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.

中文翻译：

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函数线性混合模型中的不确定性估计

功能数据分析已被证明在许多科学应用中很有用，在这些应用中，物理过程被观察为一条曲线。在许多应用中，由于多个受试者观察到多条曲线，提供了统计意义上的重复。最近的文献开发了几种在回归框架中注册曲线和估计相关模型的技术。标准回归模型忽略曲线之间的异质性。函数线性混合模型是一种流行的方法，可以组合多条曲线并通过随机效应捕获曲线之间的可变性。尽管有大量工作可用于使用混合模型分析功能数据，但对推理的关注有限。估计后，当使用函数线性混合模型进行预测时，我们专注于测量均方误差方面的不确定性。尽管在任何统计预测中测量不确定性都是最重要的，但对于功能混合效应模型没有理论上有效的表达式。理论近似值的质量取决于观察到的曲线数量。在许多实际应用中，只能观察到有限数量的曲线。在这种情况下，重要的是评估任何有效统计语句的错误率。在本文中，我们推导出一个理论上有效的预测不确定性测量近似值。我们还为模型估计提供了一些修改。通过数值例子研究了所提出方法的经验性能，并酌情与现有文献进行了比较。我们的方法计算简单，通常优于其他现有方法。