1. Space-time processes NRCSE

2. Separability Separable covariance structure: Cov(Z(x,t),Z(y,s))=C S (x,y)C T (s,t) Nonseparable alternatives Temporally varying spatial covariances Fourier approach Completely monotone functions

3. SARMAP revisited Spatial correlation structure depends on hour of the day:

4. Bruno’s seasonal nonseparability Nonseparability generated by seasonally changing spatial term (uniformly modulated at each time) Z 1 large-scale feature Z 2 separable field of local features (Bruno, 2004)

5. General stationary space-time covariances Cressie & Huang (1999): By Bochner’s theorem, a continuous, bounded, symmetric integrable C(h;u) is a space- time covariance function iff is a covariance function for all . Usage: Fourier transform of C  (u) Problem: Need to know Fourier pairs

6. Spectral density Under stationarity and separability, If spatially nonstationary, write Define the spatial coherency as Under separability this is independent of frequency τ

7. Estimation Let (variance stabilizing) where R is estimated using

8. Models-3 output

9. ANOVA results ItemdfrssP-value Between points 10.1290.68 Between freqs 511.140.0008 Residual50.346

10. Coherence plot a 3,b 3 a 6,b 6

11. A class of Matérn-type nonseparable covariances  =1: separable  =0: time is space (at a different rate) scale spatial decay temporal decay space-time interaction

13. Chesapeake Bay wind field forecast (July 31, 2002)

14. Fuentes model Prior equal weight on  =0 and  =1. Posterior: mass (essentially) 0 for  =0 for regions 1, 2, 3, 5; mass 1 for region 4.

15. Another approach Gneiting (2001): A function f is completely monotone if (-1) n f (n) ≥0 for all n. Bernstein’s theorem shows that for some non- decreasing F. In particular,is a spatial covariance function for all dimensions iff f is completely monotone. The idea is now to combine a completely monotone function and a function  with completey monotone derivative into a space-time covariance

16. Some examples

17. A particular case  =1/2,  =1/2  =1/2,  =1  =1,  =1/2  =1,  =1

18. Velocity-driven space-time covariances C S covariance of purely spatial field V (random) velocity of field Space-time covariance Frozen field model: P(V=v)=1 (e.g. prevailing wind)

19. Irish wind data Daily average wind speed at 11 stations, 1961-70, transformed to “velocity measures” Spatial: exponential with nugget Temporal: Space-time: mixture of Gneiting model and frozen field

20. Evidence of asymmetry Time lag 1 Time lag 2 Time lag 3

21. A national US health effects study

23. Trend model where V ik are covariates, such as population density, proximity to roads, local topography, etc. where the f j are smoothed versions of temporal singular vectors (EOFs) of the TxN data matrix. We will set  1 (s i ) =  0 (s i ) for now.

24. SVD computation

25. EOF 1

26. EOF 2

27. EOF 3

29. Kriging of  0

30. Kriging of  2

31. Quality of trend fits

32. Observed vs. predicted

33. A model for counts Work by Monica Chiogna, Carlo Gaetan, U. Padova Blue grama (Bouteloua gracilis)

34. The data Yearly counts of blue grama plants in a series of 1 m 2 quadrats in a mixed grass prairie (38.8N, 99.3W) in Hays, Kansas, between 1932 and1972 (41 years).

35. Some views

36. Modelling Aim: See if spatial distribution is changing with time. Y(s,t)  (s,t) ~ Po( (s,t)) log( (s,t)) = constant + fixed effect of temp & precip + trend + weighted average of principal fields

37. Principal fields

38. Coefficients

39. Years