Fast Covariance Estimation for High-Dimensional Functional Data

• Activity Number: 582
• Type: Invited
• Date/Time: Thursday, August 7, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): David Ruppert*+ and Ciprian Crainiceanu and Vadim Zipunnikov and Luo Xiao
• Companies: Cornell University and Johns Hopkins University and Johns Hopkins University and Johns Hopkins University
• Keywords: High-dimensional data ; PCA ; Functional data analysis

Abstract

High dimensional functional data are becoming increasingly common. For such data, we propose fast methods for smooth estimation of the covariance function. These methods scale up linearly with J, the number of observations per function. Most available methods and software cannot smooth covariance matrices of dimension J greater than 500; a recently introduced sandwich smoother is an exception, but it is not adapted to smooth covariance matrices of large dimensions, such as J = 10, 000. We introduce two new methods that circumvent this problem: 1) an extremely 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 smooths the eigenvectors. In high dimensions, these new approaches are at least an order of magnitude faster than standard methods and drastically reduce memory requirements. The new approaches provide instantaneous (a few seconds) smoothing for matrices of dimension J = 10,000 and very fast (< 10 minutes) smoothing for J = 100, 000.