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": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Supress Warnings\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing Housing.csv\n",
    "housing = pd.read_csv('Housing.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>yes</td>\n",
       "      <td>2</td>\n",
       "      <td>yes</td>\n",
       "      <td>furnished</td>\n",
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       "    <tr>\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": 5,
     "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": 6,
   "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": 7,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "      <th>bedrooms</th>\n",
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       "      <th>parking</th>\n",
       "      <th>prefarea</th>\n",
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       "      <td>3</td>\n",
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       "      <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>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>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",
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       "      <td>0</td>\n",
       "      <td>1</td>\n",
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       "      <td>furnished</td>\n",
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       "      <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>furnished</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\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": 7,
     "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": 8,
   "metadata": {},
   "outputs": [
    {
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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": "markdown",
   "metadata": {},
   "source": [
    "Now, you don't need three columns. You can drop the `furnished` column, as the type of furnishing can be identified with just the last two columns where — \n",
    "- `00` will correspond to `furnished`\n",
    "- `01` will correspond to `unfurnished`\n",
    "- `10` will correspond to `semi-furnished`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "      <td>furnished</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": 9,
     "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": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\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",
       "      <td>0</td>\n",
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       "    </tr>\n",
       "    <tr>\n",
       "      <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": 10,
     "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": 11,
   "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": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['area', 'bedrooms', 'bathrooms', 'stories', 'mainroad', 'guestroom',\n",
       "       'basement', 'hotwaterheating', 'airconditioning', 'parking', 'prefarea',\n",
       "       'semi-furnished', 'unfurnished'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_train.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['price', 'area', 'bedrooms', 'bathrooms', 'stories', 'mainroad',\n",
       "       'guestroom', 'basement', 'hotwaterheating', 'airconditioning',\n",
       "       'parking', 'prefarea', 'semi-furnished', 'unfurnished'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_test.columns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rescaling the Features \n",
    "\n",
    "We will use MinMax scaling."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import MinMaxScaler\n",
    "scaler = MinMaxScaler()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    }\n",
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       "    }\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",
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       "      <td>0.4</td>\n",
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       "    <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",
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       "      <td>1</td>\n",
       "      <td>0</td>\n",
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       "    <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",
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       "      <td>1</td>\n",
       "      <td>0.000000</td>\n",
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       "      <td>0</td>\n",
       "      <td>0</td>\n",
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       "    <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",
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       "      <td>0</td>\n",
       "      <td>1</td>\n",
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       "      <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": 13,
     "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": 14,
   "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": 15,
   "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": 16,
   "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": 17,
   "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": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(zip(X_train.columns,rfe.support_,rfe.ranking_))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['area', 'bedrooms', 'bathrooms', 'stories', 'mainroad', 'guestroom',\n",
       "       'hotwaterheating', 'airconditioning', 'parking', 'prefarea'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col = X_train.columns[rfe.support_]\n",
    "col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['basement', 'semi-furnished', 'unfurnished'], dtype='object')"
      ]
     },
     "execution_count": 19,
     "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": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating X_test dataframe with RFE selected variables\n",
    "X_train_rfe = X_train[col]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "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": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "lm = sm.OLS(y_train,X_train_rfe).fit()   # Running the linear model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "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:                Mon, 20 Jan 2020   Prob (F-statistic):           1.28e-82\n",
      "Time:                        05:45:34   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": 24,
   "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": 25,
   "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:                Mon, 20 Jan 2020   Prob (F-statistic):           6.67e-83\n",
      "Time:                        05:45:37   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": [
    "# Adding a constant variable \n",
    "import statsmodels.api as sm  \n",
    "X_train_lm = sm.add_constant(X_train_new)\n",
    "\n",
    "lm = sm.OLS(y_train,X_train_lm).fit()   # Running the linear model\n",
    "\n",
    "#Let's see the summary of our linear model\n",
    "print(lm.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['const', 'area', 'bathrooms', 'stories', 'mainroad', 'guestroom',\n",
       "       'hotwaterheating', 'airconditioning', 'parking', 'prefarea'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_train_new.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_train_new = X_train_new.drop(['const'], axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "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": 28,
     "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": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_train_price = lm.predict(X_train_lm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "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": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Errors')"
      ]
     },
     "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": {
      "needs_background": "light"
     },
     "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": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['price', 'area', 'bedrooms', 'bathrooms', 'stories', 'mainroad',\n",
       "       'guestroom', 'basement', 'hotwaterheating', 'airconditioning',\n",
       "       'parking', 'prefarea', 'semi-furnished', 'unfurnished'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_test.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "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": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_test = df_test.pop('price')\n",
    "X_test = df_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "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": 42,
   "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": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y_pred')"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}