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Submitted by pscully on Thu, 05/15/2008 - 14:13.
05/22/2008 - 16:10
05/22/2008 - 17:30
STA/BST 290: Debashis Paul
Principal components analysis for correlated functional and high dimensional data
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.