THE PLS MODEL AND ITS FITTING

If Y and X denote matrices of dependent and independent variablesrespectively, then the aim of PLS is to fit a bilinear model having the form T=XW, X=TP'+E and Y=TQ'+F, where W is a matrix of coefficients whose columns define the PLS factors as linear combinations of the independent variables. Successive PLS factors contained in the columns of T are selected both to minimise the residuals in E and simultaneously to have high squared covariance with a single Y variate (PLS1) or a linear combination of multiple Y variates (PLS2). The columns T are constrained to be mutually orthogonal. See Helland (1988) or Hoskuldsson (1988) for a more comprehensive description of the PLS method.

The procedure allows the calculation of PLS1 and PLS2 models with cross- validation to assist in the determination of the correct number of dimensions to include in the model. By setting the NGROUPS option the data are randomly divided into a number of groups; samples in each group are then modelled from the remaining samples only. The sum of squares of differences between these "leave out predictions" and the observed values of Y are called PRESS. Many tests of significance for determining the correct number of dimensions are based on comparing values of PRESS for PLS models of varying rank. Values of PRESS are used in the procedure to perform Osten's (1988) test of significance and may also be plotted out in a scree diagram. In addition to the factor scores, factor loadings and residuals, the procedure also calculates a leverage measure (Martens & Naes 1989 page 276) and a single linear combination of the X variables (ESTIMATES) which summarises the entire PLS model.

To use a PLS model to make predictions from new observations on the X variables, two methods are available. Either the user may do this manually by using the model as specified in the estimates matrix, or the new X data may be specified beforehand as the pointer to variates XPREDICT and the corresponding predicions obtained as YPREDICT.

Although the PLS method is often presented in terms of an iterative algorithm (Manne 1987), the X block loadings vector for the first PLS dimension (w1) is simply the eigenvector of X'YY'X corresponding to its largest eigenvalue. To find the second and subsequent dimensions, X and Y are deflated by orthogonalising with respect to the current PLS factor (t=Xw) and the eigenanalysis repeated. The above approach was adopted by Rogers (1987) in an implementation of a Genstat 4 macro. Here we adopt a very similar approach by performing a singular value decomposition on the matrix X'Y which simultaneously obtains loading vectors for both data blocks (Hoskuldsson 1988).

Partial Least Squares Regression 27.2.96 Page : 15b of 16