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), the challenge of 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 same n-dimensional subspace as the
original sample. As a result, all bootstrap principal components are limited to
the same n-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 p-dimensional bootstrap
components. Fast bootstrap PCA is applied to a dataset of sleep
electroencephalogram (EEG) recordings (p=900, n=392), and to a dataset of
brain magnetic resonance images (MRIs) (pβ 3 million, n=352). For the
brain 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.Comment: 25 pages, including 9 figures and link to R package. 2014-05-14
update: final formatting edits for journal submission, condensed figure