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# Variograms/Covariances and their estimation - PowerPoint PPT Presentation

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/Covariancesand their estimation

STAT 498B

• 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.

• 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,

• Corresponding variogram

nugget

sill

range

• 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

• (Z(x)-Z(y))2 is chi-squared for Gaussian data

• Fourth root is variance stabilizing

• Cressie and Hawkins:

• Minimize

• Alternatives:

• fourth root transformation

• weighting by 1/2

• generalized least squares

• Z~Nn(,) = [(si-sj;)] =  V()

• Maximize

• and q maximizes the profile likelihood