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This study focuses on the estimation and application of variograms and covariance functions in spatial statistics, particularly through the lens of exponential correlation. The core correlation function ( ho(v) = e^{-v/phi} ) serves as a foundation, linking to Gaussian processes with continuous yet non-differentiable paths. The analysis extends to generalized forms, including the power exponential and the Matérn class. The paper discusses various estimation techniques, highlighting the method of moments, robust estimators, and maximum likelihood approaches for accurately modeling spatial correlations.
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The exponential correlation • A commonly used correlation function is (v) = e–v/. Corresponds to a Gaussian process with continuous but not differentiable sample paths. • More generally, (v) = c(v=0) + (1-c)e–v/ has a nugget c, corresponding to measurement error and spatial correlation at small distances. • All isotropic correlations are a mixture of a nugget and a continuous isotropic correlation.
The squared exponential • Using yields • corresponding to an underlying Gaussian field with analytic paths. • This is sometimes called the Gaussian covariance, for no really good reason. • A generalization is the power(ed) exponential correlation function,
The spherical • Corresponding variogram nugget sill range
The Matérn class • where is a modified Bessel function of the third kind and order . It corresponds to a spatial field with –1 continuous derivatives • =1/2 is exponential; • =1 is Whittle’s spatial correlation; • yields squared exponential.
Estimation of variograms • Recall • Method of moments: square of all pairwise differences, smoothed over lag bins • Problems: Not necessarily a valid variogram • Not very robust
A robust empirical variogram estimator • (Z(x)-Z(y))2 is chi-squared for Gaussian data • Fourth root is variance stabilizing • Cressie and Hawkins:
Least squares • Minimize • Alternatives: • fourth root transformation • weighting by 1/2 • generalized least squares
Maximum likelihood • Z~Nn(,) = [(si-sj;)] = V() • Maximize • and q maximizes the profile likelihood
