Separable expansions for covariance estimation via the partial inner product

Abstract

The nonparametric estimation of covariance lies at the heart of functional data analysis, whether for curve or surface-valued data. The case of a two-dimensional domain poses both statistical and computational challenges, which are typically alleviated by assuming separability. However, separability is often questionable, sometimes even demonstrably inadequate. We propose a framework for the analysis of covariance operators of random surfaces that generalizes separability, while retaining its major advantages. Our approach is based on the expansion of the covariance into a series of separable terms. The expansion is valid for any covariance over a two-dimensional domain. Leveraging the key notion of the partial inner product, we generalize the power iteration method to general Hilbert spaces and show how the aforementioned expansion can be efficiently constructed in practice at the level of the surface observations. Truncation of the expansion and retention of the leading terms automatically induces a nonparametric estimator of the covariance, whose parsimony is dictated by the truncation level. The resulting estimator can be calculated, stored and manipulated with little computational overhead relative to separability. Consistency and rates of convergence are derived under mild regularity assumptions, illustrating the trade-off between bias and variance regulated by the truncation level. The merits and practical performance of the proposed methodology are demonstrated in a comprehensive simulation study.

CovNet: Covariance Networks for Functional Data on Multidimensional Domains

“…In particular, we can take the classical approach of first smoothing the observed fields and then working with these smoothed fields. A similar approach was taken by Masak et al (2022), following whom it is also possible to derive theoretical guarantees for the modified estimator.…”

Test of Partial Separability for Multivariate Functional Data

“…The common functional principal component model in Benko et al (2009) aimed at identifying common FPCs from two populations, whereas the proposed partial separability accounts for the cross-covariance among p different elements/populations where p may tend to infinity. Recently Masak et al (2023) defined a novel "R-separable" structure based on the partial inner product that generalizes the notion of separability. Unlike partial separability that yields a straightforward representation (2.2) for the multivariate functional process, "R-…”

“…However, the separable structure is demonstrably violated in this dataset (e.g. Masak et al, 2023), indicating the need for more flexible dimension reduction approaches like partial separability to simplify the cross-covariance structure effectively. Conducting the proposed statistical tests can offer insights into the appropriateness of adopting partial separability, and the test result can significantly impact subsequent model specifications, including the functional graphical model (Zapata et al, 2022) discussed in Section 2.3.…”