Principal component analysis: a review and recent developments

Abstract

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.

Keywords: dimension reduction; eigenvectors; multivariate analysis; palaeontology.

Figure 1.

The two-dimensional principal subspace for the fossil teeth data. The coordinates in either or both PCs may switch signs when different software is used.

Figure 2.

Biplot for the fossil teeth data (correlation matrix PCA), obtained using R’s biplot command. (Online version in colour.)

Figure 3.

(a,b) The first two correlation-based EOFs for the SLP data account for 21% and 13% of total variation. (Adapted from [36].)

Figure 4.

(a,b) LASSO-based simplified EOFs for the SLP data. Grey areas are grid-points with exactly zero loadings. (Adapted from [36].)