Advanced Nonparametric Methods for Function Derivatives and Related Localized Features

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dc.contributor.advisorLi, Mengen_US
dc.creatorLiu, Zejianen_US
dc.date.accessioned2024-05-21T21:47:21Zen_US
dc.date.created2024-05en_US
dc.date.issued2024-04-15en_US
dc.date.submittedMay 2024en_US
dc.date.updated2024-05-21T21:47:21Zen_US
dc.descriptionEMBARGO NOTE: This item is embargoed until 2024-11-01en_US
dc.description.abstractIn this thesis, we propose nonparametric statistical methods to study the derivatives and localized features of functions, areas of paramount importance across diverse scientific disciplines such as cosmology, environmental science, and neuroscience. We first develop a plug-in kernel ridge regression estimator for derivatives of arbitrary order. We provide both non-asymptotic and asymptotic error bounds for this estimator and show its substantial improvements over existing techniques, particularly for high-order derivatives and multi-dimensional settings. We then transition to exploring the application of Gaussian processes (GPs) in modeling derivative functionals, addressing a long-standing skepticism surrounding derivative estimation strategies using GPs. We show that the GP prior exhibits a remarkable plug-in property with minimax optimality---this, to the best of our knowledge, provides the first positive result for using GPs in estimating function derivatives. A data-driven empirical Bayes approach is studied for hyperparameter tuning, which achieves theoretical optimality and computational efficiency. Lastly, we tackle the challenging problem of identifying and analyzing localized features of functions, such as local extrema, in the presence of noise. To this end, we introduce a semiparametric Bayesian method built upon derivative-constrained GP priors. We establish large sample frequentist properties for the proposed method, from which point and interval estimates are derived, offering the advantage of fast implementation without requiring complex sampling procedures. Overall, this thesis blends theoretical and methodological innovation with practical application, aiming to equip researchers and practitioners with more powerful and versatile tools for the analysis of nonparametric functions. The advancements presented herein not only contribute to the statistical community but also have broad implications across various fields where derivative estimation and localized feature analysis are essential. Our methods are illustrated via a climate change application for analyzing the global sea-level rise, and an application to cognitive science for analyzing event-related potentials.en_US
dc.embargo.lift2024-11-01en_US
dc.embargo.terms2024-11-01en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationLiu, Zejian. Advanced Nonparametric Methods for Function Derivatives and Related Localized Features. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/116130en_US
dc.identifier.urihttps://hdl.handle.net/1911/116130en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectDerivative estimationen_US
dc.subjectGaussian processen_US
dc.subjectKernel ridge regressionen_US
dc.subjectLocal extremaen_US
dc.subjectNonparametric statisticsen_US
dc.subjectPlug-in propertyen_US
dc.titleAdvanced Nonparametric Methods for Function Derivatives and Related Localized Featuresen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentStatisticsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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