Optimal Spatial Anomaly Detection
There has been a growing interest in anomaly detection problems recently, whilst their focuses are mostly on anomalies taking place on the time index. In this work, we investigate a new anomaly-in-mean problem in multidimensional spatial lattice, that is, to detect the number and locations of anomaly ``spatial regions'' from the baseline. In addition to the classic minimization over the cost function with a $L_0$ penalization, we introduce an innovative penalty on the area of the minimum convex hull that covers the anomaly regions. We show that the proposed method yields a consistent estimation of the number and locations of spatial anomalies. Under the minimax framework, we characterize the optimal detection error for multidimensional spatial anomaly detection problem and reveal the trade-off between detection performance and the geometric flexibility of anomaly region shapes. Large-scale Monte Carlo simulations are carried out to examine the numeric performance of the method. The method has a wide range of applications in real-world problems. As an example, we apply it to detect the marine heatwaves using the sea surface temperature data from the European Space Agency.