Space-time processes

1. NRCSE Space-time processes

2. Separability Separable covariance structure: Cov(Z(x,t),Z(y,s))=CS(x,y)CT(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) Z1 large-scale feature Z2 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 w. Usage: Fourier transform of Cw(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

10. Coherence plot a3,b3 a6,b6

11. A class of Matérn-type nonseparable covariances =1: separable =0: time is space (at a different rate) spatial decay temporal decay scale 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)nf(n)≥0for 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 y with completey monotone derivative into a space-time covariance

16. Some examples

17. A particular case a=1/2,g=1/2 a=1/2,g=1 a=1,g=1/2 a=1,g=1

18. Velocity-driven space-time covariances CS 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 Vik are covariates, such as population density, proximity to roads, local topography, etc. where the fj are smoothed versions of temporal singular vectors (EOFs) of the TxN data matrix. We will set m1(si) = m0(si) for now.

24. SVD computation

25. EOF 1

26. EOF 2

27. EOF 3

29. Kriging of m0

30. Kriging of r2

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 m2 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