Abstract

We propose a new family of depth measures called the elastic depths that can be used to greatly improve shape anomaly detection in functional data. Shape anomalies are functions that have considerably different geometric forms or features from the rest of the data. Identifying them is generally more difficult than identifying magnitude anomalies because shape anomalies are often not distinguishable from the bulk of the data with visualization methods. The proposed elastic depths use the recently developed elastic distances to directly measure the centrality of functions in the amplitude and phase spaces. Measuring shape outlyingness in these spaces provides a rigorous quantification of shape, which gives the elastic depths a strong theoretical and practical advantage over other methods in detecting shape anomalies. A simple boxplot and thresholding method is introduced to identify shape anomalies using the elastic depths. We assess the elastic depth’s detection skill on simulated shape outlier scenarios and compare them against popular shape anomaly detectors. Finally, we use hurricane trajectories to demonstrate the elastic depth methodology on manifold valued functional data.

摘要

我们提出了一个新的深度度量系列，称为弹性深度，可用于大大改善功能数据中的形状异常检测。形状异常是与其余数据具有显着不同几何形式或特征的函数。识别它们通常比识别幅度异常更困难，因为形状异常通常无法通过可视化方法与大量数据区分开来。建议的弹性深度使用最近开发的弹性距离来直接测量振幅和相位空间中函数的中心性。测量这些空间中的形状无关性提供了严格的形状量化，这使弹性深度在检测形状异常方面比其他方法具有强大的理论和实践优势。引入了一种简单的箱线图和阈值方法来使用弹性深度识别形状异常。我们在模拟形状异常值场景中评估弹性深度的检测技能，并将它们与流行的形状异常检测器进行比较。最后，我们使用飓风轨迹来演示对流形有价值的功能数据的弹性深度方法。