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.

翻译：近年来，异常检测问题日益受到关注，但其研究重点多集中于时间序列上的异常。本文研究多维空间格点中的均值异常问题，即从基线中检测异常“空间区域”的数量与位置。除了对带有$L_0$惩罚项的成本函数进行经典最小化外，我们引入了一项创新性惩罚项，该惩罚项基于覆盖异常区域的最小凸包面积。我们证明所提方法能够对异常数量实现一致估计，并在极小极大框架下达到近乎最优的定位误差。我们还提出一种动态规划算法来近似求解双重惩罚的成本最小化问题，并通过大规模蒙特卡洛模拟检验其数值性能。该方法在现实问题中具有广泛的应用前景。例如，我们将其应用于利用欧洲航天局海表温度数据检测海洋热浪事件。