Estimating and testing zones of abrupt changes for spatial data: From local tests to a global test

We propose a method for detecting the zones where a variable, irregularly sampled in the plane, changes abruptly.
Our general model is that under the null hypothesis, the variable of interest is the realization of a stationary Gaussian process with constant expectation. The alternative is modeled as a discontinuity of the expectation along a set of curves.
A local test is first built using distribution properties of the estimated gradient under the null hypothesis. The points where the local test is rejected define the potential Zones of Abrupt Change (ZACs). These local tests are then aggregated into a single global test. The theory that links the local tests and the global test is based on asymptotic distributions of excursion sets of nonstationary X2 fields.
We first establish new results about the curvature at local maxima of these fields and give the asymptotic distribution of the area of the connected components of the excursion sets for high thresholds. The distribution is used to test the global significance of the potential ZACs.
Issues arising when implementing the method in practice are discussed; in particular we propose an iterative procedure for estimating simultaneously the covariance function and the ZACs. The power of the method is evaluated by a simulation study. Its application to a soil data set from an agricultural field shows how it can be used in an exploratory analysis for proposing a subdivision of the domain.