Nonparametric operator-regularized covariance function estimation for functional data

Abstract

In functional data analysis (FDA), 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. This paper develops a new class of nonparametric covariance function estimators in terms of various spectral regularizations of an operator associated with a reproducing kernel Hilbert space. Despite their nonparametric nature, the covariance estimators are automatically positive semi-definite without any additional modification steps. An unconventional representer theorem is established to provide a finite dimensional representation for this class of covariance estimators, which leads to a closed-form expression of the corresponding L 2 eigen-decomposition. Trace-norm regularization is particularly studied to further achieve a low-rank representation, another desirable property which leads to dimension reduction and is often needed in advanced FDA approaches. An efficient algorithm is developed based on the accelerated proximal gradient method. This resulted estimator is shown to enjoy an excellent rate of convergence under both fixed and random designs. The outstanding practical performance of the trace-norm-regularized covariance estimator is demonstrated by a simulation study and the analysis of a traffic dataset.

Sparsely observed functional time series: estimation and prediction

“…The rate of convergence of the kernel smoother of Yao et al [37] was later strengthened by Hall et al [11] and Li and Hsing [22]. Other methods to deal with sparsely observed functional data make use of minimizing a specific convex criterion function and expressing the estimator within a reproducing kernel Hilbert space, see Cai and Yuan [6], and Wong and Zhang [35]).…”

Fast covariance estimation for multivariate sparse functional data

“…Various nonparametric methods have now been proposed to estimate the smooth covariance function, for example, Peng and Paul (), Cai and Yuan (), Goldsmith et al. (), Xiao, Li, Checkley, and Crainiceanu (), and Wong and Zhang ().…”

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

“…Probabilistic features of and estimators for lag-h-covariance operators C X;h of stationary processes X = (X k ) k∈Z with values in L 2 [0, 1], the space of measurable, square-Lebesgue integrable real valued functions with domain [0, 1], are widely studied for fixed lag h, see, e. g., [5], [19], [22], [34], [27]. Further, [39] developed covariance estimators in the space of continuous functions C[0, 1], [48] in tensor product Sobolev-Hilbert spaces, [33] for continuous surfaces, and [18], [1] for arbitrary separable Hilbert spaces. [34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator.…”

“…[34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator. Thereby, [5], [19], [22], [1] utilized classical moment estimators, [27] estimated the integral kernels, in [18], [34] truncated spectral decompositions occured having estimated principle components, and [48] used operator regularized covariance estimators. Also, the limit distribution of the estimation errors of the lag-0-covariance operators was discussed in [26], [28].…”