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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. 

In demand of predicting new human immunodeficiency virus (HIV) diagnosis rates based on publicly available HIV data that are abundant in space but have few points in time, we propose a class of spatially varying auto-regressive models compounded with conditional auto-regressive spatial correlation structures. We then propose to use the copula approach and a flexible conditional auto-regressive formulation to model the dependence between adjacent counties. These models allow for spatial and temporal correlation as well as space–time interactions and are naturally suitable for predicting HIV cases and other spatiotemporal disease data that feature a similar data structure. We apply the proposed models to HIV data over Florida, California and New England states and compare them with a range of linear mixed models that have been recently popular for modelling spatiotemporal disease data. The results show that for such data our proposed models outperform the others in terms of prediction.

 

We propose to model a spatio-temporal random field that has nonstationary covariance structure in both space and time domains by applying the concept of the dimension expansion method in Bornn et al. (2012). Simulations are conducted for both separable and nonseparable space-time covariance models, and the model is also illustrated with a streamflow dataset. Both simulation and data analyses show that modeling nonstationarity in both space and time can improve the predictive performance over stationary covariance models or models that are nonstationary in space but stationary in time.