Functional Outlier Detection for Density-Valued Data with Application to Robustify Distribution-to-Distribution Regression

Distributional data analysis, concerned with the statistical analysis of data objects consisting of random probability distributions in the framework of functional data analysis (FDA), has received considerable interest in recent years and is increasingly applied in various fields including engineering. Outlier detection and robustness are of great practical interest; however, these aspects remain unexplored for distributional data. To this end, this study focuses on density-valued outlier detection and its application in robust distributional regression. Specifically, we propose a transformation-based approach for single-dataset outlying density detection with an emphasis on converting the less detectable shape outliers to easily detectable magnitude outliers. We also propose a distributional regression-based approach for detecting the abnormal associations of the density-valued two-tuples associated with two datasets. Then, the proposed outlier detection methods are applied to robustify a distribution-to-distribution regression method used in engineering, and we develop a robust estimator for the regression operator by downweighting the detected outliers. The proposed methods are validated and evaluated via extensive simulation studies. The relevant results reveal the superiority of our method over other competitors in distributional outlier detection. A case study in structural health monitoring demonstrates the great potential of our proposal in engineering applications. Supplementary materials for this article are available online.

分布数据分析（distributional data analysis）是在函数型数据分析（Functional Data Analysis, FDA）框架下，针对由随机概率分布组成的数据对象开展统计分析的研究领域，近年来受到广泛关注，并在包括工程在内的诸多领域得到日益广泛的应用。异常值检测与鲁棒性具有重要的实际应用价值，但目前针对分布数据的相关研究仍较为匮乏。为此，本研究聚焦于密度值型异常值检测方法及其在鲁棒分布回归中的应用。具体而言，本研究提出一种基于变换的单数据集异常密度检测方法，重点在于将难以识别的形状型异常值转换为易于检测的幅度型异常值。此外，本研究还提出一种基于分布回归的方法，用于检测两个数据集对应的密度值二元组之间的异常关联关系。随后，将所提出的异常值检测方法应用于工程领域常用的分布-分布回归方法的鲁棒化改造，并通过对检测出的异常值赋予低权重，构建了回归算子的鲁棒估计量。通过大量模拟实验对所提方法进行了验证与评估，结果表明，在分布数据异常值检测任务中，本研究方法的性能优于其他现有同类方法。一项结构健康监测领域的案例研究，验证了所提方法在工程应用中的巨大潜力。本文的补充材料可在线获取。