title: Fast Covariance Estimation for High-dimensional Functional Data.
abstract: We propose two fast covariance smoothing methods and associated software that scale up linearly with the number of observations per function. Most available methods and software cannot smooth covariance matrices of dimensionJ> 500; a recently introduced sandwich smoother is an exception but is not adapted to smooth covariance matrices of large dimensions, such asJ= 10, 000. We introduce two new methods that circumvent those problems: 1) a fast implementation of the sandwich smoother for covariance smoothing; and 2) a two-step procedure that first obtains the singular value decomposition of the data matrix and then smoothes the eigenvectors. These new approaches are at least an order of magnitude faster in high dimensions and drastically reduce computer memory requirements. The new approaches provide instantaneous (a few seconds) smoothing for matrices of dimensionJ= 10,000 and very fast (< 10 minutes) smoothing forJ= 100, 000. R functions, simulations, and data analysis provide ready to use, reproducible, and scalable tools for practical data analysis of noisy high-dimensional functional data.
authors.affiliation: Department of Biostatistics, Johns Hopkins University, Baltimore, MD.
authors.forename: Luo
authors.initials: L
authors.lastname: Xiao
authors.affiliation: Department of Biostatistics, Johns Hopkins University, Baltimore, MD.
authors.forename: Vadim
authors.initials: V
authors.lastname: Zipunnikov
authors.affiliation: Department of Biostatistics, Johns Hopkins University, Baltimore, MD.
authors.forename: David
authors.initials: D
authors.lastname: Ruppert
authors.affiliation: Department of Statistical Science and School of Operations Research and Information Engineering, Cornell University, Ithaca, NY.
authors.forename: Ciprian
authors.initials: C
authors.lastname: Crainiceanu
keywords: FACE; fPCA; penalized splines; sandwich smoother; singular value decomposition; smoothing