Why does X @ coef_ + intercept_ does not equal Y_pred for sklearn PLSRegression?

Question

I performed partial least squares regression using Python's sklearn.cross_decomposition.PLSRegression using the example data in the sklearn docs. I am surprised that X @ coef_ + intercept_ does not equal Y_pred. Can someone please explain?

from sklearn.cross_decomposition import PLSRegression
X = [[0., 0., 1.], [1.,0.,0.], [2.,2.,2.], [2.,5.,4.]]
Y = [[0.1, -0.2], [0.9, 1.1], [6.2, 5.9], [11.9, 12.3]]
pls2 = PLSRegression(n_components=2)
pls2.fit(X, Y)
PLSRegression()
Y_pred = pls2.predict(X)

[email protected]_ + pls2.intercept_

returns

array([[ 6.80991986,  6.88073249],
       [ 6.24687317,  6.24590503],
       [16.37620337, 16.72659034],
       [27.32746904, 28.13552828]])

but Y_pred is

array([[ 0.26087869,  0.15302213],
       [ 0.60667302,  0.45634164],
       [ 6.46856199,  6.48931562],
       [11.7638863 , 12.00132061]])

Answer 1

I think you need to center and scale your X matrix before multiplying by the coefficients.

Answer 2

Scikit-learn is open-source, so you always can check the source code

        # Normalize
        X -= self._x_mean
        X /= self._x_std
        # TODO(1.3): change `self._coef_` to `self.coef_`
        Ypred = X @ self._coef_.T
        return Ypred + self.intercept_

where

    @property
    def coef_(self):
        [...]
        return self._coef_.T