1. Statistical Tools for Environmental Problems peter@stat.washington.edu www.stat.washington.edu/peter NRCSE

2. Course content 1. Kriging 1. Gaussian regression 2. Simple kriging 3. Ordinary and universal kriging 4. Effect of estimated covariance 5. Bayesian kriging 2. Spatial covariance 1. Isotropic covariance in R 2 2. Covariance families 3. Parametric estimation 4. Nonparametric models 5. Fourier analysis 6. Covariance on a sphere

3. 3. Nonstationary structures I: deformations 1. Linear deformations 2.Thin-plate splines 3.Classical estimation 4.Bayesian estimation 5.Other deformations 4. Markov random fields 1.The Markov property 2.Hammersley-Clifford 3.Ising model 4.Gaussian MRF 5.Conditional autoregression 6.The non-lattice case

4. 5. Nonstationary structures II: linear combinations etc. 1.Moving window kriging 2.Integrated white noise 3.Spectral methods 4.Testing for nonstationarity 5.Kernel methods 6. Space-time models 1.Mean surface 2.Separability 3.A simple non-separable model 4.Stationary space-time processes 5.Space-time covariance models 6.Testing for separability

5. 7. Statistics, data and deterministic models 1.The kriging approach 2.Bayesian hierarchical models 3.Bayesian melding 4.Data assimilation 5.Model approximation 8. Setting air quality standards 1.Health effect studies 2.Standards as hypothesis tests 3.Potential network bias 4.What are people exposed to 5.A framework for setting standards

6. 9. Wavelet tools 1. Basic wavelet theory 2. Studying trends in climate series 3. Wavelets in nonstationary spatial covariance models 10. Statistics of compositions 1. An algebra for compositions 2. The logistic normal distribution 3. Source apportionment 4. Analysis of variance 5. A space-time model

7. Programs R geoR Fields S-Plus S+ SpatialStats

8. Kriging NRCSE

9. Research goals in air quality research Calculate air pollution fields for health effect studies Assess deterministic air quality models against data Interpret and set air quality standards Improved understanding of complicated systems

10. The geostatistical model 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

11. The problem Given observations at n locations Z(s 1 ),...,Z(s n ) estimate Z(s 0 ) (the process at an unobserved location) (an average of the process) In the environmental context often time series of observations at the locations. or

12. Some history Regression (Galton, Bartlett) Mining engineers (Krige 1951, Matheron, 60s) Spatial models (Whittle, 1954) Forestry (Matérn, 1960) Objective analysis (Grandin, 1961) More recent work Cressie (1993), Stein (1999)

13. A Gaussian formula If then

14. Simple kriging Let X = (Z(s 1 ),...,Z(s n )) T, Y = Z(s 0 ), so that  X =  1 n,  Y = ,  XX =[C(s i -s j )],  YY =C(0), and  YX =[C(s i -s 0 )]. Then This is the best unbiased linear predictor when  and C are known (simple kriging). The prediction variance is

15. Some variants Ordinary kriging (unknown  ) where Universal kriging (  (s)=A(s)  for some spatial variable A) Still optimal for known C.

16. Universal kriging variance simple kriging variance variability due to estimating 

17. The (semi)variogram Intrinsic stationarity Weaker assumption (C(0) needs not exist) Kriging predictions can be expressed in terms of the variogram instead of the covariance.

18. Ordinary kriging where and kriging variance

19. An example Ozone data from NE USA (median of daily one hour maxima June–August 1974, ppb)

20. Fitted variogram

21. Kriging surface

22. Kriging standard error

23. A better combination

24. Effect of estimated covariance structure The usual geostatistical method is to consider the covariance known. When it is estimated the predictor is not linear nor is it optimal the “plug-in” estimate of the variability often has too low mean Let. Is a good estimate of m 2 (  ) ?

25. Some results 1. Under Gaussianity, m 2  ≥ m 1 (  with equality iff p 2 (X)=p(X;  ) a.s. 2. Under Gaussianity, if is sufficient, and if the covariance is linear in  then 3. An unbiased estimator of m 2 (  is where is an unbiased estimator of m 1 (  ).

26. Better prediction variance estimator (Zimmerman and Cressie, 1992) (Taylor expansion; often approx. unbiased) A Bayesian prediction analysis takes account of all sources of variability (Le and Zidek, 1992)

27. Bayesian kriging Instead of estimating the parameters, we put a prior distribution on them, and update the distribution using the data. Model: Prior: Posterior:

28. geoR Default covariance model exponential Prior is assigned to  and. The latter assumed zero unless specified. The distributions can be discretized. Default prior on  is flat (if not specified, assumed constant). (Lots of different assignments are possible)

29. Prior/posterior of 

30. Variogram estimates mean median mode

31. Predictive mean

32. Comparison Ordinary kriging Bayesian kriging

33. Kriging variances BayesClassical range (52,307) range (170,278)

34. Practical exercises Lectures, references, and labs will be made available at http://www.stat.washington.edu/peter/brisbane