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Optimal Spatial Anomaly Detection

Baiyu Wang, Chao Zheng

TL;DR

The paper tackles spatial anomaly detection of anomaly-in-mean on a regular lattice, proposing a double-penalised least squares approach (DPLS-SAD) that jointly penalises the number of anomaly regions and the area of their minimum convex hulls to recover both the count and the geometry of irregular spatial anomalies. It establishes non-asymptotic consistency and minimax optimal localisation rates in 2D, and extends the framework to general dimensions and spatial dependence, with a fast approximate algorithm (CRS) to handle computational intractability. The method is validated via extensive simulations and is applied to marine heatwave detection from global sea surface temperature data, showing accurate, interpretable segmentation of complex regions and alignment with historical events. Together, these results demonstrate a scalable, theoretically sound tool for automatic, shape-flexible SAD in both synthetic and real-world spatial data contexts.

Abstract

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.

Optimal Spatial Anomaly Detection

TL;DR

The paper tackles spatial anomaly detection of anomaly-in-mean on a regular lattice, proposing a double-penalised least squares approach (DPLS-SAD) that jointly penalises the number of anomaly regions and the area of their minimum convex hulls to recover both the count and the geometry of irregular spatial anomalies. It establishes non-asymptotic consistency and minimax optimal localisation rates in 2D, and extends the framework to general dimensions and spatial dependence, with a fast approximate algorithm (CRS) to handle computational intractability. The method is validated via extensive simulations and is applied to marine heatwave detection from global sea surface temperature data, showing accurate, interpretable segmentation of complex regions and alignment with historical events. Together, these results demonstrate a scalable, theoretically sound tool for automatic, shape-flexible SAD in both synthetic and real-world spatial data contexts.

Abstract

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 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.
Paper Structure (17 sections, 7 theorems, 36 equations, 10 figures, 3 tables, 2 algorithms)

This paper contains 17 sections, 7 theorems, 36 equations, 10 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Detection of spatial anomalies
  3. Model setup and spatial anomalies
  4. Regional loss and penalised cost function
  5. Consistency theory
  6. Minimax optimal detection rate
  7. Extending to general dimensions and spatial dependent data
  8. Anomaly detection for general dimensional data
  9. Anomaly detection for spatially dependent data
  10. Algorithm for fast detection
  11. Simulation studies
  12. Simulation for independent data
  13. Simulation for 2D dependent and 3D data
  14. Real-world data application
  15. Non-separable spatial anomalies
  16. ...and 2 more sections

Key Result

Theorem 1

(Consistency) Suppose Assumptions assmp:data, assmp:signal and assmp:smooth hold. If we choose $\beta=C_\beta \sqrt{n}\log n$ and $\lambda = C_{\lambda}\log n/\sqrt{n}$, where $C_\beta$ and $C_\lambda$ are some absolute constants not depending on $n$ and $m^\ast$. Let $\{\hat{m};\,\widehat{R}_{1:\ha holds with probability at least $1-2\exp(-c_{\gamma}\sqrt{n}\log n)$.

Figures (10)

  • Figure 1: llustration of spatial anomaly regions (each formed by a collection of points, highlighted in same colour) on a 2D spatial lattice, and the baseline region is plotted in grey points.
  • Figure 2: Minimum convex hull (points contained within the solid line) of anomaly regions. (a) The anomaly region is concave, and its minimum convex hull is a square that encloses all the anomaly points plus a few baseline points. (b) We added 9 distant points on the right side to the original anomaly region (whose minimum convex hull is a rectangular), the resulted new minimum convex hull encompasses many baseline points.
  • Figure 3: Illustration of the regional smoothness condition (i). At the same horizontal coordinate $s_1$ (grid points on the red line), the anomaly region can be divided into 6 segments, each is a collection of consecutive points on the line. Any isolated point is counted as a single segment.
  • Figure 4: Plots of anomaly regions in three different settings (top panel) and realisations of the observed data under different combinations of $\Delta$ and $|R|$ (bottom panel), with sample size $n=2500$. (a1) Setting 1: five square anomaly regions, where $\mu_1^* = \Delta$, $\mu_2^* =\mu_3^* = 2\Delta$, and $\mu_4^* = \mu_5^*=3\Delta$; (a2) Setting 2: a ellipse anomaly, a circular anomaly with holes, and a disconnected anomaly, where $\mu_1^* = \Delta$, $\mu_2^* = 2\Delta$, and $\mu_3^*=3\Delta$. (a3) Setting 3: a concave anomaly and a disconnected anomaly, where $\mu_1^* = \mu_2^*=\Delta$. (b1)-(b3): one time data realisation under different $\Delta$ and $|R|$ (top to bottom, $\Delta$ increasing; left to right, $|R|$ increasing.)
  • Figure 5: Frequency of points detected as anomalies, varying by 3 settings, with sample sizes $n=400$ (top panel) and $n=2500$ (bottom panel). Each setting and sample size includes 9 combinations of $\Delta$ and $|R|$ (top to bottom, $\Delta$ increasing; left to right, $|R|$ increasing).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Definition 3
  • Theorem 4
  • ...and 5 more