Efficiently estimating some common geostatistical models by ‘energy–variance matching’ or its randomized ‘conditional–mean’ versions
Résumé
We consider the problem of fitting an isotropic zero-mean stationary Gaussian field model to (possibly noisy) observations, when the model belongs to the Matérn family with known regularity index ν > 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 grid with missing values are also analyzed.
This study empirically demonstrates that, for all the (possibly random) uniform designs, the statistical efficiency CGEM−EV 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 and ν too large, this SNR restriction being alleviated as ν decreases. For the “very non-uniform” design, a simple weighting of EV restores this efficiency. In the less favorable cases, the statistical loss remains in fact acceptable: e.g. for the largest considered index (ν = 3/2 ) and a “not strong enough” SNR, it may happen (in fact only for large ranges) that CGEM-EV 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 CGEM-EV does not significantly degrade this statistical efficiency.
This study empirically demonstrates that, for all the (possibly random) uniform designs, the statistical efficiency CGEM−EV 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 and ν too large, this SNR restriction being alleviated as ν decreases. For the “very non-uniform” design, a simple weighting of EV restores this efficiency. In the less favorable cases, the statistical loss remains in fact acceptable: e.g. for the largest considered index (ν = 3/2 ) and a “not strong enough” SNR, it may happen (in fact only for large ranges) that CGEM-EV 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 CGEM-EV does not significantly degrade this statistical efficiency.
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