1. Nonstationary covariance structures II NRCSE

2. Drawbacks with deformation approach Oversmoothing of large areas Local deformations not local part of global fits Covariance shape does not change over space Limited class of nonstationary covariances

3. Thetford revisited Features depend on spatial location

4. Kernel averaging Fuentes (2000): Introduce uncorrelated stationary processes Z k (s), k=1,...,K, defined on disjoint subregions S k and construct where w k (s) is a weight function related to dist(s,S k ). Then

5. Spectral version so where Hence

6. Estimating spectrum Asymptotically

7. Details K = 9; h = 687 km Mixture of Matérn spectra

8. An example Models-3 output, 81x87 grid, 36km x 36km. Hourly averaged nitric acid concentration week of 950711.

9. Models-3 fit

10. A spectral approach to nonstationary processes Spectral representation:  s slowly varying square integrable, Y uncorrelated increments Hence is the space- varying spectral density Observe at grid; use FFT to estimate in nbd of s

11. Testing for nonstationarity U(s,w) = log has approximately mean f(s,  ) = log f s (  ) and constant variance in s and .Taking s 1,...,s n and  1,...,  m sufficently well separated, we have approximately U ij = U(s i,  j ) = f ij +  ij with the  ij iid. We can test for interaction between location and frequency (nonstationarity) using ANOVA.

12. Details The general model has The hypothesis of no interaction (  ij =0) corresponds to additivity on log scale: (uniformly modulated process: Z 0 stationary) Stationarity corresponds to Tests based on approx  2 -distribution (known variance)

13. Models-3 revisited Sourcedfsum of squares 22 Between spatial points 826.55663.75 Between frequencies 8366.849171 Interaction6430.54763.5 Total80423.9310598.25

14. Moving averages A simple way of constructing stationary sequences is to average an iid sequence. A continuous time counterpart is, where  is a random measure which is stationary and assigns independent random variables to disjoint sets, i.e., has stationary and independent increments.

15. Lévy-Khinchine is the Lévy measure, and  t is the Lévy process. W can construct it from a Poisson measure H(du,ds) on R 2 with intensity E(H(du,ds))= (du)ds and a standard Brownian motion B t as

16. Examples Brownian motion with drift:  t ~N(  t,  2 t) (du)=0. Poisson process:  t ~Po( t)  =  2 =0, (du)=  {1} (du) Gamma process:  t ~  (  t,  )  =  2 =0, (du)=  e -  u 1(u>0)du/u Cauchy process:  =  2 =0, (du)=  u -2 du/ 

17. Spatial moving averages We can replace R for t with R 2 (or a general metric space) We can replace R for s with R 2 (or a general metric space) We can replace b(t-s) by b t (s) to relax stationarity We can let the intensity measure for H be an arbitrary measure (ds,du)

18. Gaussian moving averages Higdon (1998), Swall (2000): Let  be a Brownian motion without drift, and. This is a Gaussian process with correlogram Account for nonstationarity by letting the kernel b vary with location:

19. Details yields an explicit covariance function which is squared exponential with parameter changing with spatial location. The prior distribution describes “local ellipses,” i.e., smoothly changing random kernels.

20. Local ellipses Foci

21. Prior parametrization Foci chosen independently Gaussian with isotropic squared exponential covariance Another parameter describes the range of influence of a given ellipse. Prior gamma.

22. Example Piazza Road Superfund cleanup. Dioxin applied to road seeped into groundwater.

23. Posterior distribution

24. Estimating nonstationary covariance using wavelets 2-dimensional wavelet basis obtained from two functions  and  : First generation scaled translates of all four; subsequent generations scaled translates of the detail functions. Subsequent generations on finer grids. detail functions

25. W-transform

26. Karhunen-Loeve expansion and where A i are iid N(0,1) Idea: use wavelet basis instead of eigenfunctions, allow for dependent A i

27. Covariance expansion For covariance matrix  write Useful if D close to diagonal. Enforce by thresholding off-diagonal elements (set all zero on finest scales)

28. Surface ozone model ROM, daily average ozone 48 x 48 grid of 26 km x 26 km centered on Illinois and Ohio. 79 days summer 1987. 3x3 coarsest level (correlation length is about 300 km) Decimate leading 12 x 12 block of D by 90%, retain only diagonal elements for remaining levels.

29. ROM covariance