Fast, Exact Bootstrap Principal Component Analysis forp> 1 million.
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 (p) is much larger than the number of subjects (n), calculating and storing the leading principal components from each bootstrap sample can be computationally infeasible. To address this, we outline methods for fast, exact calculation of bootstrap principal components, eigenvalues, and scores. Our methods leverage the fact that all bootstrap samples occupy the samen-dimensional subspace as the original sample. As a result, all bootstrap principal components are limited to the samen-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 thep-dimensional bootstrap components. Fast bootstrap PCA is applied to a dataset of sleep electroencephalogram recordings (p= 900,n= 392), and to a dataset of brain magnetic resonance images (MRIs) (p≈ 3 million,n= 352). For the MRI dataset, our method allows for standard errors for the first 3 principal components based on 1000 bootstrap samples to be calculated on a standard laptop in 47 minutes, as opposed to approximately 4 days with standard methods.
Aaron
A
Fisher
Brian
B
Caffo
Brian
B
Schwartz
Vadim
V
Zipunnikov
PCA; SVD; functional data analysis; image analysis; singular value decomposition