Mathematics Colloquia and Seminars

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Description

The talk will begin with an introduction on "Functional Data Analysis", an emerging approach to analyze high dimensional data often recorded over a time period on subjects. The results are a sample of curves (or functions), which are infinitely dimensional. Principal components analysis for multivariate data has been extended to functional data as a dimension reduction tool and termed "Functional Principal Components Analysis (PCA)". We review such an approach briefly when the entire curve can be observed from each subject. This is often the case when the data are recorded by a machine or in a controlled experimental environment. However, in longitudinal studies data are often recorded intermittently, causing different measurement schedules and numbers of measurements among subjects. In addition, longitudinal data are often sparse and subject to measurement errors. All these deviations from traditional functional data setting call for adjustments on the functional PCA approach to accommodate such longitudinal data. In this talk, we explain the need, at least in the initial data analysis phase, to employ nonparametric methods originally developed for functional data to longitudinal data that consist of noisy measurements with underlying smooth random trajectories for each subject in a sample. We show how to employ functional principal component analysis to sparse and noisy longitudinal data. The performance of the methods was illustrated through a simulation study and a longitudinal CD4 data in AIDS patients and a time-course microarray data. Asymptotic properties are also investigated. One of the advantages of such an approach is to unveil the individual trajectories of the underlying longitudinal process. This provides guidance to modeling this process. The approach has many other applications including functional regression where both the predictor and response are longitudinal data.