1. Spatial Statistics IV Stat 518 Sp08

2. Recall Method of moments: square of all pairwise differences, smoothed over lag bins Problems: Not necessarily a valid variogram Not very robust Estimation of variograms

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

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

5. Maximum likelihood Z~N n ( ,  )  =  [  (s i -s j ;  )] =  V(  ) Maximize and  maximizes the profile likelihood

6. Parana data ml ls

7. A peculiar ml fit

8. Some more fits

9. All together now...

10. 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.

11. The deformation idea In the geometric anisotropic case, write where f(x) = Ax. This suggests using a general nonlinear transformation. Usually d=2 or 3. G-plane D-space We do not want f to fold. Do a Bayesian implementation using thin plate splines

12. California ozone

13. Posterior samples