1. Modelling non-stationarity in space and time for air quality data Peter Guttorp University of Washington peter@stat.washington.edu NRCSE

2. Outline Lecture 1: Geostatistical tools Gaussian predictions Kriging and its neighbours The need for refinement Lecture 2: Nonstationary covariance estimation The deformation approach Other nonstationary models Extensions to space-time Lecture 3: Putting it all together Estimating trends Prediction of air quality surfaces Model assessment

3. Research goals in air quality modeling Create exposure fields for health effects modeling Assess deterministic air quality models Interpret environmental standards Enhance understanding of complex systems

4. The geostatistical setup Gaussian process  (s)=EZ(s) Var Z(s) < ∞ Z is strictly stationary if Z is weakly stationary if Z is isotropic if weakly stationary and

5. The problem Given observations at n sites Z(s 1 ),...,Z(s n ) estimate Z(s 0 ) (the process at an unobserved site) or (a weighted average of the process)

6. A Gaussian formula If then

7. Simple kriging Let X = (Z(s 1 ),...,Z(S n )) T, Y = Z(s 0 ), so  X =  1 n,  Y = ,  XX =[C(s i -s j )],  YY =C(0), and  YX =[C(s i -s 0 )]. Thus This is the best linear unbiased predictor for known  and C (simple kriging). Variants: ordinary kriging (unknown  ) universal kriging (  =A  for some covariate A) Still optimal for known C. Prediction error is given by

8. The (semi)variogram Intrinsic stationarity Weaker assumption (C(0) need not exist) Kriging can be expressed in terms of variogram

9. Method of moments: square of all pairwise differences, smoothed over lag bins Problem: Not necessarily a valid variogram Estimation of covariance functions

10. Least squares Minimize Alternatives: fourth root transformation weighting by 1/  2 generalized least squares

11. Fitted variogram

12. Kriging surface

13. Kriging standard error

14. A better combination

15. Maximum likelihood Z~N n ( ,  )  =  [  (s i -s j ;  )] =  V(  ) Maximize and maximizes the profile likelihood

16. A peculiar ml fit

17. Some more fits

18. All together now...

19. Effect of estimating covariance structure Standard geostatistical practice is to take the covariance as known. When it is estimated, optimality criteria are no longer valid, and “plug-in” estimates of variability are biased downwards. (Zimmerman and Cressie, 1992) A Bayesian prediction analysis takes proper account of all sources of uncertainty (Le and Zidek, 1992)

20. Violation of isotropy

21. 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)

22. 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.

23. 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.

24. Implementation Consider observations at sites x 1,...,x n. Let be the empirical covariance between sites x i and x j. Minimize where J(f) is a penalty for non-smooth transformations, such as the bending energy

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

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

27. Thin-plate splines Linear part

28. A Bayesian implementation Likelihood: Prior: Linear part: –fix two points in the G-D mapping –put a (proper) prior on the remaining two parameters Posterior computed using Metropolis-Hastings

29. California ozone

30. Posterior samples

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

32. Isotropic covariances on the sphere Isotropic covariances on a sphere are of the form where p and q are directions,  pq the angle between them, and P i the Legendre polynomials. Example: a i =(2i+1)  i

33. A class of global transformations Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude). (Das, 2000)

34. Three iterations

35. Global temperature Global Historical Climatology Network 7280 stations with at least 10 years of data. Subset with 839 stations with data 1950-1991 selected.

36. Isotropic correlations

37. Deformation

38. Assessing uncertainty

39. 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:

40. Kernel averaging Fuentes (2000): Introduce orthogonal local 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 A continuous version has

41. Simplifying assumptions in space-time models Temporal stationarity seasonality decadal oscillations Spatial stationarity orographic effects meteorological forcing Separability C(t,s)=C 1 (t)C 2 (s)

42. SARMAP revisited Spatial correlation structure depends on hour of the day (non-separable):

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

44. A non-separable class of stationary space-time covariance functions Cressie & Huang (1999): Fourier domain Gneiting (2001): f is completely monotone if (-1) n f (n) ≥ 0 for all n. Bernstein’s theorem : for some non- decreasing F. Combine a completely monotone function and a function  with completely monotone derivative into a space-time covariance

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

46. Uses for surface estimation Compliance –exposure assessment –measurement Trend Model assessment –comparing (deterministic) model to data –approximating model output Health effects modeling

47. Health effects Personal exposure (ambient and non- ambient) Ambient exposure outdoor time infiltration Outdoor concentration model for individual i at time t

48. 2 years, 26 10-day sessions A total of 167 subjects: 56 COPD subjects 40 CHD subjects 38 healthy subjects (over 65 years old, non-smokers) 33 asthmatic kids A total of 108 residences: 55 private homes 23 private apartments 30 group homes Seattle health effects study

49. pDR PUF HPEM Ogawa sampler

50. HI Ogawa sampler T/RH logger Nephelometer Quiet Pump Box CO 2 monitor CAT

52. PM 2.5 measurements

53. Where do the subjects spend their time? Asthmatic kids: – 66% at home – 21% indoors away from home – 4% in transit – 6% outdoors Healthy (CHD, COPD) adults: – 83% (86,88) at home – 8% (7,6) indoors away from home – 4% (4,3) in transit – 3% (2,2) outdoors

54. Panel results Asthmatic children not on anti- inflammatory medication: decrease in lung function related to indoor and to outdoor PM 2.5, not to personal exposure Adults with CV or COPD: increase in blood pressure and heart rate related to indoor and personal PM 2.5

57. 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.

58. SVD computation

59. EOF 1

60. EOF 2

61. EOF 3

65. Kriging of  0

66. Kriging of  2

67. Quality of trend fits

68. Observed vs. predicted

69. Observed vs. predicted, cont.

70. Conclusions Good prediction of day-to-day variability seasonal shape of mean Not so good prediction of long-term mean Need to try to estimate

71. Other difficulties Missing data Multivariate data Heterogenous (in space and time) geostatistical tools Different sampling intervals (particularly a PM problem)

72. Southern California PM 2.5 data