{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "651019ae-1bd8-41b3-be75-474a841f1f8c",
   "metadata": {},
   "source": [
    "# Example of Multiple Linear Regression in Python\n",
    "https://datatofish.com/multiple-linear-regression-python/"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de3b5be7-792e-4a94-8cd0-a17df627c419",
   "metadata": {},
   "source": [
    "## About Linear Regression\n",
    "Linear regression is used as a predictive model that assumes a linear relationship between the dependent variable (which is the variable we are trying to predict/estimate) and the independent variable/s (input variable/s used in the prediction)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efd67629-b49f-47e2-9305-a7abb6f10366",
   "metadata": {},
   "source": [
    "## Example of Multiple Linear Regression in Python\n",
    "\n",
    "In the following example, we will perform multiple linear regression for a fictitious economy, where the index_price is the dependent variable, and the 2 independent/input variables are:\n",
    "\n",
    "* interest_rate\n",
    "* unemployment_rate\n",
    "\n",
    "Please note that you will have to validate that several assumptions are met before you apply linear regression models. Most notably, you have to make sure that a linear relationship exists between the dependent variable and the independent variable/s (more on that under the checking for linearity section)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3dacd2dc-aa3e-40f3-a66d-252b7512bef7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    year  month  interest_rate  unemployment_rate  index_price\n",
      "0   2017     12           2.75                5.3         1464\n",
      "1   2017     11           2.50                5.3         1394\n",
      "2   2017     10           2.50                5.3         1357\n",
      "3   2017      9           2.50                5.3         1293\n",
      "4   2017      8           2.50                5.4         1256\n",
      "5   2017      7           2.50                5.6         1254\n",
      "6   2017      6           2.50                5.5         1234\n",
      "7   2017      5           2.25                5.5         1195\n",
      "8   2017      4           2.25                5.5         1159\n",
      "9   2017      3           2.25                5.6         1167\n",
      "10  2017      2           2.00                5.7         1130\n",
      "11  2017      1           2.00                5.9         1075\n",
      "12  2016     12           2.00                6.0         1047\n",
      "13  2016     11           1.75                5.9          965\n",
      "14  2016     10           1.75                5.8          943\n",
      "15  2016      9           1.75                6.1          958\n",
      "16  2016      8           1.75                6.2          971\n",
      "17  2016      7           1.75                6.1          949\n",
      "18  2016      6           1.75                6.1          884\n",
      "19  2016      5           1.75                6.1          866\n",
      "20  2016      4           1.75                5.9          876\n",
      "21  2016      3           1.75                6.2          822\n",
      "22  2016      2           1.75                6.2          704\n",
      "23  2016      1           1.75                6.1          719\n"
     ]
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "data = {'year': [2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016],\n",
    "        'month': [12,11,10,9,8,7,6,5,4,3,2,1,12,11,10,9,8,7,6,5,4,3,2,1],\n",
    "        'interest_rate': [2.75,2.5,2.5,2.5,2.5,2.5,2.5,2.25,2.25,2.25,2,2,2,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75],\n",
    "        'unemployment_rate': [5.3,5.3,5.3,5.3,5.4,5.6,5.5,5.5,5.5,5.6,5.7,5.9,6,5.9,5.8,6.1,6.2,6.1,6.1,6.1,5.9,6.2,6.2,6.1],\n",
    "        'index_price': [1464,1394,1357,1293,1256,1254,1234,1195,1159,1167,1130,1075,1047,965,943,958,971,949,884,866,876,822,704,719]        \n",
    "        }\n",
    "\n",
    "df = pd.DataFrame(data) \n",
    "\n",
    "print(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "545f4540-581f-4fc9-89e8-02150f40cabd",
   "metadata": {},
   "source": [
    "## Checking for Linearity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "9a3bfbae-0877-442b-86fa-d477cf6153a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.scatter(df['interest_rate'], df['index_price'], color='red')\n",
    "plt.title('Index Price Vs Interest Rate', fontsize=14)\n",
    "plt.xlabel('Interest Rate', fontsize=14)\n",
    "plt.ylabel('Index Price', fontsize=14)\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61835a39-e63e-430a-b5df-e614d96776d0",
   "metadata": {},
   "source": [
    "## Performing the Multiple Linear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "4658f05a-3adc-44e4-a576-eed94de204d0",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = df[['interest_rate','unemployment_rate']]\n",
    "y = df['index_price']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b7443b5-1c4a-49ac-89e4-4d3bd29f584a",
   "metadata": {},
   "source": [
    "### The Python Code using Statsmodels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "9d2f8b51-ee14-4e3e-bd86-75343644d1e1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:            index_price   R-squared:                       0.898\n",
      "Model:                            OLS   Adj. R-squared:                  0.888\n",
      "Method:                 Least Squares   F-statistic:                     92.07\n",
      "Date:                Sun, 09 Jul 2023   Prob (F-statistic):           4.04e-11\n",
      "Time:                        17:15:55   Log-Likelihood:                -134.61\n",
      "No. Observations:                  24   AIC:                             275.2\n",
      "Df Residuals:                      21   BIC:                             278.8\n",
      "Df Model:                           2                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "=====================================================================================\n",
      "                        coef    std err          t      P>|t|      [0.025      0.975]\n",
      "-------------------------------------------------------------------------------------\n",
      "const              1798.4040    899.248      2.000      0.059     -71.685    3668.493\n",
      "interest_rate       345.5401    111.367      3.103      0.005     113.940     577.140\n",
      "unemployment_rate  -250.1466    117.950     -2.121      0.046    -495.437      -4.856\n",
      "==============================================================================\n",
      "Omnibus:                        2.691   Durbin-Watson:                   0.530\n",
      "Prob(Omnibus):                  0.260   Jarque-Bera (JB):                1.551\n",
      "Skew:                          -0.612   Prob(JB):                        0.461\n",
      "Kurtosis:                       3.226   Cond. No.                         394.\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "x = sm.add_constant(x)\n",
    " \n",
    "model = sm.OLS(y, x).fit()\n",
    "predictions = model.predict(x) \n",
    " \n",
    "print_model = model.summary()\n",
    "print(print_model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80f3744a-6b0f-4fef-bd77-587bdf7bbed0",
   "metadata": {},
   "source": [
    "### The Python Code using Sklearn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "3c6d8a8d-ba24-477a-bda4-c54e74e76be4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression()"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn import linear_model\n",
    "\n",
    "regr = linear_model.LinearRegression()\n",
    "regr.fit(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "cd5a8b11-a49c-4fcd-bfdb-059b6b3e2ab5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept: \n",
      " 1798.4039776258558\n",
      "Coefficients: \n",
      " [   0.          345.54008701 -250.14657137]\n"
     ]
    }
   ],
   "source": [
    "print('Intercept: \\n', regr.intercept_)\n",
    "print('Coefficients: \\n', regr.coef_)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "722a577e-9e89-4808-841e-0c9535b8fd99",
   "metadata": {},
   "source": [
    "### Conclusion\n",
    "\n",
    "Linear regression is often used in Machine Learning. You have seen some examples of how to perform multiple linear regression in Python using both sklearn and statsmodels.\n",
    "\\\n",
    "Before applying linear regression models, make sure to check that a linear relationship exists between the dependent variable (i.e., what you are trying to predict) and the independent variable/s (i.e., the input variable/s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0b683e7d-d9ca-4ab2-87f9-c8c0d992ada3",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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