Partial least squares regression

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In statistics, the method of partial least squares regression (PLS-regression) bears some relation to principal component analysis; instead of finding the hyperplanes of maximum variance, it finds a linear model describing some predicted variables in terms of other observable variables.

It is used to find the fundamental relations between two matrices (X and Y), i.e. a latent variable approach to modeling the covariance structures in these two spaces. A PLS model will try to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the Y space. Partial least squares is particularly suited when the matrix of predictors has more variables that observations (see mulricolinearity). By contrast, standard regression will fail in these cases.

It was first introduced by the Swedish statistician Herman Wold. An alternative (and arguably, more correct, according to Wold) long form for PLS is projection to latent structures but the term partial least squares is still dominant in some areas. It is widely applied in the field of chemometrics, in sensory evaluation, and more recently, in chemical engineering process data (see John F. MacGregor) and the analysis of functional brain imaging data(see [Randy McIntosh]).

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[edit] References

Frank, Ildiko and Jerome Friedman (1993). "A Statistical View of Some Chemometrics Regression Tools, Technometrics, 35(2), pp 109-148".
Haenlein, Michael and Andreas M. Kaplan (2004). "A Beginner's Guide to Partial Least Squares Analysis, Understanding Statistics, 3(4), 283-297".
Henseler, Joerg and Georg Fassott (2005). "Testing Moderating Effects in PLS Path Models. An Illustration of Available Procedures".
Tenenhaus Michel (1998). La Regression PLS: Theorie et Pratique. Paris: Technip..