Extension of conventional eigenvector analysis to three-way sets of data is possible through three-mode principal component analysis. First introduced in the 1960's, this method gives three sets of loadings corresponding to the three ways in the data, e.g., variable, location, and time. A core matrix relates loadings across modes. Data must be centered and scaled before analysis, and as in conventional two-way analysis, preprocessing options affect the reduction in dimensionality and the appearance of the results. An example using water quality data illustrates the method and preprocessing effects. Although three-way tables can be studied through conventional analysis of a two-way table created by combining two modes of the data, three-mode analysis treats each mode separately and with the same weight. In addition, a restricted three-mode principal component model avoids problems in rotational indeterminacy, and results in a particularly simple model. Factor analysis or principal component analysis begins with a two-way table with samples along one margin and variables along the other. For instance, samples may be arranged as rows and variables as columns. R-mode analysis of the columns displays interdependencies among variables and Q-mode analysis displays similarities among samples. The analysis increases in complexity if a set of variables is repeatedly observed on the same samples; each set of measurements might represent a different experimental condition, chemical or sedimentological fraction, or simply geologic time. The resulting table of data can be visualized as a three-dimensional block: horizontal slices represent samples, vertical slices parallel with the front represent variables, and vertical slices parallel with the ends represent different conditions, times, or fractions. For example, Oudin (1970) performed elemental analyses of organic extracts from Jurassic shales in the Paris basin. Samples represented very different depths of maximum burial. Each was fractionated into several extracts according to solubility in organic solvents. Data published by Oudin (1970) have the three ways: locality, fraction, and element. Multivariate analysis of these data was presented by Hohn (1979). Hohn and Friberg (1979) applied principal component analysis to petrographic data in which the three modes were sample, mineral phase, and chemical component.
Higher-order principal component analysis for the approximation of tensors in tree-based low-rank formats