The Gaussian Process (GP) assumption is often used in functional data analysis. We propose a method to assess departures from the GP assumption, both in terms of the shape of the distribution and its potential dependence on covariates, using a sequence of functional moment regressions. Our methods are inspired by and applied to objectively measured minute-level physical activity data from the National Health and Nutrition Examination Survey (NHANES) 2011-2014 study. In this setting, we find that the GP assumption is not satisfied, quantify the associations between functional moments and covariates, and show that standard data transformations, such as the log transformation, do not resolve the discrepancy between assumptions and reality. We further show that when the effect sizes are moderate, inference on the functional fixed effects is largely unaffected by departures from the GP assumption. However, when effect sizes are small, both inference and prediction of subject-level data can be strongly affected. Extensive simulations support these findings. This pragmatic paper presents new methods for real data analysis, with implications for statistical methodology and for understanding human activity and health.

翻译：高斯过程（GP）假设在函数型数据分析中常被使用。我们提出了一种方法，用于评估数据对GP假设的偏离程度——既包括分布形态的偏离，也包括其与协变量潜在相关性的偏离——该方法基于一系列函数型矩回归。我们的方法受美国国家健康与营养调查（NHANES）2011-2014研究中客观测量的分钟级体力活动数据启发，并应用于该数据。在此场景下，我们发现GP假设并不成立，量化了函数型矩与协变量之间的关联，并表明标准数据变换（如对数变换）无法消除假设与现实的差异。我们进一步发现：当效应量适中时，对函数型固定效应的推断基本不受GP假设偏离的影响；但当效应量较小时，对受试者层面数据的推断和预测均会受到强烈影响。大量模拟研究支持了这些结论。这篇面向实际应用的论文提出了真实数据分析的新方法，对统计方法论以及理解人类活动与健康具有启示意义。