ABSTRACT
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.
