1. Continuity and covariance Recall that for stationary processes so if C is continuous then Z is mean square continuous. To get results about sample paths need stronger conditions. Isotropic case: if then fractal dimension is 3-  /2  2m times differentiable at origin yields paths having m derivatives

2. Nugget and smoothness Matérn covariance,  =3/2,  =1. Nugget  2 = 0 (left); small (right) Simulated using RandomFields in R.

3. Nonstationary covariance structure I: Deformations

4. Evidence of anisotropy 15 o red 60 o green 105 o blue 150 o brown

5. Another view of anisotropy

6. General setup Z(x,t) =  (x,t) + (x) 1/2 E(x,t) +  (x,t) trend + smooth + error We shall assume that  is known or constant t = 1,...,T indexes temporal replications E is L 2 -continuous, mean 0, variance 1, independent of the error  C(x,y) = Cor(E(x,t),E(y,t)) D(x,y) = Var(E(x,t)-E(y,t)) (dispersion)

7. Geometric anisotropy Recall that 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.

8. 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.

9. Implementation Consider observations at sites x 1,...,x n. Let be the empirical covariance between sites x i and x j. Minimize where P(f) is a penalty for non-smooth transformations, such as the bending energy Solution: pair of thin-plate splines

10. SARMAP An ozone monitoring exercise in California, summer of 1990, collected data on some 130 sites.

11. Transformation This is for hr. 16 in the afternoon

12. Identifiability Perrin and Meiring (1999): Let If (1) f and f -1 are differentiable in R n (2)  (u) is differentiable for u>0 then (f,  ) is unique up to a scaling for  and a homothetic transformation for f (scaling, rotation, reflection)

13. Richness Perrin & Senoussi (2000): Let f and f -1 be differentiable, and let C(x,y) be continuously differentiable. Then (stationarity) iff Let f(0)=0,c i i th column of. Then (isotropy) iff and

14. Levy’s spatial Brownian motion

15. Deformation of Brownian process Let and Then So this Gaussian process can be thought of as a stationary deformation. It is however not an istropic deformation.

16. Estimating variability Resample time slices with replacement from the original data (to maintain spatial structure). Re-estimate deformation based on each bootstrap sample. Kriging estimates can be made based on each of the bootstrap estimates, to get a better sense of the variability.

17. French rainfall data Altitude-adjusted 10-day aggregated rainfall data Nov-Dec 1975-1992 for 39 sites from Languedoc-Rousillon region of France.

18. Estimated deformation

19. G-plane equicorrelation contours

20. D-plane Equicorrelation Contours

21. Uncertainty in deformation Model refitted using 24 of the 36 sites.

22. The smoothing parameter Cross-validation: Leave out sampling station i, estimate(θ,f) from remaining n-1 stations. Minimize the prediction error for site i, summed over i. Together with bootstrap estimate of variability, very computer intensive.

23. Thin-plate splines

24. A Bayesian implementation Likelihood: (Integrated) likelihood:

25. Prior on deformation Linear part: fix two points in the G-D mapping put a (proper) prior on the remaining two parameters smoothing parameter

26. Computation Metropolis-Hastings algorithm for sampling from highly multidimensional posterior. Given estimates of D-plane locations, f(x i ), the transformation is extrapolated to the whole domain using thin plate splines. Predictive distributions for (a) temporal variance at unobserved sites, (b) the spatial covariance for pairs of observed and/or unobserved sites, (c) the observation process at unobserved sites.

27. California ozone

28. Posterior samples

29. Application to US presidential elections NY Times election map 2004

30. Other applications Point process deformation (Jensen & Nielsen, Bernoulli, 2000) Deformation of brain images (Worseley et al., 1999)