Jorge Cadima

Principal component analysis: a review and recent developments.

Large datasets are increasingly common and are often difficult to interpret. Principal component analysis (PCA) is a technique for reducing the dimensionality of such datasets, increasing interpretability but at the same time minimizing information loss. It does so by creating new uncorrelated variables that successively maximize variance. Finding such new variables, the principal components, reduces to solving an eigenvalue/eigenvector problem, and the new variables are defined by the dataset at hand, not a priori, hence making PCA an adaptive data analysis technique. It is adaptive in another sense too, since variants of the technique have been developed that are tailored to various different data types and structures. This article will begin by introducing the basic ideas of PCA, discussing what it can and cannot do. It will then describe some variants of PCA and their application.

Computational aspects of algorithms for variable selection in the context of principal components

Variable selection consists in identifying a k-subset of a set of original variables that is optimal for a given criterion of adequate approximation to the whole data set. Several algorithms for the optimization problems resulting from three different criteria in the context of principal components analysis are considered, and computational results are presented.

Variable selection and the interpretation of principal subspaces

Principal component analysis is widely used in the analysis of multivariate data in the agricultural, biological, and environmental sciences. The first few principal components (PCs) of a set of variables are derived variables with optimal properties in terms of approximating the original variables. This paper considers the problem of identifying subsets of variables that best approximate the full set of variables or their first few PCs, thus stressing dimensionality reduction in terms of the original variables rather than in terms of derived variables (PCs) whose definition requires all the original variables. Criteria for selecting variables are often ill defined and may produce inappropriate subsets. Indicators of the performance of different subsets of the variables are discussed and two criteria are defined. These criteria are used in stepwise selection-type algorithms to choose good subsets. Examples are given that show, among other things, that the selection of variable subsets should not be based only on the PC loadings of the variables.

Size- and Shape-Related Principal Component Analysis

The separation of morphometric variation into a component related to size and other components associated with shape is of considerable interest and has generated much discussion in the literature. One class of approaches to achieving this separation is based on principal component analysis. A new technique is proposed within this class, which overcomes some of the disadvantages of existing approaches.

The eigenstructure of block-structured correlation matrices and its implications for principal component analysis

Block-structured correlation matrices are correlation matrices in which the p variables are subdivided into homogeneous groups, with equal correlations for variables within each group, and equal correlations between any given pair of variables from different groups. Block-structured correlation matrices arise as approximations for certain data sets’ true correlation matrices. A block structure in a correlation matrix entails a certain number of properties regarding its eigendecomposition and, therefore, a principal component analysis of the underlying data. This paper explores these properties, both from an algebraic and a geometric perspective, and discusses their robustness. Suggestions are also made regarding the choice of variables to be subjected to a principal component analysis, when in the presence of (approximately) block-structured variables.

Predicting degradability parameters of diets for ruminants using regressions on chemical components

BACKGROUND Dry matter degradability (DMD) parameters (a, b and c in the Ørskov and McDonald model) are usually determined by the nylon bag technique. The aim of this study was to estimate DMD parameters of ruminant mixed diets, which are in general unavailable, through multiple linear regressions on their chemical composition (ash, crude protein, neutral detergent fibre (NDF), acid detergent fibre (ADF) and acid detergent lignin (ADL)). The regressions were based on data from 77 feeds. RESULTS The prediction model for a was reduced to a simple linear regression on NDF (adjusted R(2) = 0.727, F test P < 0.001). A regression model for b was obtained with ADL as the sole predictor (adjusted R(2) = 0.691, P < 0.001). The models upper asymptote (a + b) was predicted from ADL, NDF and ash (adjusted R(2) = 0.908, P < 0.001). Modelling c proved more difficult (adjusted R(2) with all five predictors = 0.481, P < 0.001). CONCLUSION Regressing model parameters on feed chemical composition is a promising method for estimating the degradability of mixed diets, providing an alternative to invasive and expensive laboratory techniques.