Bridge–PLS regression: two‐block bilinear regression without deflation

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Abstract
Functional MRI (fMRI) represents experiments with experimental design in the time domain, and yields a very high number of response variables. In this paper an fMRI data set is analyzed for temporal response delays relative to the design, and for spatial response patterns. Two families of two‐block PLS are compared, namely PLS Regression (PLSR) and a method developed by F.L. Bookstein, here called Bookstein PLS (BPLS). In BPLS, all components are found simultaneously, making the deflation of X and/or y superfluous. When reformulated as a regression method y = f(X), the BPLS performs badly compared with PLSR in the case of only one dependent variable, y, because only one component can be calculated. Contrary to the asymmetrical PLSR, the symmetrical BPLS lacks a predictive direction from X to y, and is therefore difficult to assess, e.g. by cross‐validation. A new PLS method, here called bridge–PLS regression, is presented. It combines elements from both PLSR (predictive asymmetry) and BPLS (all components determined by one single eigen‐decomposition). Copyright © 2004 John Wiley & Sons, Ltd.

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