The Saga of PLS

by Gaston Sanchez

Conclusions and Final Remarks

What we know today as Partial Least Squares is the result of a long period of evolution, with a vast range of methods and techniques proposed since the late 1960s / early 1970s. They come from different disciplines and fields of application, motivated to solve a number of multivariate problems—involving relations among one or more blocks of variables. This means that PLS methods have grown organically over several decades, gradually mutating both in form and substance, and migrating from one field of study to another.

Even though PLS methods have been built on the NIPALS computational foundations of the mid-1960s, historically—and ideologically—we can distinguish two main branches of Partial Least Squares approaches: 1) the “Path Modeling” branch, and 2) the “Regression” branch—both based on works originally introduced by Herman Wold. The recognition of these two large categories has to do with the way in which they have subsequently unfolded. By taking different directions, the branches have produced two major movements that for the most part, have grown apart, remained disconnected, and even unaware of one another in some areas of application.

Overall, the two PLS branches, accompanied with their respective analytical movements, reflect the works emerged from Herman Wold (the father) and Svante Wold (the son). So far I’ve tried to offer an account that allows us to get a better understanding of the differences between both works. I firmly believe that additional comprehension can be gained if we compare them specifically from a generational perspective, jointly with the respective expertise of each author.

The way and style in which the two Wolds presented their works have left a profound footprint in their ulterior developments. At first glance, the Path Modeling movement, emerged from systems of equations with latent variables, seems to have insurmountable disparities with the Regression Models stemmed from chemometrics. Both branches, with its various subdivisions, show contrasting differences at various levels, notably at the area of application, but also at a philosophical, ideological, conceptual, language, technical, spread and diffusion levels. Albeit their common mathematical and operational elements, and even their shared genes, the contrasting physical appearance between path and regression models puts an illusory divide between them that can easily mislead all inexperienced users, and even some well versed PLS connoiseurs.

While it is true that such differences are not negligible, most of them are at the format level (i.e. the presentation). It is surprinsing how their outer layers pull them apart more than bringing them closer together. Despite all the differences, there are still strong ties and similarities. The most important common denominator is the mathematical and algorithmic principles. These common traits can be exploited to link them back together and see them both under the same glass. Fortunately, this separation has been reduced considerably in the last years, thanks to the organization of PLS symposiums, and the active work of researchers who have taken care of bridging the existing gaps.

Despite all their differences and constrasts, it seems a bit unfair to keep isolated PLS-PM from PLS-R. This should not be the case anymore, specially when it can be shown the connection between both approaches. Similarly, there is also a drawback with the way PLS Regression methods haven presented in the chemometrics-based literature. As much attractive as it can be their minimal style, it misses so much of the richness of PLS-PM.

A Third PLS Culture

Today we can say that a contemporary conciliatory culture has taken the best of both schools, keeping the path modeling roots of Herman Wold, the very pragmatic side of the chemometricians, and adding a refreshing view with a strong multivariate data analysis flavor.

Regardless of whether you are a newcomer or an experienced user of some PLS methods, I think it is important to talk about these issues. They help us be aware of the differences in the literature, be conscious about difficulties, but also be open to new opportunities and advantageous analytical possibilities.

Enough talking. It’s time to have some pudding!