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1.8. Cross decomposition

The cross decomposition module contains supervised estimators for dimensionality reduction and regression, belonging to the “Partial Least Squares” family.

Cross decomposition algorithms find the fundamental relations between two matrices (X and Y). They are latent variable approaches to modeling the covariance structures in these two spaces. They will try to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the Y space. In other words, PLS projects both X and Y into a lower-dimensional subspace such that the covariance between transformed(X) and transformed(Y) is maximal.

PLS draws similarities with Principal Component Regression (PCR), where the samples are first projected into a lower-dimensional subspace, and the targets y are predicted using transformed(X). One issue with PCR is that the dimensionality reduction is unsupervised, and may lose some important variables: PCR would keep the features with the most variance, but it’s possible that features with a small variances are relevant from predicting the target. In a way, PLS allows for the same kind of dimensionality reduction, but by taking into account the targets y. An illustration of this fact is given in the following example: * Principal Component Regression vs Partial Least Squares Regression.

Apart from CCA, the PLS estimators are particularly suited when the matrix of predictors has more variables than observations, and when there is multicollinearity among the features. By contrast, standard linear regression would fail in these cases unless it is regularized.

Classes included in this module are PLSRegression, PLSCanonical, CCA and PLSSVD

1.8.1. PLSCanonical

We here describe the algorithm used in PLSCanonical. The other estimators use variants of this algorithm, and are detailed below. We recommend section [1] for more details and comparisons between these algorithms. In [1], PLSCanonical corresponds to “PLSW2A”.

Given two centered matrices $X\in {\mathbb{R}}^{n\times d}$ and $Y\in {\mathbb{R}}^{n\times t}$, and a number of components $K$, PLSCanonical proceeds as follows:

Set $X_1$ to $X$ and $Y_1$ to $Y$. Then, for each $k\in [1,K]$:

• a) compute $u_k\in {\mathbb{R}}^d$ and $v_k\in {\mathbb{R}}^t$, the first left and right singular vectors of the cross-covariance matrix $C=X_k^T{Y}_k$. $u_k$ and $v_k$ are called the weights. By definition, $u_k$ and $v_k$ are chosen so that they maximize the covariance between the projected $X_k$ and the projected target, that is $\text{Cov}(X_k{u}_k,Y_k{v}_k)$.

• b) Project $X_k$ and $Y_k$ on the singular vectors to obtain scores: ${\xi }_k=X_k{u}_k$ and ${\omega }_k=Y_k{v}_k$

• c) Regress $X_k$ on ${\xi }_k$, i.e. find a vector ${\gamma }_k\in {\mathbb{R}}^d$ such that the rank-1 matrix ${\xi }_k{\gamma }_k^T$ is as close as possible to $X_k$. Do the same on $Y_k$ with ${\omega }_k$ to obtain ${\delta }_k$. The vectors ${\gamma }_k$ and ${\delta }_k$ are called the loadings.

• d) deflate $X_k$ and $Y_k$, i.e. subtract the rank-1 approximations: $X_{k+1}=X_k-{\xi }_k{\gamma }_k^T$, and $Y_{k+1}=Y_k-{\omega }_k{\delta }_k^T$.

At the end, we have approximated $X$ as a sum of rank-1 matrices: $X=\mathrm{\Xi }{\mathrm{\Gamma }}^T$ where $\mathrm{\Xi }\in {\mathbb{R}}^{n\times K}$ contains the scores in its columns, and ${\mathrm{\Gamma }}^T\in {\mathbb{R}}^{K\times d}$ contains the loadings in its rows. Similarly for $Y$, we have $Y=\mathrm{\Omega }{\mathrm{\Delta }}^T$.

Note that the scores matrices $\mathrm{\Xi }$ and $\mathrm{\Omega }$ correspond to the projections of the training data $X$ and $Y$, respectively.

Step a) may be performed in two ways: either by computing the whole SVD of $C$ and only retain the singular vectors with the biggest singular values, or by directly computing the singular vectors using the power method (cf section 11.3 in [1]), which corresponds to the 'nipals' option of the algorithm parameter.

Transforming data Click for more details

To transform $X$ into $\overline{X}$, we need to find a projection matrix $P$ such that $\overline{X}=XP$. We know that for the training data, $\mathrm{\Xi }=XP$, and $X=\mathrm{\Xi }{\mathrm{\Gamma }}^T$. Setting $P=U({\mathrm{\Gamma }}^{T}U{)}^{-1}$ where $U$ is the matrix with the $u_k$ in the columns, we have $XP=XU({\mathrm{\Gamma }}^{T}U{)}^{-1}=\mathrm{\Xi }({\mathrm{\Gamma }}^{T}U)({\mathrm{\Gamma }}^{T}U{)}^{-1}=\mathrm{\Xi }$ as desired. The rotation matrix $P$ can be accessed from the x_rotations_ attribute.

Similarly, $Y$ can be transformed using the rotation matrix $V({\mathrm{\Delta }}^{T}V{)}^{-1}$, accessed via the y_rotations_ attribute.

Predicting the targets Y Click for more details

To predict the targets of some data $X$, we are looking for a coefficient matrix $\beta \in R^{d\times t}$ such that $Y=X\beta$.

The idea is to try to predict the transformed targets $\mathrm{\Omega }$ as a function of the transformed samples $\mathrm{\Xi }$, by computing $\alpha \in \mathbb{R}$ such that $\mathrm{\Omega }=\alpha \mathrm{\Xi }$.

Then, we have $Y=\mathrm{\Omega }{\mathrm{\Delta }}^T=\alpha \mathrm{\Xi }{\mathrm{\Delta }}^T$, and since $\mathrm{\Xi }$ is the transformed training data we have that $Y=X\alpha P{\mathrm{\Delta }}^T$, and as a result the coefficient matrix $\beta =\alpha P{\mathrm{\Delta }}^T$.

$\beta$ can be accessed through the coef_ attribute.

1.8.2. PLSSVD

PLSSVD is a simplified version of PLSCanonical described earlier: instead of iteratively deflating the matrices $X_k$ and $Y_k$, PLSSVD computes the SVD of $C=X^{T}Y$ only once, and stores the n_components singular vectors corresponding to the biggest singular values in the matrices U and V, corresponding to the x_weights_ and y_weights_ attributes. Here, the transformed data is simply transformed(X) = XU and transformed(Y) = YV.

If n_components == 1, PLSSVD and PLSCanonical are strictly equivalent.

1.8.3. PLSRegression

The PLSRegression estimator is similar to PLSCanonical with algorithm='nipals', with 2 significant differences:

• at step a) in the power method to compute $u_k$ and $v_k$, $v_k$ is never normalized.

• at step c), the targets $Y_k$ are approximated using the projection of $X_k$ (i.e. ${\xi }_k$) instead of the projection of $Y_k$ (i.e. ${\omega }_k$). In other words, the loadings computation is different. As a result, the deflation in step d) will also be affected.

These two modifications affect the output of predict and transform, which are not the same as for PLSCanonical. Also, while the number of components is limited by min(n_samples, n_features, n_targets) in PLSCanonical, here the limit is the rank of $X^{T}X$, i.e. min(n_samples, n_features).

PLSRegression is also known as PLS1 (single targets) and PLS2 (multiple targets). Much like Lasso, PLSRegression is a form of regularized linear regression where the number of components controls the strength of the regularization.

1.8.4. Canonical Correlation Analysis

Canonical Correlation Analysis was developed prior and independently to PLS. But it turns out that CCA is a special case of PLS, and corresponds to PLS in “Mode B” in the literature.

CCA differs from PLSCanonical in the way the weights $u_k$ and $v_k$ are computed in the power method of step a). Details can be found in section 10 of [1].

Since CCA involves the inversion of $X_k^T{X}_k$ and $Y_k^T{Y}_k$, this estimator can be unstable if the number of features or targets is greater than the number of samples.

Reference Click for more details

[1] (1,2,3,4)

A survey of Partial Least Squares (PLS) methods, with emphasis on the two-block case JA Wegelin

Examples:

• Compare cross decomposition methods

• Principal Component Regression vs Partial Least Squares Regression