Functional Moments Regression

TL;DR

This paper introduces a new method to evaluate the Gaussian Process assumption in functional data analysis, revealing its limitations and impacts on inference and prediction in real-world health data.

Contribution

It proposes a sequence of functional moment regressions to assess departures from the GP assumption, especially considering covariate effects and data transformations.

Findings

GP assumption is often violated in real data.

Standard transformations like log do not fix assumption violations.

Departures from GP affect inference and prediction when effect sizes are small.

Abstract

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.