Fast, Exact Bootstrap Principal Component Analysis for <i>p</i> > 1 Million

<p>Many have suggested a bootstrap procedure for estimating the sampling variability of principal component analysis (PCA) results. However, when the number of measurements per subject (<i>p</i>) is much larger than the number of subjects (<i>n</i>), calculating and storing the leading principal components (PCs) from each bootstrap sample can be computationally infeasible. To address this, we outline methods for fast, exact calculation of bootstrap PCs, eigenvalues, and scores. Our methods leverage the fact that all bootstrap samples occupy the same <i>n</i>-dimensional subspace as the original sample. As a result, all bootstrap PCs are limited to the same <i>n</i>-dimensional subspace and can be efficiently represented by their low-dimensional coordinates in that subspace. Several uncertainty metrics can be computed solely based on the bootstrap distribution of these low-dimensional coordinates, without calculating or storing the <i>p</i>-dimensional bootstrap components. Fast bootstrap PCA is applied to a dataset of sleep electroencephalogram recordings (<i>p</i> = 900, <i>n</i> = 392), and to a dataset of brain magnetic resonance images (MRIs) (<i>p</i> ≈ 3 million, <i>n</i> = 352). For the MRI dataset, our method allows for standard errors for the first three PCs based on 1000 bootstrap samples to be calculated on a standard laptop in 47 min, as opposed to approximately 4 days with standard methods. Supplementary materials for this article are available online.</p

Analysis of Group ICA-Based Connectivity Measures from fMRI: Application to Alzheimer's Disease

<div><p>Functional magnetic resonance imaging (fMRI) is a powerful tool for the in vivo study of the pathophysiology of brain disorders and disease. In this manuscript, we propose an analysis stream for fMRI functional connectivity data and apply it to a novel study of Alzheimer's disease. In the first stage, spatial independent component analysis is applied to group fMRI data to obtain common brain networks (spatial maps) and subject-specific mixing matrices (time courses). In the second stage, functional principal component analysis is utilized to decompose the mixing matrices into population-level eigenvectors and subject-specific loadings. Inference is performed using permutation-based exact logistic regression for matched pairs data. The method is applied to a novel fMRI study of Alzheimer's disease risk under a verbal paired associates task. We found empirical evidence of alternative ICA-based metrics of connectivity when comparing subjects evidencing mild cognitive impairment relative to carefully matched controls.</p> </div

Regions with over 20% overlap with the specified spatial maps.

<p>Red areas load positively, blue negatively, yellow have partial volumes loading positively and negatively. Abbreviations: Amyg. = Amygdala, Cer. = Cerebellum, Fr. = Frontal, Hippo = hippocampus, Inf. = Inferior, Ins. = Insula, L. = Left, Olf. = Olfactory, Op. = Opercular part, Pal. = pallium, PHG = Para-Hippocampal Gyrus, Put. = putamen, R. = Right, Sup. = Superior, Temp. = Temporal, Tri. = triangularis.</p

Decliner and control characteristics.

<p>Decliner and control characteristics.</p

AD study: regression results using the functional connectivity as the predictors.

a<p>The between network connectivity of spatial maps 19 and 28.</p>b<p>The between network connectivity of spatial maps 22 and 28.</p>c<p>The within network connectivity of spatial map 26.</p

AD study: univariate regression analysis results.

<p>AD study: univariate regression analysis results.</p

Plots of eigenfunctions associated with the significant predictors.

<p>Plots of eigenfunctions associated with the significant predictors.</p

Plots (A)∼(F) are heatmaps for time courses modulating spatial maps 11, 13, 19, 22, 26, 28 respectively.

<p>The subjects are grouped by matched pairs.</p

Three-D rendering of thresholded spatial maps associated with the significant predictors.

<p>Red areas load positively while blue areas load negatively. The figures from the upper left to the upper right are spatial maps of IC 11, 13 and 19 respectively. The figures from the lower left to the lower right are spatial maps of IC 22, 26 and 28 respectively.</p