Lennart Eriksson

PLS-REGRESSION: A BASIC TOOL OF CHEMOMETRICS

PLS-regression (PLSR) is the PLS approach in its simplest, and in chemistry and technology, most used form (two-block predictive PLS). PLSR is a method for relating two data matrices, X and Y, by a linear multivariate model, but goes beyond traditional regression in that it models also the structure of X and Y. PLSR derives its usefulness from its ability to analyze data with many, noisy, collinear, and even incomplete variables in both X and Y. PLSR has the desirable property that the precision of the model parameters improves with the increasing number of relevant variables and observations.This article reviews PLSR as it has developed to become a standard tool in chemometrics and used in chemistry and engineering. The underlying model and its assumptions are discussed, and commonly used diagnostics are reviewed together with the interpretation of resulting parameters.Two examples are used as illustrations: First, a Quantitative Structure-Activity Relationship (QSAR)/Quantitative Structure-Property Relationship (QSPR) data set of peptides is used to outline how to develop, interpret and refine a PLSR model. Second, a data set from the manufacturing of recycled paper is analyzed to illustrate time series modelling of process data by means of PLSR and time-lagged X-variables.

Multivariate analysis of aquatic toxicity data with PLS

A common task in data analysis is to model the relationships between two sets of variables, the descriptor matrixX and the response matrixY. A typical example in aquatic science concerns the relationships between the chemical composition of a number of samples (X) and their toxicity to a number of different aquatic species (Y). This modelling is done in order to understand the variation ofY in terms of the variation ofX, but also to lay the ground for predictingY of unknown observations based on their knownX-data. Correlations of this type are usually expressed as regression models, and are rather common in aquatic science. Often, however, the multivariateX andY matrices invalidate the use of multiple linear regression (MLR) and call for methods which are better suited for collinear data. In this context, multivariate projection methods represent a highly useful alternative, in particular, partial least squares projections to latent structures (PLS). This paper introduces PLS, highlights its strengths and presents applications of PLS to modelling aquatic toxicity data. A general discussion of regression, comparing MLR and PLS, is provided.

MULTIVARIATE DESIGN AND MODELING IN QSAR

Abstract Quantitative structure-activity relationship (QSAR) modeling provides a rational basis for understanding mechanisms of biological performance and how to alter chemical structures to achieve improved performance. However, in the quest for a valid QSAR model, several critical problems must be dealt with in an appropriate manner. Chemometric techniques are relevant in QSAR development and will help the inexperienced QSAR analyst in avoiding trivial mistakes. The chemometric QSAR strategy, applicable both in drug design and environmentally related sciences, highlights some crucial steps that otherwise often are neglected. These steps include how to select the proper data analytical method, how to design the training set, how to describe chemical and biological properties of compounds and analyze these data, and finally how to validate the relevance of an established QSAR model. A discussion of these steps is given, using four illustrative examples.

On the selection of the training set in environmental QSAR analysis when compounds are clustered

In QSAR analysis in environmental sciences, adverse effects of chemicals released to the environment are modelled and predicted as a function of the chemical properties of the pollutants. Usually the set of compounds under study contains several classes of substances, i.e. a more or less strongly clustered set. It is then needed to ensure that the selected training set comprises compounds representing all those chemical classes. Multivariate design in the principal properties of the compound classes is usually appropriate for selecting a meaningful training set. However, with clustered data, often seen in environmental chemistry and toxicology, a single multivariate design may be suboptimal because of the risk of ignoring small classes with few members and only selecting training set compounds from the largest classes. Recently a procedure for training set selection recognizing clustering was proposed by us. In this approach, when non‐selective biological or environmental responses are modelled, local multivariate designs are constructed within each cluster (class). The chosen compounds arising from the local designs are finally united in the overall training set, which thus will contain members from all clusters. The proposed strategy is here further tested and elaborated by applying it to a series of 351 chemical substances for which the soil sorption coefficient is available. These compounds are divided into 14 classes containing between 10 and 52 members. The training set selection is discussed, followed by multivariate QSAR modelling, model interpretation and predictions for the test set. Various types of statistical experimental designs are tested during the training set selection phase. Copyright

Orthogonal signal correction, wavelet analysis, and multivariate calibration of complicated process fluorescence data

In this paper, multivariate calibration of complicated process fluorescence data is presented. Two data sets related to the production of white sugar are investigated. The first data set comprises 106 observations and 571 spectral variables, and the second data set 268 observations and 3997 spectral variables. In both applications, a single response, ash content, is modelled and predicted as a function of the spectral variables. Both data sets contain certain features making multivariate calibration efforts non-trivial. The objective is to show how principal component analysis (PCA) and partial least squares (PLS) regression can be used to overview the data sets and to establish predictively sound regression models. It is shown how a recently developed technique for signal filtering, orthogonal signal correction (OSC), can be applied in multivariate calibration to enhance predictive power. In addition, signal compression is tested on the larger data set using wavelet analysis. It is demonstrated that a compression down to 4% of the original matrix size - in the variable direction - is possible without loss of predictive power. It is concluded that the combination of OSC for pre-processing and wavelet analysis for compression of spectral data is promising for future use.

Megavariate analysis of environmental QSAR data. Part I – A basic framework founded on principal component analysis (PCA), partial least squares (PLS), and statistical molecular design (SMD)

This paper introduces principal component analysis (PCA), partial least squares projections to latent structures (PLS), and statistical molecular design (SMD) as useful tools in deriving multi- and megavariate quantitative structure-activity relationship (QSAR) models. Two QSAR data sets from the fields of environmental toxicology and environmental chemistry are worked out in detail, showing the benefits of PCA, PLS and SMD. PCA is useful when overviewing a data set and exploring relationships among compounds and relationships among variables. PLS is the regression extension of PCA and is used for establishing QSARs. SMD is essential for selecting informative training and test sets of compounds for QSAR calibration and validation.

A strategy for ranking environmentally occurring chemicals

A systematic methodology for quantitative structure-activity relationship (QSAR) development in environmental toxicology is provided. The methodology is summarized in a strategy with six sequential ...

Variable influence on projection (VIP) for orthogonal projections to latent structures (OPLS)

A new approach for variable influence on projection (VIP) is described, which takes full advantage of the orthogonal projections to latent structures (OPLS) model formalism for enhanced model interpretability. This means that it will include not only the predictive components in OPLS but also the orthogonal components. Four variants of variable influence on projection (VIP) adapted to OPLS have been developed, tested and compared using three different data sets, one synthetic with known properties and two real‐world cases. Copyright

Multivariate process and quality monitoring applied to an electrolysis process. : Part II - Multivariate time-series analysis of lagged latent variables

Abstract Multivariate time series analysis is applied to understand and model the dynamics of an electrolytic process manufacturing copper. Here, eight metal impurities were measured, twice daily, over a period of one year, to characterize the quality of the copper. In the data analysis, these eight variables were summarized by means of principal component analysis (PCA). Two principal component (PC) scores were sufficient to well summarize the eight measured variables ( R 2 =0.67). Subsequently, the dynamics of these PC-scores (latent variables) were investigated using multivariate time series analysis, i.e., partial least squares (PLS) modelling of the lagged latent variables. Stochastic models of the auto-regressive moving average (ARMA) family were appropriate for both PC-scores. Hence, the dynamics of both scores make the exponentially weighted moving average (EWMA) control chart suitable for process monitoring.

A chemometrics toolbox based on projections and latent variables

A personal view is given about the gradual development of projection methods—also called bilinear, latent variable, and more—and their use in chemometrics. We start with the principal components analysis (PCA) being the basis for more elaborate methods for more complex problems such as soft independent modeling of class analogy, partial least squares (PLS), hierarchical PCA and PLS, PLS‐discriminant analysis, Orthogonal projection to latent structures (OPLS), OPLS‐discriminant analysis and more.