N-way principal component analysis theory, algorithms and applications

Abstract

Due to sophisticated experimental designs and to modern instrumental constellations the investigation of N-dimensional (or N-way or N-mode) data arrays is attracting more and more attention. Three-dimensional arrays may be generated by collecting data tables with a fixed set of objects and variables under different experimental conditions, at different sampling times, etc. Stacking all the tables along varying conditions provides a cubic arrangement of data. Accordingly the three index sets or modes spanning a three-way array are called objects, variables and conditions. In many situations of practical relevance even higher-dimensional arrays have to be considered. Among numerous extensions of multivariate methods to the three-way case the generalization of principal component analysis (PCA) has central importance. There are several simplified approaches of three-way PCA by reduction to conventional PCA. One of them is unfolding of the data array by combining two modes to a single one. Such a procedure seems reasonable in some specific situations like multivariate image analysis, but in general combined modes do not meet the aim of data reduction. A more advanced way of unfolding which yields separate component matrices for each mode is the Tucker 1 method. Some theoretically based models of reduction to two-way PCA impose some specific structure on the array. A proper model of three-way PCA was first formulated by Tucker (so-called Tucker 3 model among other proposals). Unfortunately the Tucker 1 method is not optimal in the least squares sense of this model. Kroonenberg and De Leeuw demonstrated that the optimal solution of Tucker's model obeys an interdependent system of eigenvector problems and they proposed an iterative scheme (alternating least squares algorithm) for solving it. With appropriate notation Tucker's model as well as the solution algorithm are easily generalized to the N-way case (N > 3). There are some specific aspects of three-way PCA, such as complicated ways of data scaling or interpretation and simple-structure-transformation of a so-called core matrix, which make it more difficult to understand than classical PCA. An example from water chemistry serves as an illustration. Additionally, there is an application section demonstrating several rules of interpretation of loading plots with examples taken from environmental chemistry, analysis of complex round robin tests and contamination analysis in tungsten wire production.