Geometrical description of the PLS method

The principles behind PLS are simple and easily understood from a geometrical illustration. The method is based upon projections, similar to principal components analysis. The two blocks of variables are given by the matrices X and Y. The following notations will be used  

The number of independent variables (e.g. descriptors) in the X block is given by K, and the number of dependent variables in the Y block is given by M. The number of objects (e.g. experiments) both in the X block and the Y block is N. The X block is a (N x K) matrix and the Y block is a N x M) matrix. 

It is a common misunderstanding that in a statistical analysis the number of objects must exceed the number of variables. This is true of regression analysis but it is not true of PLS. Since PLS is based upon projections, we can easily have more variables than objects. Any object included in the analysis can be characterized by a large number of descriptors, and the outcome of the experiment can be 

As was discussed in Chapter 15, any matrix can be represented geometrically by a swarm of points in a multidimensional space. The K variables in the X space will thus define a X-dimensional X space, and the M response variables an Af-dimensional Y space. The N objects define a swarm of N points in each space. 

In PLS modelling as much as possible of the variation in the Y space should be modelled with a simultaneous modelling of the variations in the X space in such a way that these variations can be related to each other. 

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