TY - JOUR
T1 - Fast, Exact Bootstrap Principal Component Analysis for p > 1 Million
AU - Fisher, Aaron
AU - Caffo, Brian
AU - Schwartz, Brian
AU - Zipunnikov, Vadim
N1 - Publisher Copyright:
© 2016, © American Statistical Association.
PY - 2016/4/2
Y1 - 2016/4/2
N2 - 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 (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 n-dimensional subspace as the original sample. As a result, all bootstrap PCs 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 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 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.
AB - 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 (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 n-dimensional subspace as the original sample. As a result, all bootstrap PCs 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 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 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.
KW - Functional data analysis
KW - Image analysis
KW - PCA
KW - SVD
KW - Singular value decomposition
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U2 - 10.1080/01621459.2015.1062383
DO - 10.1080/01621459.2015.1062383
M3 - Article
C2 - 27616801
AN - SCOPUS:84983331305
SN - 0162-1459
VL - 111
SP - 846
EP - 860
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 514
ER -