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003MultiLinear_Regression.py
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003MultiLinear_Regression.py
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import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.preprocessing import LabelEncoder,OneHotEncoder
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
import statsmodels.formula.api as sm
"""
5 methods of building multiple linear regression model:
1. All in
2. Backward elimination(fastest)
3. Forward selection
4. Bidirectional elimination(slowest)
5. Score compare(worst as it compares 2^n models if there are n columns)
Multiple linear regression does not need feature scaling as it does it by itself.
"""
# Loading datasets
dataset = pd.read_csv("50_Startups.csv")
#print(dataset)
# Creating datasets
x = dataset.iloc[:,:-1].values
y = dataset.iloc[:,4].values
# Encoding categorical data
le_x = LabelEncoder()
x[:,3] = le_x.fit_transform(x[:,3])
ohe = OneHotEncoder(categorical_features=[3])
x = ohe.fit_transform(x).toarray()
# Avoiding dummy variable trap
x = x[:, 1:]
#print(x)
# Training and testing datasets
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
"""print(x_train)
print(x_test)
print(y_train)
print(y_test)"""
# Fitting multiple linear regression to training set
regr = LinearRegression()
regr.fit(x_train,y_train)
# Predicting the test set results
y_pred = regr.predict(x_test)
#print(y_pred)
#print(y_test)
# Building the optimal model using Backward Elimination
x = np.append(arr = np.ones([50, 1]).astype(int), values = x, axis = 1)
#print(x)
x_opt = x[:, [0,1,2,3,4,5]]
mregr = sm.OLS(endog = y, exog = x_opt).fit()
mregr.summary()
"""
OLS Regression Results
Dep. Variable: y R-squared: 0.951
Model: OLS Adj. R-squared: 0.945
Method: Least Squares F-statistic: 169.9
Date: Sun, 06 Jan 2019 Prob (F-statistic): 1.34e-27
Time: 18:39:15 Log-Likelihood: -525.38
No. Observations: 50 AIC: 1063.
Df Residuals: 44 BIC: 1074.
Df Model: 5
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 5.013e+04 6884.820 7.281 0.000 3.62e+04 6.4e+04
x1 198.7888 3371.007 0.059 0.953 -6595.030 6992.607
x2 -41.8870 3256.039 -0.013 0.990 -6604.003 6520.229
x3 0.8060 0.046 17.369 0.000 0.712 0.900
x4 -0.0270 0.052 -0.517 0.608 -0.132 0.078
x5 0.0270 0.017 1.574 0.123 -0.008 0.062
Omnibus: 14.782 Durbin-Watson: 1.283
Prob(Omnibus): 0.001 Jarque-Bera (JB): 21.266
Skew: -0.948 Prob(JB): 2.41e-05
Kurtosis: 5.572 Cond. No. 1.45e+06
"""
x_opt = x[:, [0,1,3,4,5]]
mregr = sm.OLS(endog = y, exog = x_opt).fit()
mregr.summary()
"""
OLS Regression Results
Dep. Variable: y R-squared: 0.951
Model: OLS Adj. R-squared: 0.946
Method: Least Squares F-statistic: 217.2
Date: Sun, 06 Jan 2019 Prob (F-statistic): 8.49e-29
Time: 18:43:12 Log-Likelihood: -525.38
No. Observations: 50 AIC: 1061.
Df Residuals: 45 BIC: 1070.
Df Model: 4
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 5.011e+04 6647.870 7.537 0.000 3.67e+04 6.35e+04
x1 220.1585 2900.536 0.076 0.940 -5621.821 6062.138
x2 0.8060 0.046 17.606 0.000 0.714 0.898
x3 -0.0270 0.052 -0.523 0.604 -0.131 0.077
x4 0.0270 0.017 1.592 0.118 -0.007 0.061
Omnibus: 14.758 Durbin-Watson: 1.282
Prob(Omnibus): 0.001 Jarque-Bera (JB): 21.172
Skew: -0.948 Prob(JB): 2.53e-05
Kurtosis: 5.563 Cond. No. 1.40e+06
"""
x_opt = x[:, [0,3,4,5]]
mregr = sm.OLS(endog = y, exog = x_opt).fit()
mregr.summary()
"""
OLS Regression Results
Dep. Variable: y R-squared: 0.951
Model: OLS Adj. R-squared: 0.948
Method: Least Squares F-statistic: 296.0
Date: Sun, 06 Jan 2019 Prob (F-statistic): 4.53e-30
Time: 18:45:23 Log-Likelihood: -525.39
No. Observations: 50 AIC: 1059.
Df Residuals: 46 BIC: 1066.
Df Model: 3
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 5.012e+04 6572.353 7.626 0.000 3.69e+04 6.34e+04
x1 0.8057 0.045 17.846 0.000 0.715 0.897
x2 -0.0268 0.051 -0.526 0.602 -0.130 0.076
x3 0.0272 0.016 1.655 0.105 -0.006 0.060
Omnibus: 14.838 Durbin-Watson: 1.282
Prob(Omnibus): 0.001 Jarque-Bera (JB): 21.442
Skew: -0.949 Prob(JB): 2.21e-05
Kurtosis: 5.586 Cond. No. 1.40e+06
"""
x_opt = x[:, [0,3,5]]
mregr = sm.OLS(endog = y, exog = x_opt).fit()
mregr.summary()
"""
OLS Regression Results
Dep. Variable: y R-squared: 0.950
Model: OLS Adj. R-squared: 0.948
Method: Least Squares F-statistic: 450.8
Date: Sun, 06 Jan 2019 Prob (F-statistic): 2.16e-31
Time: 18:46:51 Log-Likelihood: -525.54
No. Observations: 50 AIC: 1057.
Df Residuals: 47 BIC: 1063.
Df Model: 2
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 4.698e+04 2689.933 17.464 0.000 4.16e+04 5.24e+04
x1 0.7966 0.041 19.266 0.000 0.713 0.880
x2 0.0299 0.016 1.927 0.060 -0.001 0.061
Omnibus: 14.677 Durbin-Watson: 1.257
Prob(Omnibus): 0.001 Jarque-Bera (JB): 21.161
Skew: -0.939 Prob(JB): 2.54e-05
Kurtosis: 5.575 Cond. No. 5.32e+05
"""
"""
We will not remove the Marketing spend though the p-value is more than 5% because the adjusted R^2 value decreases
instead of increasing which shows that the our model predicts less accurately than the model before this step.
Hence We will not include this step.
Formula for R^2 and adjusted R^2:
-> R^2 = 1 - (SSres / SStot)
-> Adjusted R^2 = 1 - (1 - R^2)*((n - 1)/(n - p - 1))
where n = sample size
p = no. of regresses
x_opt = x[:, [0,3]]
mregr = sm.OLS(endog = y, exog = x_opt).fit()
mregr.summary()
OLS Regression Results
Dep. Variable: y R-squared: 0.947
Model: OLS Adj. R-squared: 0.945
Method: Least Squares F-statistic: 849.8
Date: Sun, 06 Jan 2019 Prob (F-statistic): 3.50e-32
Time: 18:48:12 Log-Likelihood: -527.44
No. Observations: 50 AIC: 1059.
Df Residuals: 48 BIC: 1063.
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 4.903e+04 2537.897 19.320 0.000 4.39e+04 5.41e+04
x1 0.8543 0.029 29.151 0.000 0.795 0.913
Omnibus: 13.727 Durbin-Watson: 1.116
Prob(Omnibus): 0.001 Jarque-Bera (JB): 18.536
Skew: -0.911 Prob(JB): 9.44e-05
Kurtosis: 5.361 Cond. No. 1.65e+05
"""