Elastic depths for detecting shape anomalies in functional data

We propose a new depth metric called elastic depth 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 than the rest of the data. Identifying them is far more difficult than identifying magnitude anomalies because shape anomalies are often not separable 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 in turn gives the elastic depths a strong practical advantage over other methods in detecting anomalies. A simple boxplot and thresholding method are introduced to identify shape anomalies using the elastic depths. We assess the elastic depth's detection skill on simulated shape outlier scenarios and compare it against popular shape anomaly detectors. Finally, bond yields, image outlines, and hurricane tracks are used to demonstrate our methods applicability to functional data on three different manifolds.