Multivariate Decompositions in Brain Imaging
Hernando Ombao, Martin Lindquist, Wesley Thompson, John Aston · 2016
Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 15.2 Principal Component Analysis and Singular Value Decomposition . . . . . . . . . . . 400 15.2.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 15.2.2 Principal Components Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 15.2.3 PCA in Brain Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 15.3 Structured PCA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 15.3.1 Calculation of High-Dimensional PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 15.4 Independent Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 15.4.1 ICA in Brain Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 15.4.2 Homotopic Group ICA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 15.4.3 Computation of High-Dimensional ICA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 15.5 Discussion of Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 15.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 In brain imaging studies, information rich, high-dimensional data are collected, typically along with important research or clinical demographic and covariate information. A variety of brain imaging modalities exist to probe brain structure, function, and chemical composition. One, in particular, is functional magnetic resonance imaging (fMRI), where four-dimensional images of the brain are collected with three dimensions corresponding to space and the fourth to time. In structural magnetic resonance imaging (sMRI) and other static imaging techniques, such as static positron emission tomography (PET), the images are three dimensional for each session. In each case, another dimension is created with multiple scanning sessions per subject, such as in a longitudinal or crossover study.