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 minimisation over the cost function with a $L_0$ penalisation, 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 of anomalies, and it achieves near optimal localisation error under the minimax framework. We also propose a dynamic programming algorithm to solve the double penalised cost minimisation approximately, and carry out large-scale Monte Carlo simulations to examine its numeric performance. 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.