Meaning/interpretation of intercept_ in partial least squares

Question

After using sklearn library for Partial Least Squares, I have doubts about the interpretation of the "intercept" of the model.

As you can see in the code that follows, and its corresponding ouput, the intercept_ and the prediction when all the predictors are zero differ.

From the documentation here https://scikit-learn.org/stable/modules/generated/sklearn.cross_decomposition.PLSRegression.html

intercept_ndarray of shape (n_targets,) The intercepts of the linear model such that Y is approximated as Y = X @ coef_.T + intercept_.

Both values should be the same, I'm right?

pd: I verified that I'm using scikit-learn version 1.5.0

Code

import pandas as pd
from sklearn.cross_decomposition import PLSRegression
import numpy as np

data = pd.DataFrame(data = [
[6.0,10.086797,3.535027,0.341043,8.3502],
[6.0,10.088014,3.535453,0.341084,5.8764],
[6.0,10.272183,2.704227,0.199019,3.7959],
[6.0,10.507442,3.169033,0.228000,4.6900],
[6.0,11.086873,3.693520,0.225735,1.0480],
[0.0,10.131321,3.526459,0.347951,2.0241],
[0.0,10.103844,3.516895,0.347008,2.8899],
[0.0,10.167639,3.539101,0.349199,1.4600],
[0.0,10.236161,3.562951,0.351552,1.4361],
[15.0,7.576252,3.859195,0.356888,8.6559],
[15.0,7.842521,3.851668,0.357062,7.8487]
], columns = ['q','x', 'y', 'z', 'u'])

plsModel = PLSRegression(n_components = 3, scale = False)
plsReg = plsModel.fit(data[['q','x', 'y', 'z']], data['u'])
 
print(f'Intercept value: {plsReg.intercept_[0]:.4f}')

p0 = pd.DataFrame(data = [[0,0,0,0]], columns = ['q','x', 'y', 'z'])
uPredicted = plsReg.predict(p0)

print(f'Predicted value: {uPredicted[0]:.4f}')

Output

Intercept value: 4.3705
Predicted value: 14.7594

Answer 1

Both values should be the same, I'm right?

Yeah I'd think so too, but apparently this method always mean centers even with scale = False. This kind of makes sense because the mean doesn't have a natural place in the transformations that happen here.

Anyway 4.3705 is just the mean of data['u'] and the prediction values agree if we mean center the fit.

#how to mean center:  https://stackoverflow.com/a/34954441/7840119
plsReg = plsModel.fit(data[['q','x', 'y', 'z']].apply(lambda x: x-x.mean()), data['u'])
 #I had problems with your print-statements. I don't know why
print('Intercept value:' + str(plsReg.intercept_))
print('coefs value:' + str(plsReg.coef_))
print('mean value:' + str(np.mean(data['u'])))
p0 = pd.DataFrame(data = [[0,0,0,0]], columns = ['q','x', 'y', 'z'])
print('transform value:' + str(plsReg.transform(p0)))
uPredicted = plsReg.predict(p0)
print('Predicted value:' + str(uPredicted))

result

Intercept value:[4.37047273]
coefs value:[[ 0.31427577 -0.84705601 -1.13213912  0.5839974 ]]
mean value:4.370472727272728
transform value:[[-7.01497764e-16  1.92786050e-16 -4.46294642e-16]]
Predicted value:[[4.37047273]]