sklearn_MLR_2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiple Linear Regression with sklearn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "data located in  real_estate_price_size_year.csv \n",
    "\n",
    "To do:\n",
    "-  Create a multiple linear regression \n",
    "-  Display the intercept and coefficient(s)\n",
    "-  Find the R-squared and Adjusted R-squared\n",
    "-  Compare the R-squared and the Adjusted R-squared\n",
    "-  Compare the R-squared of this regression and the simple linear regression where only 'size' was used\n",
    "-  Using the model make a prediction about an apartment with size 750 sq.ft. from 2009\n",
    "-  Find the univariate / multivariate p-values of the two variables. Describe.\n",
    "-  Create a summary table\n",
    "\n",
    "In this exercise, the dependent variable is 'price', while the independent variables are 'size' and 'year'.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set()\n",
    "\n",
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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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>size</th>\n",
       "      <th>year</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>234314.144</td>\n",
       "      <td>643.09</td>\n",
       "      <td>2015</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>228581.528</td>\n",
       "      <td>656.22</td>\n",
       "      <td>2009</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>281626.336</td>\n",
       "      <td>487.29</td>\n",
       "      <td>2018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>401255.608</td>\n",
       "      <td>1504.75</td>\n",
       "      <td>2015</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>458674.256</td>\n",
       "      <td>1275.46</td>\n",
       "      <td>2009</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        price     size  year\n",
       "0  234314.144   643.09  2015\n",
       "1  228581.528   656.22  2009\n",
       "2  281626.336   487.29  2018\n",
       "3  401255.608  1504.75  2015\n",
       "4  458674.256  1275.46  2009"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = pd.read_csv('real_estate_price_size_year.csv')\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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>size</th>\n",
       "      <th>year</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>100.000000</td>\n",
       "      <td>100.000000</td>\n",
       "      <td>100.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>292289.470160</td>\n",
       "      <td>853.024200</td>\n",
       "      <td>2012.600000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>77051.727525</td>\n",
       "      <td>297.941951</td>\n",
       "      <td>4.729021</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>154282.128000</td>\n",
       "      <td>479.750000</td>\n",
       "      <td>2006.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>234280.148000</td>\n",
       "      <td>643.330000</td>\n",
       "      <td>2009.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>280590.716000</td>\n",
       "      <td>696.405000</td>\n",
       "      <td>2015.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>335723.696000</td>\n",
       "      <td>1029.322500</td>\n",
       "      <td>2018.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>500681.128000</td>\n",
       "      <td>1842.510000</td>\n",
       "      <td>2018.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               price         size         year\n",
       "count     100.000000   100.000000   100.000000\n",
       "mean   292289.470160   853.024200  2012.600000\n",
       "std     77051.727525   297.941951     4.729021\n",
       "min    154282.128000   479.750000  2006.000000\n",
       "25%    234280.148000   643.330000  2009.000000\n",
       "50%    280590.716000   696.405000  2015.000000\n",
       "75%    335723.696000  1029.322500  2018.000000\n",
       "max    500681.128000  1842.510000  2018.000000"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Declare the dependent and the independent variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = data[['size','year']]\n",
    "y = data['price']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg = LinearRegression()\n",
    "reg.fit(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Find the intercept"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-5772267.01746328"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.intercept_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Find the coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 227.70085401, 2916.78532684])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.coef_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculate the R-squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7764803683276792"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.score(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculate the Adjusted R-squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Let's use the handy function we created\n",
    "def adj_r2(x,y):\n",
    "    r2 = reg.score(x,y)\n",
    "    n = x.shape[0]\n",
    "    p = x.shape[1]\n",
    "    adjusted_r2 = 1-(1-r2)*(n-1)/(n-p-1)\n",
    "    return adjusted_r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7718717161282499"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "adj_r2(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compare the R-squared and the Adjusted R-squared"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems the the R-squared is only slightly larger than the Adjusted R-squared, implying that we were not penalized a lot for the inclusion of 2 independent variables. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compare the Adjusted R-squared with the R-squared of the simple linear regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparing the Adjusted R-squared with the R-squared of the simple linear regression (when only 'size' was used - a couple of lectures ago), we realize that 'Year' is not bringing too much value to the result."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Making predictions\n",
    "\n",
    "Find the predicted price of an apartment that has a size of 750 sq.ft. from 2009."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([258330.34465995])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.predict([[750,2009]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculate the univariate p-values of the variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.feature_selection import f_regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([285.92105192,   0.85525799]), array([8.12763222e-31, 3.57340758e-01]))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_regression(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8.12763222e-31, 3.57340758e-01])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_values = f_regression(x,y)[1]\n",
    "p_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.   , 0.357])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_values.round(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create a summary table with your findings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Features</th>\n",
       "      <th>Coefficients</th>\n",
       "      <th>p-values</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>size</td>\n",
       "      <td>227.700854</td>\n",
       "      <td>0.000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>year</td>\n",
       "      <td>2916.785327</td>\n",
       "      <td>0.357</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      ],
      "text/plain": [
       "  Features  Coefficients  p-values\n",
       "0     size    227.700854     0.000\n",
       "1     year   2916.785327     0.357"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg_summary = pd.DataFrame(data = x.columns.values, columns=['Features'])\n",
    "reg_summary ['Coefficients'] = reg.coef_\n",
    "reg_summary ['p-values'] = p_values.round(3)\n",
    "reg_summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems that 'Year' is not event significant, therefore we should remove it from the model."
   ]
  }
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