6-Multiple Linear Regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import os\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 50 entries, 0 to 49\n",
      "Data columns (total 5 columns):\n",
      " #   Column           Non-Null Count  Dtype  \n",
      "---  ------           --------------  -----  \n",
      " 0   R&D Spend        50 non-null     float64\n",
      " 1   Administration   50 non-null     float64\n",
      " 2   Marketing Spend  50 non-null     float64\n",
      " 3   State            50 non-null     object \n",
      " 4   Profit           50 non-null     float64\n",
      "dtypes: float64(4), object(1)\n",
      "memory usage: 2.1+ KB\n"
     ]
    }
   ],
   "source": [
    "dataset=pd.read_csv('C:/Users/H_jam/OneDrive/Uni-MA/Term2/DataMining/DataSets/Regression/Multiple Linear Regression/50_Startups.csv')\n",
    "dataset.info()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining Independent and Dependent Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x25cafc66b80>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=dataset.iloc[:,:-1].values\n",
    "y=dataset.iloc[:,4].values\n",
    "plt.scatter(x[:,0],y)\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Encoding Categorical Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import OneHotEncoder\n",
    "from sklearn.compose import ColumnTransformer\n",
    "onehotencoder=ColumnTransformer(transformers=[('one_hot_encoder',OneHotEncoder(categories='auto'),[3])],remainder='passthrough')\n",
    "x=onehotencoder.fit_transform(x)\n",
    "x=x[:,1:]"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Spliting the Dataset into the Training set and Test set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split\n",
    "x_train,x_test,y_train,y_test=train_test_split(x,y,test_size=0.2,random_state=0)\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fitting Multiple Linear Regression to the Training set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "regressor=LinearRegression()\n",
    "regressor.fit(x_train,y_train)\n",
    "\n",
    "# Predict the test set results\n",
    "y_pred=regressor.predict(x_test)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "New data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([180607.63973379])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_new=np.array([[1,0,160000,130000,300000]])\n",
    "regressor.predict(x_new)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evalaution by R-Squared\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9347068473282405\n"
     ]
    }
   ],
   "source": [
    "r_squared_test=regressor.score(x_test,y_test)\n",
    "print(r_squared_test)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Building the optimal model using Backward Elimination"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "# append a column of ones to the x matrix\n",
    "x=np.append(arr=np.ones((50,1)).astype(int),values=x,axis=1)\n",
    "\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find the optimal model using Backward Eliminationq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.951</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.945</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   169.9</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Tue, 18 Apr 2023</td> <th>  Prob (F-statistic):</th> <td>1.34e-27</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>16:46:20</td>     <th>  Log-Likelihood:    </th> <td> -525.38</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1063.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    44</td>      <th>  BIC:               </th> <td>   1074.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     5</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td> 5.013e+04</td> <td> 6884.820</td> <td>    7.281</td> <td> 0.000</td> <td> 3.62e+04</td> <td>  6.4e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>  198.7888</td> <td> 3371.007</td> <td>    0.059</td> <td> 0.953</td> <td>-6595.030</td> <td> 6992.607</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>  -41.8870</td> <td> 3256.039</td> <td>   -0.013</td> <td> 0.990</td> <td>-6604.003</td> <td> 6520.229</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>    <td>    0.8060</td> <td>    0.046</td> <td>   17.369</td> <td> 0.000</td> <td>    0.712</td> <td>    0.900</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x4</th>    <td>   -0.0270</td> <td>    0.052</td> <td>   -0.517</td> <td> 0.608</td> <td>   -0.132</td> <td>    0.078</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x5</th>    <td>    0.0270</td> <td>    0.017</td> <td>    1.574</td> <td> 0.123</td> <td>   -0.008</td> <td>    0.062</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>14.782</td> <th>  Durbin-Watson:     </th> <td>   1.283</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  21.266</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.948</td> <th>  Prob(JB):          </th> <td>2.41e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.572</td> <th>  Cond. No.          </th> <td>1.45e+06</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 1.45e+06. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.951\n",
       "Model:                            OLS   Adj. R-squared:                  0.945\n",
       "Method:                 Least Squares   F-statistic:                     169.9\n",
       "Date:                Tue, 18 Apr 2023   Prob (F-statistic):           1.34e-27\n",
       "Time:                        16:46:20   Log-Likelihood:                -525.38\n",
       "No. Observations:                  50   AIC:                             1063.\n",
       "Df Residuals:                      44   BIC:                             1074.\n",
       "Df Model:                           5                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       5.013e+04   6884.820      7.281      0.000    3.62e+04     6.4e+04\n",
       "x1           198.7888   3371.007      0.059      0.953   -6595.030    6992.607\n",
       "x2           -41.8870   3256.039     -0.013      0.990   -6604.003    6520.229\n",
       "x3             0.8060      0.046     17.369      0.000       0.712       0.900\n",
       "x4            -0.0270      0.052     -0.517      0.608      -0.132       0.078\n",
       "x5             0.0270      0.017      1.574      0.123      -0.008       0.062\n",
       "==============================================================================\n",
       "Omnibus:                       14.782   Durbin-Watson:                   1.283\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               21.266\n",
       "Skew:                          -0.948   Prob(JB):                     2.41e-05\n",
       "Kurtosis:                       5.572   Cond. No.                     1.45e+06\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 1.45e+06. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# x_opt is x optimized\n",
    "x_opt=np.array(x[:,[0,1,2,3,4,5]], dtype=float)\n",
    "regressor_OLS=sm.OLS(endog=y,exog=x_opt).fit()\n",
    "regressor_OLS.summary()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.951</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.948</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   296.0</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Tue, 18 Apr 2023</td> <th>  Prob (F-statistic):</th> <td>4.53e-30</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>16:47:27</td>     <th>  Log-Likelihood:    </th> <td> -525.39</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1059.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    46</td>      <th>  BIC:               </th> <td>   1066.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     3</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td> 5.012e+04</td> <td> 6572.353</td> <td>    7.626</td> <td> 0.000</td> <td> 3.69e+04</td> <td> 6.34e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    0.8057</td> <td>    0.045</td> <td>   17.846</td> <td> 0.000</td> <td>    0.715</td> <td>    0.897</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>   -0.0268</td> <td>    0.051</td> <td>   -0.526</td> <td> 0.602</td> <td>   -0.130</td> <td>    0.076</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>    <td>    0.0272</td> <td>    0.016</td> <td>    1.655</td> <td> 0.105</td> <td>   -0.006</td> <td>    0.060</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>14.838</td> <th>  Durbin-Watson:     </th> <td>   1.282</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  21.442</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.949</td> <th>  Prob(JB):          </th> <td>2.21e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.586</td> <th>  Cond. No.          </th> <td>1.40e+06</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 1.4e+06. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.951\n",
       "Model:                            OLS   Adj. R-squared:                  0.948\n",
       "Method:                 Least Squares   F-statistic:                     296.0\n",
       "Date:                Tue, 18 Apr 2023   Prob (F-statistic):           4.53e-30\n",
       "Time:                        16:47:27   Log-Likelihood:                -525.39\n",
       "No. Observations:                  50   AIC:                             1059.\n",
       "Df Residuals:                      46   BIC:                             1066.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       5.012e+04   6572.353      7.626      0.000    3.69e+04    6.34e+04\n",
       "x1             0.8057      0.045     17.846      0.000       0.715       0.897\n",
       "x2            -0.0268      0.051     -0.526      0.602      -0.130       0.076\n",
       "x3             0.0272      0.016      1.655      0.105      -0.006       0.060\n",
       "==============================================================================\n",
       "Omnibus:                       14.838   Durbin-Watson:                   1.282\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               21.442\n",
       "Skew:                          -0.949   Prob(JB):                     2.21e-05\n",
       "Kurtosis:                       5.586   Cond. No.                     1.40e+06\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 1.4e+06. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# با توجه به نتیجه قبلی، ستون 1 و 2 را حذف میکنیم\n",
    "\n",
    "x_opt=np.array(x[:,[0,3,4,5]], dtype=float)\n",
    "regressor_OLS=sm.OLS(endog=y,exog=x_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.950</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.948</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   450.8</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Tue, 18 Apr 2023</td> <th>  Prob (F-statistic):</th> <td>2.16e-31</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>16:49:03</td>     <th>  Log-Likelihood:    </th> <td> -525.54</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1057.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    47</td>      <th>  BIC:               </th> <td>   1063.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td> 4.698e+04</td> <td> 2689.933</td> <td>   17.464</td> <td> 0.000</td> <td> 4.16e+04</td> <td> 5.24e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    0.7966</td> <td>    0.041</td> <td>   19.266</td> <td> 0.000</td> <td>    0.713</td> <td>    0.880</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>    0.0299</td> <td>    0.016</td> <td>    1.927</td> <td> 0.060</td> <td>   -0.001</td> <td>    0.061</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>14.677</td> <th>  Durbin-Watson:     </th> <td>   1.257</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  21.161</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.939</td> <th>  Prob(JB):          </th> <td>2.54e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.575</td> <th>  Cond. No.          </th> <td>5.32e+05</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 5.32e+05. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.950\n",
       "Model:                            OLS   Adj. R-squared:                  0.948\n",
       "Method:                 Least Squares   F-statistic:                     450.8\n",
       "Date:                Tue, 18 Apr 2023   Prob (F-statistic):           2.16e-31\n",
       "Time:                        16:49:03   Log-Likelihood:                -525.54\n",
       "No. Observations:                  50   AIC:                             1057.\n",
       "Df Residuals:                      47   BIC:                             1063.\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       4.698e+04   2689.933     17.464      0.000    4.16e+04    5.24e+04\n",
       "x1             0.7966      0.041     19.266      0.000       0.713       0.880\n",
       "x2             0.0299      0.016      1.927      0.060      -0.001       0.061\n",
       "==============================================================================\n",
       "Omnibus:                       14.677   Durbin-Watson:                   1.257\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               21.161\n",
       "Skew:                          -0.939   Prob(JB):                     2.54e-05\n",
       "Kurtosis:                       5.575   Cond. No.                     5.32e+05\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 5.32e+05. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# با توجه به نتیجه قبلی، ستون 4 را حذف میکنیم\n",
    "x_opt=np.array(x[:,[0,3,5]], dtype=float)\n",
    "regressor_OLS=sm.OLS(endog=y,exog=x_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We recognize to remove the x2 from Adjusted R-Squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.947</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.945</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   849.8</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Tue, 18 Apr 2023</td> <th>  Prob (F-statistic):</th> <td>3.50e-32</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>16:52:48</td>     <th>  Log-Likelihood:    </th> <td> -527.44</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1059.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    48</td>      <th>  BIC:               </th> <td>   1063.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td> 4.903e+04</td> <td> 2537.897</td> <td>   19.320</td> <td> 0.000</td> <td> 4.39e+04</td> <td> 5.41e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    0.8543</td> <td>    0.029</td> <td>   29.151</td> <td> 0.000</td> <td>    0.795</td> <td>    0.913</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>13.727</td> <th>  Durbin-Watson:     </th> <td>   1.116</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  18.536</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.911</td> <th>  Prob(JB):          </th> <td>9.44e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.361</td> <th>  Cond. No.          </th> <td>1.65e+05</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 1.65e+05. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.947\n",
       "Model:                            OLS   Adj. R-squared:                  0.945\n",
       "Method:                 Least Squares   F-statistic:                     849.8\n",
       "Date:                Tue, 18 Apr 2023   Prob (F-statistic):           3.50e-32\n",
       "Time:                        16:52:48   Log-Likelihood:                -527.44\n",
       "No. Observations:                  50   AIC:                             1059.\n",
       "Df Residuals:                      48   BIC:                             1063.\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       4.903e+04   2537.897     19.320      0.000    4.39e+04    5.41e+04\n",
       "x1             0.8543      0.029     29.151      0.000       0.795       0.913\n",
       "==============================================================================\n",
       "Omnibus:                       13.727   Durbin-Watson:                   1.116\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               18.536\n",
       "Skew:                          -0.911   Prob(JB):                     9.44e-05\n",
       "Kurtosis:                       5.361   Cond. No.                     1.65e+05\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 1.65e+05. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# با توجه به نتیجه قبلی، ستون 5 را حذف میکنیم\n",
    "x_opt=np.array(x[:,[0,3]], dtype=float)\n",
    "regressor_OLS=sm.OLS(endog=y,exog=x_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "نتیجه میگیریم که متغیر x2  که حذف کردیم، مهم بوده و باید برگردد\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "پس مدل بهینه ما ، مدل 0 3 5 بوده است"
   ]
  }
 ],
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