The STOCHASTICA SNIP seminar will be held online every two weeks starting on January 14. Talks of 30-45 minutes will be followed by 15-30 minutes of open discussion. Initially, seminars will be focused on introducing new problems and applications, to stimulate further discussions and ideas within individual working groups.
The seventh seminar was given by , Imperial College London, and chaired by Dr C贸nall Kelly.
Parameter estimation for Mat茅rn random fields from local measurements
Gaussian random fields with Mat茅rn covariance structure are fundamental models in spatial statistics and widely used in applications. Given discrete observations on a regular grid, we study the parametric estimation of key parameters governing variance, range, and smoothness. Our approach is based on the representation of the random field as the solution to an elliptic stochastic partial differential equation (SPDE), which provides a structural framework for inference. A key insight from this representation is that the model parameters exhibit distinct local scaling behavior, reflecting the local structure of the underlying differential operator.
Building on this idea, we construct spatially localized linear combinations of the data, referred to as local measurements, which approximate localized features of the field. By analyzing how these quantities scale across resolutions, we develop a novel class of estimators based on localized quadratic functionals. This can be viewed as a multi-dimensional extension of quadratic variation techniques from time series analysis. By carefully designing test functions with prescribed moment cancellation properties, we obtain explicit estimators for the variance, range, and smoothness parameters. We establish asymptotic normality under infill asymptotics and derive explicit expressions for the estimator variances using Gaussian moment identities.
The resulting methods are computationally efficient, with linear complexity in the number of observations. Numerical experiments demonstrate that the proposed estimators are competitive with existing approaches while offering substantial computational advantages.
