Stats with Python: Multiple Linear Regression
March 31, 2021 | 4 min read | 28 views
The last post discussed simple linear regression, which explains the objective variable by a single explanatory variable . This post introduces regression with multiple explanatory variables : multiple linear regression.
Just like simple linear regression, multiple linear regression also finds their parameters by the method of least squares. Given the paired data , the sum of the squared error between the predicted and the actual is written in a matrix form:
Therefore, the least squares estimates are:
Coefficient of determination is defined in exactly the same way as for simple linear regression.
In multiple linear regression, the correlation coefficient is defined as the square root of the coefficient of determination, that is, .
As extra explanatory variables are added, spuriously increases. For feature selection, you might want to use adjusted coefficient of determination to complement this effect.
Here, is the degree of freedom of the totals and is the degree of freedom of the residuals .
It’s easy to use multple linear regression because scikit-learn’s
LinearRegression supports it by default. You can use it exactly the same way as the last post.
First, let’s prepare the dataset.
import numpy as np import matplotlib.pyplot as plt import seaborn as sns sns.set_style("darkgrid") n = 100 m = 2 X = np.random.rand(n,2) y = X[:,0] - 2*X[:,1] + 0.5*np.random.rand(n) fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.scatter(X[:,0], X[:,1], y, color='k') plt.show()
LinearRegression, fit the model, and plot the regression plane in 3D space.
from sklearn.linear_model import LinearRegression from sklearn.metrics import r2_score model = LinearRegression() model.fit(X, y.reshape(-1, 1)) xx, yy = np.meshgrid(np.linspace(0, 1, 20), np.linspace(0, 1, 20)) model_viz = np.array([xx.flatten(), yy.flatten()]).T predicted = model.predict(model_viz) fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.scatter(xx.flatten(), yy.flatten(), predicted, facecolor=(0,0,0,0), s=20, edgecolor='#70b3f0') ax.scatter(X[:,0], X[:,1], y, color='k') plt.show()
is also calculated in the same way.
r2_score(y, model.predict(X)) # >> 0.9640837457111405
 倉田 博史, 星野 崇宏. ”入門統計解析“（第9章）. 新世社. 2009.
 Multiple Linear Regression and Visualization in Python | Pythonic Excursions
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