On uncertainty estimation in functional linear mixed models

Functional data analysis has proven useful in many scientific applications where a physical process is observed as a curve. In many applications, several curves are observed due to multiple subjects, providing replicates in the statistical sense. Recent literature develops several techniques for registering curves and estimating associated models in a regression framework. Standard regression models ignore heterogeneity among curves. Functional linear mixed models are one popular way to combine several curves and capture variability among curves via random effects. Although there is a good amount of work available for analyzing functional data using mixed models, limited attention has been paid to inference. After estimation, we concentrate on measuring uncertainty in terms of mean squared error when functional linear mixed models are used for prediction. Although measuring uncertainty is of paramount interest in any statistical prediction, there is no theoretically valid expression available for functional mixed effect models. The quality of theoretical approximations depends on the number of curves observed. In many real life applications, only a finite number of curves can be observed. In such situations, it is important to asses the error rate for any valid statistical statement. In this article, we derive a theoretically valid approximation of uncertainty measurements for prediction. We also provide some modifications for model estimation. The empirical performance of the proposed method is investigated by numerical examples and is compared with existing literature as appropriate. Our method is computationally simple and often outperforms other existing methods.