sklearn - Feature Selection with F-regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiple linear regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Import the relevant libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# For these lessons we will need NumPy, pandas, matplotlib and seaborn\n",
    "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",
    "# and of course the actual regression (machine learning) module\n",
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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>SAT</th>\n",
       "      <th>Rand 1,2,3</th>\n",
       "      <th>GPA</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1714</td>\n",
       "      <td>1</td>\n",
       "      <td>2.40</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1664</td>\n",
       "      <td>3</td>\n",
       "      <td>2.52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1760</td>\n",
       "      <td>3</td>\n",
       "      <td>2.54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1685</td>\n",
       "      <td>3</td>\n",
       "      <td>2.74</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1693</td>\n",
       "      <td>2</td>\n",
       "      <td>2.83</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "    SAT  Rand 1,2,3   GPA\n",
       "0  1714           1  2.40\n",
       "1  1664           3  2.52\n",
       "2  1760           3  2.54\n",
       "3  1685           3  2.74\n",
       "4  1693           2  2.83"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Load the data from a .csv in the same folder\n",
    "data = pd.read_csv(r'D:\\Stu\\Data Science\\Exercises Regression\\1.02. Multiple linear regression.csv')\n",
    "\n",
    "# Let's explore the top 5 rows of the df\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>SAT</th>\n",
       "      <th>Rand 1,2,3</th>\n",
       "      <th>GPA</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>1845.273810</td>\n",
       "      <td>2.059524</td>\n",
       "      <td>3.330238</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>104.530661</td>\n",
       "      <td>0.855192</td>\n",
       "      <td>0.271617</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1634.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>2.400000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>1772.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>3.190000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1846.000000</td>\n",
       "      <td>2.000000</td>\n",
       "      <td>3.380000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>1934.000000</td>\n",
       "      <td>3.000000</td>\n",
       "      <td>3.502500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>2050.000000</td>\n",
       "      <td>3.000000</td>\n",
       "      <td>3.810000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      ],
      "text/plain": [
       "               SAT  Rand 1,2,3        GPA\n",
       "count    84.000000   84.000000  84.000000\n",
       "mean   1845.273810    2.059524   3.330238\n",
       "std     104.530661    0.855192   0.271617\n",
       "min    1634.000000    1.000000   2.400000\n",
       "25%    1772.000000    1.000000   3.190000\n",
       "50%    1846.000000    2.000000   3.380000\n",
       "75%    1934.000000    3.000000   3.502500\n",
       "max    2050.000000    3.000000   3.810000"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This method gives us very nice descriptive statistics. We don't need this for now, but will later on!\n",
    "data.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create the multiple linear regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Declare the dependent and independent variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# There are two independent variables: 'SAT' and 'Rand 1,2,3'\n",
    "x = data[['SAT','Rand 1,2,3']]\n",
    "\n",
    "# and a single depended variable: 'GPA'\n",
    "y = data['GPA']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regression itself"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We start by creating a linear regression object\n",
    "reg = LinearRegression()\n",
    "\n",
    "# The whole learning process boils down to fitting the regression\n",
    "reg.fit(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00165354, -0.00826982])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Getting the coefficients of the regression\n",
    "reg.coef_\n",
    "# Note that the output is an array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.29603261264909486"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Getting the intercept of the regression\n",
    "reg.intercept_\n",
    "# Note that the result is a float as we usually expect a single value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the R-squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4066811952814285"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the R-squared of the regression\n",
    "reg.score(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Formula for Adjusted R^2\n",
    "\n",
    "$R^2_{adj.} = 1 - (1-R^2)*\\frac{n-1}{n-p-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(84, 2)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the shape of x, to facilitate the creation of the Adjusted R^2 metric\n",
    "x.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.39203134825134023"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# If we want to find the Adjusted R-squared we can do so by knowing the r2, the # observations, the # features\n",
    "r2 = reg.score(x,y)\n",
    "# Number of observations is the shape along axis 0\n",
    "n = x.shape[0]\n",
    "# Number of features (predictors, p) is the shape along axis 1\n",
    "p = x.shape[1]\n",
    "\n",
    "# We find the Adjusted R-squared using the formula\n",
    "adjusted_r2 = 1-(1-r2)*(n-1)/(n-p-1)\n",
    "adjusted_r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Feature selection\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import the feature selection module from sklearn\n",
    "# This module allows us to select the most appopriate features for our regression\n",
    "# There exist many different approaches to feature selection, however, we will use one of the simplest\n",
    "from sklearn.feature_selection import f_regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([56.04804786,  0.17558437]), array([7.19951844e-11, 6.76291372e-01]))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We will look into: f_regression\n",
    "# f_regression finds the F-statistics for the *simple* regressions created with each of the independent variables\n",
    "# In our case, this would mean running a simple linear regression on GPA where SAT is the independent variable\n",
    "# and a simple linear regression on GPA where Rand 1,2,3 is the indepdent variable\n",
    "# The limitation of this approach is that it does not take into account the mutual effect of the two features\n",
    "f_regression(x,y)\n",
    "\n",
    "# There are two output arrays\n",
    "# The first one contains the F-statistics for each of the regressions\n",
    "# The second one contains the p-values of these F-statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.19951844e-11, 6.76291372e-01])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Since we are more interested in the latter (p-values), we can just take the second array\n",
    "p_values = f_regression(x,y)[1]\n",
    "p_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.   , 0.676])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# To be able to quickly evaluate them, we can round the result to 3 digits after the dot\n",
    "p_values.round(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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