CovNet: Covariance Networks for Functional Data on Multidimensional Domains

Abstract

Covariance estimation is ubiquitous in functional data analysis. Yet, the case of functional observations over multidimensional domains introduces computational and statistical challenges, rendering the standard methods effectively inapplicable. To address this problem, we introduce Covariance Networks (CovNet) as a modeling and estimation tool. The CovNet model is universal -it can be used to approximate any covariance up to desired precision. Moreover, the model can be fitted efficiently to the data and its neural network architecture allows us to employ modern computational tools in the implementation. The CovNet model also admits a closed-form eigen-decomposition, which can be computed efficiently, without constructing the covariance itself. This facilitates easy storage and subsequent manipulation in the context of the CovNet. Moreover, we establish consistency of the proposed estimator and derive its rate of convergence. The usefulness of the proposed method is demonstrated by means of an extensive simulation study.

Lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces

“…Thereby, [1,7,28,31] utilized classical moment estimators, [36] estimated the integral kernels, [25,44] used truncated spectral decompositions with estimated FPCs, and [59] used operator regularized covariance estimates. Moreover, [51] studied covariance networks for functional data on multidimensional domains, and [41,60] dealt with covariance estimation of sparse (multivariate) functional data. The limit distribution of the covariance operator's estimation errors was discussed in [35,37].…”

Registration for Incomplete Non-Gaussian Functional Data

“…Standard techniques for covariance estimation quickly become computationally infeasible in high-dimensional (Li et al, 2020) or irregular (Cederbaum et al, 2018) data settings and algorithmic innovations are required. Recently, Sarkar and Panaretos (2021) introduced promising neural network architectures for the efficient, nonparametric approximation of (multidimensional) covariance operators and their eigen-decomposition.…”

“…-Computational efficiency A practical constraint for the application of the evaluated methods remains their computational efficiency in large-scale data settings. In this regard, a promising strain of research are recently proposed neural network based frameworks like Nunez et al (2021) and Chen and Srivastava (2021) for registration and Sarkar and Panaretos (2021) for covariance estimation.…”