# Ergodic Subspace Analysis

Abstract

Properties of psychological variables at the mean or variance level can differ between persons and within persons across multiple time points. For example, cross-sectional findings between persons of different ages do not necessarily reflect the development of a single person over time. Recently, there has been an increased interest in the difference between covariance structures, expressed by covariance matrices, that evolve between persons and within a single person over multiple time points. If these structures are identical at the population level, the structure is called ergodic. However, recent data confirms that ergodicity is not generally given, particularly not for cognitive variables. For example, the g factor that is dominant for cognitive abilities between persons seems to explain far less variance when concentrating on a single person’s data. However, other subdimensions of cognitive abilities seem to appear both between and within persons; that is, there seems to be a lower-dimensional subspace of cognitive abilities in which cognitive abilities are in fact ergodic. In this article, we present ergodic subspace analysis (ESA), a mathematical method to identify, for a given set of variables, which subspace is most important within persons, which is most important between person, and which is ergodic. Similar to the common spatial patterns method, the ESA method first whitens a joint distribution from both the between and the within variance structure and then performs a principle component analysis (PCA) on the between distribution, which then automatically acts as an inverse PCA on the within distribution. The difference of the eigenvalues allows a separation of the rotated dimensions into the three subspaces corresponding to within, between, and ergodic substructures. We apply the method to simulated data and to data from the COGITO study to exemplify its usage.