Fast covariance estimation for high-dimensional functional data

Luo Xiao, Vadim Zipunnikov, David Ruppert, Ciprian Crainiceanu

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


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 dimension $$J>500$$J>500; a recently introduced sandwich smoother is an exception but is not adapted to smooth covariance matrices of large dimensions, such as $$J= 10{,}000$$J=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 dimension $$J=10{,}000$$J=10,000 and very fast ($$<$$<10 min) smoothing for $$J=100{,}000$$J=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.

Original languageEnglish (US)
Pages (from-to)409-421
Number of pages13
JournalStatistics and Computing
Issue number1-2
StatePublished - Jan 1 2016


  • FACE
  • Penalized splines
  • Sandwich smoother
  • Singular value decomposition
  • Smoothing
  • fPCA

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics


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