Spatial Statistics IV

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Spatial Statistics IV

• Stat 518 Sp08

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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

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A robust empirical variogram estimator

• (Z(x)-Z(y))2 is chi-squared for Gaussian data
• Fourth root is variance stabilizing
• Cressie and Hawkins

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Least squares

• Minimize
• Alternatives
• fourth root transformation
• weighting by 1/?2
• generalized least squares

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Maximum likelihood

• ZNn(?,?) ? ??(si-sj?) ? V(?)
• Maximize
• and q maximizes the profile likelihood

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Parana data
ls
ml
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A peculiar ml fit
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Some more fits
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All together now...
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Geometric anisotropy

• If we have an isotropic covariance
  (circular isocorrelation curves).
• If for a linear transformation A, we have
  geometric anisotropy (elliptical isocorrelation
  curves).
• General nonstationary correlation structures are
  typically locally geometrically anisotropic.

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The deformation idea

• In the geometric anisotropic case, write
• where f(x) Ax. This suggests using a general
  nonlinear transformation . Usually d2 or 3.
• G-plane D-space
• We do not want f to fold.
• Do a Bayesian implementation using thin plate
  splines

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California ozone
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Posterior samples