Nonparametric operator-regularized covariance function estimation for functional data

Raymond K.W. Wong and Xiaoke Zhang

Computational Statistics & Data Analysis, 2019, vol. 131, issue C, 131-144

Abstract: In functional data analysis (FDA), the covariance function is fundamental not only as a critical quantity for understanding elementary aspects of functional data but also as an indispensable ingredient for many advanced FDA methods. A new class of nonparametric covariance function estimators in terms of various spectral regularizations of an operator associated with a reproducing kernel Hilbert space is developed. Despite their nonparametric nature, the covariance estimators are automatically positive semi-definite, which is an essential property of covariance functions, via a one-step procedure. An unconventional representer theorem is established to provide a finite dimensional representation for this class of covariance estimators based on data, although the solutions are searched over infinite dimensional functional spaces. To further achieve a low-rank representation, another desirable property, e.g., for dimension reduction and easy interpretation, the trace-norm regularization is particularly studied, under which an efficient algorithm is developed based on the accelerated proximal gradient method. The outstanding practical performance of the trace-norm-regularized covariance estimator is demonstrated by a simulation study and the analysis of a traffic dataset. Under both fixed and random designs, an excellent rate of convergence is established for a broad class of operator-regularized covariance function estimators, which generalizes both the trace-norm-regularized covariance estimator and other popular alternatives.

Keywords: Functional data analysis; Low-rank estimation; Reproducing kernel Hilbert space; Spectral regularization (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:csdana:v:131:y:2019:i:c:p:131-144

DOI: 10.1016/j.csda.2018.05.013

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