Data depth is an efficient tool for robustly summarizing the distribution of functional data and detecting potential magnitude and shape outliers. Commonly used functional data depth notions, such as the modified band depth and extremal depth, are estimated from pointwise depth for each observed functional observation. However, these techniques require calculating one single depth value for each functional observation, which may not be sufficient to characterize the distribution of the functional data and detect potential outliers. This paper presents an innovative approach to make the best use of pointwise depth. We propose using the pointwise depth distribution for magnitude outlier visualization and the correlation between pairwise depth for shape outlier detection. Furthermore, a bootstrap-based testing procedure has been introduced for the correlation to test whether there is any shape outlier. The proposed univariate methods are then extended to bivariate functional data. The performance of the proposed methods is examined and compared to conventional outlier detection techniques by intensive simulation studies. In addition, the developed methods are applied to simulated solar energy datasets from a photovoltaic system. Results revealed that the proposed method offers superior detection performance over conventional techniques. These findings will benefit engineers and practitioners in monitoring photovoltaic systems by detecting unnoticed anomalies and outliers.

翻译：数据深度是稳健地总结函数型数据分布并检测潜在幅度异常值与形状异常值的有效工具。常用的函数型数据深度概念，如修正带深度和极值深度，均通过每个观测函数在单点上的深度值进行估计。然而，这些技术需要为每个函数观测值计算单一深度值，这可能不足以充分刻画函数型数据的分布特征并检测潜在异常值。本文提出了一种创新方法，以充分利用单点数据深度。我们建议采用单点深度分布来可视化幅度异常值，并利用成对深度之间的相关性检测形状异常值。此外，针对该相关性引入了一种基于自助法的检验程序，以检验是否存在形状异常值。所提出的单变量方法随后被扩展至双变量函数型数据。通过大量模拟研究，将所提方法的性能与传统异常值检测技术进行了比较。同时，这些方法被应用于光伏系统模拟太阳能数据集。结果表明，与传统技术相比，所提方法具有更优的检测性能。这些发现将有助于工程师和实践者在监测光伏系统时检测未被注意的异常与离群点。