Inference for Functional Data under Markov Constraints

TL;DR

This paper introduces a new method for inference in functional data analysis by enforcing Markovianity constraints on the covariance, offering an alternative to traditional smoothness assumptions.

Contribution

It develops a Markov transform-based estimator for covariance that is adaptive, requires minimal regularity, and includes a new test for the Markov property.

Findings

The estimator improves prediction performance in simulations.

Markovianity serves as a falsifiable structural constraint.

The proposed test efficiently assesses the Markov property.

Abstract

Smoothness has long been the dominant form of parsimony in functional data analysis, to the point of occasionally being conflated with the very notion of functional data. However, many core inferential tasks depend on the inverse covariance, where sparsity--rather than smoothness--emerges as the more natural structural constraint. In this paper, we explore Markovianity as an alternative to smoothness. Focusing on the Gaussian case as a central motivating setting, we exploit the fact that Markovianity induces a shape constraint on the covariance kernel. Building on this observation, we introduce a Markov transform of the empirical covariance together with a corresponding estimator that enforces the Markov structure. The estimator is adaptive and requires no regularity of the underlying covariance beyond continuity. In simulation experiments, it is seen to improve prediction performance even under model misspecification. Unlike smoothness-based assumptions, Markovianity is falsifiable. To assess its validity, we further propose a novel and computationally efficient test for the Markov property based on a new characterization of continuous graphical structure.