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. Nonstationary spatial covariance: Basic idea: the parameters of a local variogram model–-nugget, range, sill, and anisotropy–vary spatially. Consider some pictures of applications from recent methodology publications.

4. Major approaches: Haas, 1990, Moving window kriging Kim, Mallock & Holmes, 2005, Piecewise Gaussian modeling Nott & Dunsmuir, 2002, Biometrika, Average of locally stationary processes Fuentes, 2002, Kernel averaging of orthogonal, locally stationary processes. Pintore & Holmes, 2005, Fourier and Karhunen-Loeve expansions Higdon & Swall, 1998, 2000, Gaussian moving averages or “process convolution” model Nychka, Wikle & Royle, 2002. Wavelet expansion.

5. Kim, Mallock & Holmes, JASA 2005 Piecewise Gaussian model for groundwater permeability data

9. Fig. 2. Sydney wind pattern data. Contours of equal estimated correlation with two different fixed sites, shown by open squares: (a) location 33·85°S, 151·22°E, and (b) location 33·74°S, 149·88°E. The sites marked by dots show locations of the 45 monitored sites. Nott & Dunsmuire, 2002, Biometrika.

11. Pintore & Holmes: Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

13. Thetford revisited Features depend on spatial location

14. 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

15. Spectral version so where Hence

16. Estimating spectrum Asymptotically

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

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

19. Models-3 fit

20. 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

21. 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.

22. 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)

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

24. 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.

26. Lévy-Khinchine is the Lévy measure, and  t is the Lévy process. We 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

27. 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/ 

28. 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)

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

30. 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.

31. Local ellipses Foci

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

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

34. Posterior distribution