The Degrees of Freedom of Partial Least Squares Regression

TL;DR

This paper derives an unbiased estimate of the Degrees of Freedom for Partial Least Squares regression, revealing its dependence on predictor collinearity and aiding model selection.

Contribution

It introduces a novel unbiased Degrees of Freedom estimate for PLS regression, linking it to matrix decompositions and Krylov subspaces, and demonstrates its practical utility.

Findings

Degrees of Freedom increases with lower predictor collinearity.

The estimate is higher than the naive component count approach.

Using the estimate with information criteria improves model selection.

Abstract

The derivation of statistical properties for Partial Least Squares regression can be a challenging task. The reason is that the construction of latent components from the predictor variables also depends on the response variable. While this typically leads to good performance and interpretable models in practice, it makes the statistical analysis more involved. In this work, we study the intrinsic complexity of Partial Least Squares Regression. Our contribution is an unbiased estimate of its Degrees of Freedom. It is defined as the trace of the first derivative of the fitted values, seen as a function of the response. We establish two equivalent representations that rely on the close connection of Partial Least Squares to matrix decompositions and Krylov subspace techniques. We show that the Degrees of Freedom depend on the collinearity of the predictor variables: The lower the collinearity is, the higher the Degrees of Freedom are. In particular, they are typically higher than the naive approach that defines the Degrees of Freedom as the number of components. Further, we illustrate how the Degrees of Freedom approach can be used for the comparison of different regression methods. In the experimental section, we show that our Degrees of Freedom estimate in combination with information criteria is useful for model selection.