Geometrical Description of PLS

As was shown in Sect. 4.3.1, any matrix can be represented by a swarm of points in a multidimensional space. Thus, the X matrix will define a /c-dimensionaFspace, and the Y matrix a m-dimensional space. The n objects will define swarms of n points in each space. 

In PLS modelling, as much as possible of the variation in the Y space is modelled through projections to PLS(Y) components, with a simultaneous modelling of the variations in the X space through projections to PLS(X) components, with the constraint that these variations should be related to each other by a maximum correlation between the scores of the Y block (denoted ,) and the scores from the X block (deontes fi) for each PLS dimension i. Fig. 11 illustrates the principles for one PLS dimension (a one-component PLS model). 

Predictions for a new object by the PLS model can be understood from Fig. 11 as follows A new object defines a point, i in the X space and its projection on the PLS(X) component gives the score tu. The corresponding score, uu, along the PLS(Y) component is obtained from the correlation (called the inner relation) between and uv The calculated uu score corresponds to a point along the PLS(Y) component, and the coordinates of this point yield the predicted values j j of each response. 

more than one PLS dimension is necessary to account for the systematic variation in the Y space. As for PCA, the PLS dimensions can be peeled off, one dimension at a time, until the systematic variation has been described. The models can be established by cross-validation to ensure valid predictions. 

The residual variance after fitting the model can be used to compute tolerance limits around the model. This is illustrated by cylinders in Fig. 11. For obtaining reliable predictions with new objects, these should be within the the tolerance limits in the X space. 

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