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.

翻译：平滑性长期以来一直是函数型数据分析中简约性的主导形式，甚至有时被与函数型数据的概念本身混为一谈。然而，许多核心推断任务依赖于逆协方差，其中稀疏性——而非平滑性——是更自然的结构约束。在本文中，我们探讨马尔可夫性作为平滑性的替代方案。以高斯情形作为核心激励背景，我们利用马尔可夫性会在协方差核上诱导形状约束这一事实。基于此观察，我们引入经验协方差的马尔可夫变换，并给出相应的保持马尔可夫结构的估计量。该估计量具有自适应性，且除连续性外不对底层协方差作正则性要求。仿真实验表明，即使在模型误设下，该估计量仍能提升预测性能。与基于平滑性的假设不同，马尔可夫性是可证伪的。为评估其有效性，我们进一步基于连续图结构的新刻画，提出了一种新颖且计算高效的马尔可夫性质检验方法。