Partial Least Squares Weights and Four Types of Loadings

The concept of loadings when using PLS is considerably more complex than for PCA. Unfortunately, different concepts have been called by the same names, causing confusion in the literature. The concept of loadings for PLS is considerably more complex than for PCA. Unfortunately, different concepts have been called by the same names, causing confusion in the literature. All common PLS algorithms relate to one of these two algorithms (one with orthogonal scores and the other with nonorthogonal scores), but the loadings and scores differ in scale between methods. Due to space constraints, we will restrict our description to just these two algorithms, but all common PLS algorithms belong to one of these two classes. We will now illustrate this with the data of Table 1 of the previous article 2. For brevity, we will only model variable c1—the first column of the c block. Major aims of the last article were to show that there are different PLS models according to c values, and that for the Martens algorithm, one can rotate the scores through different angles according to whether we use PCA or PLS, and that the rotation angle differs for each c value, when there is more than one c (concentration or property) value characterising a dataset. This is best illustrated by a multivariate example as in Table 2, where X is a 15 × 5 matrix, with a maximum of 5 PLS components. The features discussed above are valid no matter how many components in the model, and can be checked by readers via the numerical data of Table 3. Data are column centred before performing PLS. It can be seen that, although there are differences between x loadings, using both algorithms and the weights, and for the Martens algorithm, they also differ according to the number of components in the model, they are not enormous. Weights or loadings are commonly used in chemometrics to determine the significance of variables in a model. For example, which variables are more important markers for a metabolic process? Usually, the ones with the highest magnitude are considered most important, so they can give a clue as to which metabolites or spectral peaks are indicative of the process of interest. We will discuss the interpretation of multivariate PLS models in later articles. The Wold algorithm is also commonly called the NIPALS algorithm and is very widespread in chemometrics and usually the default. However, it is always worth checking any software to ensure this is the method employed. The advantage of the Martens algorithm is that the data in scores space is a rotation of the original data, unlike the Wold algorithm. As we will see in the next article, if used for estimation, both algorithms provide identical answers, so whether this difference is important depends in part on the purpose of PLS. If it is for estimation, it makes no difference, which is used, but if it is to determine the significance of variables, the different algorithms may come to different conclusions, although they will not normally be large if using the x loadings. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.