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Variograms/Covariances and their estimation

Variograms/Covariances and their estimation. STAT 498B. The exponential correlation. A commonly used correlation function is  (v) = e –v/  . Corresponds to a Gaussian process with continuous but not differentiable sample paths.

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Variograms/Covariances and their estimation

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  1. Variograms/Covariancesand their estimation STAT 498B

  2. 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.

  3. 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,

  4. The spherical • Corresponding variogram nugget sill range

  5. 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.

  6. Some other covariance/variogram families

  7. 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

  8. A robust empirical variogram estimator • (Z(x)-Z(y))2 is chi-squared for Gaussian data • Fourth root is variance stabilizing • Cressie and Hawkins:

  9. Least squares • Minimize • Alternatives: • fourth root transformation • weighting by 1/2 • generalized least squares

  10. Maximum likelihood • Z~Nn(,) = [(si-sj;)] =  V() • Maximize • and q maximizes the profile likelihood

  11. A peculiar ml fit

  12. Some more fits

  13. All together now...

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