An Overview of Univariate and Multivariate Karhunen Loève Expansions in Statistics - Daw, Ranadeep; Simpson, Matthew; Wikle, Christopher K.; Holan, Scott H.; Bradley, Jonathan R. (Journal of the Indian Society for Probability and Statistics, 2022)
Dependent data are ubiquitous in statistics and across various subject matter domains, with dependencies across space, time, and variables. Basis expansions have proven quite effective in modeling such processes, particularly in the context of functional data and high-dimensional spatial, temporal, and spatio-temporal data. One of the most useful basis function representations is given by the Karhunen-Loève expansion (KLE), which is derived from the covariance kernel that controls the dependence of a random process, and can be expressed in terms of reproducing kernel Hilbert spaces. The KLE has been used in a wide variety of disciplines to solve many different types of problems, including dimension reduction, covariance estimation, and optimal spatial regionalization. Despite its utility in the univariate context, the multivariate KLE has been used much less frequently in statistics. This manuscript provides an overview of the KLE, with the goal of illustrating the utility of the univariate KLE and bringing the multivariate version to the attention of a wider audience of statisticians and data scientists. After deriving the KLE from a univariate perspective, we derive the multivariate version and illustrate the implementation of both via simulation and data examples.