A simple ``energy-variance matching'' method (or its fast randomized version in the noisy case) can efficiently estimate common isotropic Matern-type autocovariances
Résumé
We consider the problem of fitting an isotropic zero-mean stationary Gaussian field model to noisy observations, when the model belongs to the Matern family with known regularity index $\nu \geq 0$, or to the spherical family.
For estimating the correlation range (also called ``decorrelation length'') and the variance of the field, two simple estimating functions based on the so-called ``conditional Gaussian Gibbs-energy mean'' (CGEM) and the empirical variance (EV) were recently introduced.
This article presents an extensive Monte Carlo simulation study for problems with around a thousand observations and settings including large, moderate, and even ``small'', correlation ranges. The known observation sites are either on a 2D grid (including a case of ``very non-uniform'' grid spacings) or randomly uniformly distributed on a simple 2D region.
Some experiments for a $256 \times 256$ grid with missing values are also analyzed.
This study empirically demonstrates that, for all the (possibly random) uniform designs, the statistical efficiency of $\blockCGEMEV$ compared to exact maximum likelihood (ML) is globally very satisfactory (except a degradation for the very extremal ranges in some contexts)
provided the signal-to-noise ratio (SNR) is strong enough or $\nu$ is not too large (for the ``very non-uniform'' case, a simple weighting of EV restores this efficiency). In the less favorable cases, the loss remains in fact acceptable : e.g. for the largest considered index ($\nu=3/2$) and a ``not strong enough'' SNR, it may happen (in fact only for large ranges) that $\blockCGEMEV$ almost doubles the mean squared error for the range parameter or for the widely used combination of the two parameters known as microergodic-parameter. Furthermore an important conclusion for computational efficiency is that the use of the natural fast randomized-trace version of $\blockCGEMEV$ does not significantly degrade this statistical efficiency.
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