Variograms covariances and their estimation
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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

Variograms/Covariancesand their estimation

STAT 498B


The exponential correlation
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
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
The spherical

  • Corresponding variogram

nugget

sill

range


The mat rn class
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
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
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
Least squares

  • Minimize

  • Alternatives:

  • fourth root transformation

  • weighting by 1/2

  • generalized least squares


Maximum likelihood
Maximum likelihood

  • Z~Nn(,) = [(si-sj;)] =  V()

  • Maximize

  • and q maximizes the profile likelihood





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