Basic Principles of PLS

PLS is a method which is based on projections, similar to PCA. The underlying principles are easy to understand from a geometrical description of the method. 

The blocks of variables are given by the matrices X and Y. The following notations will be used The number of objects (e.g. experiments) in both blocks is given by n the number of variables in the X blocks is given by k, xx. .. xk the number of variables in the Y block is given by m, j. .. yra. The X block is a (n x k) matrix, the Y block is a (n x m) matrix. 

It is a common misunderstanding that a statistical analysis is only possible when the number of experiments exceeds the number of variables. This is true of multiple regression, but it is note true of PLS. As PLS is based on projections, it can handle any number of variables provided that the number of underlying latent variables (cf. principal components) is less than the number of objects. 

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In an earlier review of the applications of photoluminescence (PL) techniques to the characterization of adsorption, catalysis, and photocatalysis (Anpo and Che, 1999), we addressed the basic principles of PL and the importance of PL measurements for understanding of (photo)catalytic processes. This chapter describes more recent developments and focuses on investigations of catalysts in the working state, with an emphasis on the role of local structure on photocatalytic reactions determined with PL and related techniques. 

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