Partial Least Squares Scores

There are a very large number of ways of introducing PLS. In fact, over several decades, people constantly develop papers and new approaches to the method, and it is envisaged there will be new insights and ways of introducing it for many decades into the future. This article is one of many thousands but fits into the theme of this series, using simple numerical examples to illustrate common approaches. Over the past few articles, we have discussed PCA 1 introducing the concepts of scores and loadings. The scores are fundamental to interpreting the principal components. A similar concept, confusingly also commonly called scores and often denoted by the same terminology, can be developed when performing PLS. This article will be focused on PLS scores; later articles will discuss estimation, regression coefficients, weights and loadings. There are two fundamental approaches to determining PLS components: the Wold algorithm with orthogonal scores but non-orthogonal x loadings and the Martens algorithm with non-orthogonal scores but orthonormal x loadings. In order to process this data further, we will centre both the x and c blocks 2. All calculations below will be on the centred data. PLS is used to improve this correlation. Using one component should improve estimates. As we will see in later articles, including a second component results in even better estimates. There are two common types of PLS algorithms. Most other described algorithms are related to one of these two and differ just by scaling or how they achieve the result. The scores and x loadings for both algorithms and both c variables (using PLS1) are presented in Tables 3 and 4, and we will refer to these below. In this article, we see that there are two fundamental approaches to determining PLS components: the Wold (NIPALS) algorithm with orthogonal scores but non-orthogonal x loadings and the Martens algorithm with non-orthogonal scores but orthonormal x loadings. In our example, we see that there are also separate PLS models according to which c variable we choose. In the case of the Martens algorithm, this corresponds to different rotations in variable space to better model the c value using the scores of the first PLS component. In the Wold algorithm, the first PLS component still is the best way to model the c variable, but the score space is distorted, so it cannot be represented by a rotation in variable space. Sometimes when there is more than one c block variable, people use the PLS2 algorithm, for which there is a single scores matrix for a model involving all (or in our case both) c block variables, but we restrict the current discussion to PLS1 and see that we can form two separate models for each of the two c block variables. The author received no specific funding for this work. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.