Detecting changes in the mean of spatial random fields on a regular grid

We propose statistical procedures for detecting changes in the mean of spatial random fields observed on regular grids. The proposed framework provides a general approach to change detection in spatial processes. Extending a block-based method originally developed for time series, we introduce two test statistics, one based on Gini’s mean difference and a novel variance-based variant. Under mild moment conditions, we derive asymptotic normality of the variance-based statistic and prove its consistency against almost all non-constant mean functions (in a sense of positive Lebesgue measure). To accommodate spatial dependence, we further develop a de-correlation algorithm based on estimated autocovariances. Monte Carlo simulations demonstrate that both tests maintain appropriate size and power for both independent and dependent data. In an application to satellite images, especially our variance-based test reliably detects regions undergoing deforestation.

💡 Research Summary

This paper presents a novel statistical framework for detecting changes in the mean function of spatial random fields observed on regular grids. The core problem is testing whether the underlying mean structure is constant or exhibits spatial variation, which is crucial in applications like monitoring deforestation via satellite imagery, quality control in manufacturing, or analyzing medical scans.

The authors extend a block-based methodology originally developed for time series change detection to the spatial domain. The data, modeled as X_ij = μ(i/n, j/m) + Y_ij, is divided into non-overlapping blocks of size l_n x l_m. The arithmetic means of these blocks are computed, creating a set of summary statistics. The variability among these block means is then quantified to test the null hypothesis of a constant mean against the alternative of a non-constant mean.

Two specific test statistics are proposed. The first is based on Gini’s Mean Difference (GMD), directly adapting the time-series approach. The second, and primary theoretical contribution, is a novel variance-based statistic (Var). This statistic is essentially the sample variance of the block means, analogous to the treatment sum of squares in ANOVA.

Under the assumption of independent and identically distributed (i.i.d.) errors with finite (4+ε)-th moments, the paper establishes the asymptotic normality of a properly normalized version of the variance-based statistic (Theorem 1). A significant strength of the proof is showing that the term arising from centering by the global mean is asymptotically negligible. Furthermore, the authors prove the test’s consistency against “almost all” non-constant mean functions, in the sense of having power converging to one for alternatives where the change region has positive Lebesgue measure.

Recognizing that real-world spatial data (e.g., satellite pixels) are often correlated, the authors develop a practical de-correlation algorithm. This pre-processing step uses estimated spatial autocovariances to whiten the data, allowing the theoretical results derived for i.i.d. cases to be applied to dependent data scenarios.

Monte Carlo simulations validate the methods’ performance. Both the GMD-based and variance-based tests maintain the nominal significance level (e.g., α=0.05) for both independent and spatially dependent error structures. In terms of power, the variance-based test generally outperforms the GMD-based test, showing higher sensitivity to larger change regions and greater shift magnitudes.

The practical utility of the framework is demonstrated through an application to satellite imagery of the Amazon rainforest. Using a time series of Normalized Difference Vegetation Index (NDVI) images, the variance-based test successfully identifies regions with statistically significant declines in NDVI, which correspond to areas undergoing deforestation. This application underscores the method’s effectiveness beyond theoretical simulation.

In summary, the paper provides a well-founded, general, and practical approach to spatial change-point detection. It offers solid asymptotic theory for a new variance-based statistic, practical tools for handling spatial dependence, and empirical evidence of its efficacy through simulations and a real-world case study. The work opens avenues for extension to higher-dimensional data and other types of structural changes, such as in the variance.