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Variograms/Covariances and their estimationPowerPoint Presentation

Variograms/Covariances and their estimation

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### Variograms/Covariancesand 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.
- 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 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

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