Variograms/Covariances and their estimation

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

6. 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.

8. Some other covariance/variogram families

9. 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

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

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

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

13. A peculiar ml fit

14. Some more fits

15. All together now...