THURSDAY, May 22nd, 2008 at 4.10pm, MSB
1147 (Colloquium Room)
Refreshments: 3.30pm, MSB 4110 (Statistics
Lounge)
Speaker: Debashis Paul, (Statistics, UC Davis)
Title: Principal components analysis for
correlated functional and high dimensional data
Abstract: In the
first part of the talk, based on joint work with Professor Jie Peng, we
consider the problem of estimating the covariance kernel and its eigenvalues
and eigenfunctions from sparse, irregularly observed, noisy functional data. We
present a method based on pre-smoothing of individual sample curves through an
appropriate kernel. We show that the "naive" estimate based on sample
covariance of the pre-smoothed sample curves gives highly biased estimator of
the covariance kernel along its diagonal. We attend to this problem by
estimating the diagonal and off-diagonal parts of the covariance kernel
separately and then merging these two estimates together. We then present a
practical and efficient method for
choosing the bandwidth for the kernel by using an approximation to the
leave-one-curve-out cross validation score. We prove that under standard
regularity conditions on the covariance kernel, even when the noiseless sample
curves are "weakly" correlated, with a separable covariance
structure, the proposed method gives consistent estimates of the covariance
surface and its eigenfunctions.
Moreover, the rate of convergence of estimating the
eigenfunctions is optimal when the number of measurements per curve is bounded,
under an appropriate choice of bandwidth.
In the second part of the talk, based on joint work
with Kun Chen and Professor Jane-Ling Wang, we consider a variant of the above
problem, where the observations are high dimensional Gaussian vectors, and the
principal component scores are correlated across subjects. Under this setting
we study the asymptotic behavior of PCA.
