'] = X.columns\n",
    "vif['VIF'] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]\n",
    "vif['VIF'] = round(vif['VIF'], 2)\n",
    "vif = vif.sort_values(by = \"VIF\", ascending = False)\n",
    "vif"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Residual Analysis of the train data\n",
    "\n",
    "So, now to check if the error terms are also normally distributed (which is infact, one of the major assumptions of linear regression), let us plot the histogram of the error terms and see what it looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_train_price = lm.predict(X_train_lm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing the required libraries for plots.\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\admin\\Anaconda3\\lib\\site-packages\\matplotlib\\axes\\_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,0,'Errors')"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the histogram of the error terms\n",
    "fig = plt.figure()\n",
    "sns.distplot((y_train - y_train_price), bins = 20)\n",
    "fig.suptitle('Error Terms', fontsize = 20)                  # Plot heading \n",
    "plt.xlabel('Errors', fontsize = 18)                         # X-label"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Making Predictions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Applying the scaling on the test sets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_vars = ['area', 'bedrooms', 'bathrooms', 'stories', 'parking','price']\n",
    "\n",
    "df_test[num_vars] = scaler.transform(df_test[num_vars])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Dividing into X_test and y_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_test = df_test.pop('price')\n",
    "X_test = df_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Now let's use our model to make predictions.\n",
    "\n",
    "# Creating X_test_new dataframe by dropping variables from X_test\n",
    "X_test_new = X_test[X_train_new.columns]\n",
    "\n",
    "# Adding a constant variable \n",
    "X_test_new = sm.add_constant(X_test_new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Making predictions\n",
    "y_pred = lm.predict(X_test_new)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'y_pred')"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting y_test and y_pred to understand the spread.\n",
    "fig = plt.figure()\n",
    "plt.scatter(y_test,y_pred)\n",
    "fig.suptitle('y_test vs y_pred', fontsize=20)              # Plot heading \n",
    "plt.xlabel('y_test', fontsize=18)                          # X-label\n",
    "plt.ylabel('y_pred', fontsize=16)                          # Y-label"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.11.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}