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High-dimensional functional data/time series analysis: finite-sample theory, adaptive functional thresholding and prediction

Fang, Qin (2022) High-dimensional functional data/time series analysis: finite-sample theory, adaptive functional thresholding and prediction. PhD thesis, London School of Economics and Political Science.

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Identification Number: 10.21953/lse.00004491


Statistical analysis of high-dimensional functional data/times series arises in various applications. Examples include different types of brain imaging data in neuroscience (Zhu et al., 2016; Li and Solea, 2018), agespecific mortality rates for different prefectures (Gao et al., 2019a) and intraday energy consumption trajectories (Cho et al., 2013) for thousands of households, to list a few. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of independent or serially dependent observations, posing new challenges to existing work. In this thesis, we consider three fundamental tasks in high-dimensional functional data/times series analysis: finite sample theory, covariance function estimation (with a new class of adaptive functional thresholding operators) and modelling/prediction. In the first chapter, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functional cross-spectral stability measure to characterize the effect of dependence on these estimated cross terms, which are essential in the estimates for additive functional linear regressions. With the proposed functional cross-spectral stability measure, we develop useful concentration inequalities for estimated cross- (auto)covariance matrix functions to accommodate more general sub- Gaussian functional linear processes and, furthermore, establish finite sample theory for relevant estimated terms under a commonly adopted functional principal component analysis framework. Using our derived non-asymptotic results, we investigate the convergence properties of the regularized estimates for two additive functional linear regression applications under sparsity assumptions including functional linear lagged regression and partially functional linear regression in the context of high-dimensional functional/scalar time series.

Item Type: Thesis (PhD)
Additional Information: © 2022 Qin Fang
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Statistics
Supervisor: Qiao, Xinghao

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