PhD Dissertation Abstracts: 2010

PhD Dissertation Abstracts: 2010PhD Dissertation Abstracts: 2010

Statistics PhD Alumni 2010:

Dong Chen(2010)

ADVISER: Hans-Georg Müller

TITLE: Manifold models for functional data

ABSTRACT: For functional data on a nonlinear low-dimensional space, we propose the notions of manifold mean, of manifold modes of functional variation and of functional manifold components, which constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. These new tools can be used to summarize functional manifold data and for dimension reduction. In simulations and applications, we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross-sectional mean and functional principal components. Our estimating procedures borrow ideas from existing nonlinear dimension reduction methods, which we modify to address functional data settings. We also explore functional regression problems with these methods and discuss consistency properties.

Shuang Wu(2010)

ADVISER: Hans-Georg Müller

TITLE: Functional modeling of recurrent events, with application to the analysis of bid arrival times in on-line auctions.

ABSTRACT: Repeated event times where for each studied subject in a sample one observes a relatively small number of repeated event times are commonly observed in various applications, including the study of demographic events such as child births per woman or bid placement in an on-line auction. For sparse repeated events data there are no flexible methods available that can be applied when the shapes of the intensity functions that generate the observed events are not known or vary substantially between subjects. We model the intensity functions as nonparametric random functions, specific for each series of event times. The eigenfunctions of the underlying random process generating these functions provide insights into the dynamics of the underlying point process, which generates the observed event times. Subject-specific density functions that reflect the distribution of events are recovered by novel functional data analysis techniques. We present asymptotic consistency results and demonstrate in simulations that the proposed functional approach is superior to conventional nonparametric methods, as it increases efficiency by borrowing strength from the entire sample of subjects rather than aiming at the estimation of each density separately. We illustrate these methods with a study of the dynamics of bid arrival times in online e-Bay auctions.

Wenjing Yang (2010)

ADVISER: Hans-Georg Müller

TITLE: Functional Correlation and Dynamic Relations for Sparsely Sampled Random Processes

ABSTRACT: Aiming at quantifying the dependency of pairs of functional data (X, Y), we develop the concept of functional singular value decomposition for covariance and functional singular component analysis, building on the concept of canonical expansion of compact operators in functional analysis. We demonstrate the estimation of the resulting singular values, functions and components for the practically relevant case of sparse and noise-contaminated longitudinal data and provide asymptotic consistency results. Expanding bivariate functional data into singular functions emerges as a natural extension of the popular functional principal component analysis for single processes to the case of paired processes. A natural application of the functional singular value decomposition is a measure of functional correlation. Due to the involvement of an inverse operation, most previously considered functional correlation measures are plagued by numerical instabilities and strong sensitivity to the choice of smoothing parameters. These problems are acerbated for the case of sparse longitudinal data, on which we focus. The functional correlation measure derived from the functional singular value decomposition behaves well with respect to numerical stability and statistical error, as we demonstrate in a simulation study. Practical feasibility for applications to longitudinal data is illustrated with examples from a study on aging and online auctions.

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