In the LinearRegression method in sklearn, what exactly is the fit_intercept parameter doing?

Question

In the sklearn.linear_model.LinearRegression method, there is a parameter that is fit_intercept = TRUE or fit_intercept = FALSE. I am wondering if we set it to TRUE, does it add an additional intercept column of all 1's to your dataset? If I already have a dataset with a column of 1's, does fit_intercept = FALSE account for that or does it force it to fit a zero intercept model?

Update: It seems people do not get my question. The question is basically, what IF I had already a column of 1's in my dataset of predictors (the 1's are for the intercept). THEN,

1) if I use fit_intercept = FALSE, will it remove the column of 1's?

2) if I use fit_intercept = TRUE, will it add an EXTRA column of 1's?

Answer 1

fit_intercept=False sets the y-intercept to 0. If fit_intercept=True, the y-intercept will be determined by the line of best fit.

from sklearn.linear_model import LinearRegression
from sklearn.datasets import make_regression
import numpy as np
import matplotlib.pyplot as plt

bias = 100

X = np.arange(1000).reshape(-1,1)
y_true = np.ravel(X.dot(0.3) + bias)
noise = np.random.normal(0, 60, 1000)
y = y_true + noise

lr_fi_true = LinearRegression(fit_intercept=True)
lr_fi_false = LinearRegression(fit_intercept=False)

lr_fi_true.fit(X, y)
lr_fi_false.fit(X, y)

print('Intercept when fit_intercept=True : {:.5f}'.format(lr_fi_true.intercept_))
print('Intercept when fit_intercept=False : {:.5f}'.format(lr_fi_false.intercept_))

lr_fi_true_yhat = np.dot(X, lr_fi_true.coef_) + lr_fi_true.intercept_
lr_fi_false_yhat = np.dot(X, lr_fi_false.coef_) + lr_fi_false.intercept_

plt.scatter(X, y, label='Actual points')
plt.plot(X, lr_fi_true_yhat, 'r--', label='fit_intercept=True')
plt.plot(X, lr_fi_false_yhat, 'r-', label='fit_intercept=False')
plt.legend()

plt.vlines(0, 0, y.max())
plt.hlines(bias, X.min(), X.max())
plt.hlines(0, X.min(), X.max())

plt.show()

This example prints:

Intercept when fit_intercept=True : 100.32210
Intercept when fit_intercept=False : 0.00000

Visually it becomes clear what fit_intercept does. When fit_intercept=True, the line of best fit is allowed to "fit" the y-axis (close to 100 in this example). When fit_intercept=False, the intercept is forced to the origin (0, 0).

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What happens if I include a column of ones or zeros and set fit_intercept to True or False?

Below shows an example of how to inspect this.

from sklearn.linear_model import LinearRegression
from sklearn.datasets import make_regression
import numpy as np
import matplotlib.pyplot as plt

np.random.seed(1)
bias = 100

X = np.arange(1000).reshape(-1,1)
y_true = np.ravel(X.dot(0.3) + bias)
noise = np.random.normal(0, 60, 1000)
y = y_true + noise

# with column of ones
X_with_ones = np.hstack((np.ones((X.shape[0], 1)), X))

for b,data in ((True, X), (False, X), (True, X_with_ones), (False, X_with_ones)):
  lr = LinearRegression(fit_intercept=b)
  lr.fit(data, y)

  print(lr.intercept_, lr.coef_)

Take-away:

# fit_intercept=True, no column of zeros or ones
104.156765787 [ 0.29634031]
# fit_intercept=False, no column of zeros or ones
0.0 [ 0.45265361]
# fit_intercept=True, column of zeros or ones
104.156765787 [ 0.          0.29634031]
# fit_intercept=False, column of zeros or ones
0.0 [ 104.15676579    0.29634031]