  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. Supplementary materials,
including additional simulations, data examples, and an R-package are available
online.
