DEPARTMENT OF STATISTICS
Guest Speaker: Prof. Wensheng Guo
Department of Biostatistics and Epidemiology,
University of Pennsylvania School of Medicine
Title: "Multivariate Spectral Analysis Using Cholesky Decomposition"
Date: Thursday, December 4, 2003
Place: Room 2 Illini Hall
Time: 4:00 p.m. – 5:00 p.m.
Abstract
In multivariate
spectral analysis, traditional methods first calculate the periodogram and then
smooth it to obtain a consistent estimate of the multivariate spectrum. In order
to guarantee that the final estimate is positive semi-definite, some constraints
have to be imposed such as using the same smoothing parameter for all elements
in the spectral matrix. This is a very restrictive constraint as different
elements may have different smoothness and require different smoothing
parameters. We propose to smooth the Cholesky decomposition of a raw spectral
estimate instead, which allows different smoothness for different elements. The
final spectral estimate is reconstructed from the smoothed Cholesky elements,
which is consistent and positive definite. More importantly, the Cholesky
decomposition matrix of the spectrum can be used as a transfer function in
generating time series whose spectrum is identical to the given spectrum at the
Fourier frequencies. This not only provides us much flexibility in simulations,
but also allows us to construct bootstrap confidence intervals on the
multivariate spectrum by generating bootstrap samples using the Cholesky
decomposition of the spectral estimate. We then extend this approach to
multivariate locally stationary time series whose spectrum is assumed to be
smooth in both time and frequency. A numerical example and an application to an
EEG data set recorded during an epileptic seizure are used as illustrations.
*This is joint
work with Ming Dai