NRCSE

1. NRCSE STAT 592A(UW) 526 (UBC-V) 890-4(SFU) Spatial Statistical Methodspeter@stat.washington.eduwww.stat.washington.edu/peter/592

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 R2 2. Covariance families 3. Parametric estimation 4. Nonparametric models 5. Fourier analysis 6. Covariance on a sphere

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

4. 5. Nonstationary structures II: linear combinations etc. • Moving window kriging • Integrated white noise • Spectral methods • Testing for nonstationarity • Wavelet methods 6. Space-time models • Mean surface • Separability • A simple non-separable model • Stationary space-time processes • Space-time covariance models • Testing for separability • The Le-Zidek approach

5. 7. Statistics, data and deterministic models • The kriging approach • Bayesian hierarchical models • Bayesian melding • Data assimilation • Model approximation 8. Statistics of compositions • An algebra for compositions • The logistic normal distribution • Source apportionment • Analysis of variance • A space-time model

6. 9. Wavelet tools • Basic wavelet theory • Multiscale analysis • Longterm memory models • Wavelet analysis of trends 10. Setting air quality standards • Bayesian model averaging • Standards as hypothesis tests • Potential network bias • Maxima of spatial processes • Operational evaluation of air quality standards

7. Programs R geoR fields spBayes RandomFields GMRFLib: a C-library for fast and exact simulation of Gaussian Markov random fields

8. Course requirements Submit at least 8 homework problems (3 can be replaced by an approved project) Submit at least three lab reports Every other Thursday will be a lab day. Virtual lab machine (get Remote Desktop Connection)

9. Office hours MTh 10-11 B213 Padelford Skype name: guttorp Homework solutions and lab reports must be submitted electronically

10. Kriging NRCSE

11. Research goals in environmental research Calculate pollution fields for health effect studies Assess deterministic models against data Interpret and set environmental standards Improve understanding of complicated systems

12. The geostatistical model Gaussian process (s)=EZ(s) Var Z(s) < ∞ Z isstrictly stationary if Z isweakly stationary if Z isisotropic if weakly stationary and

13. The problem Given observations at n locations Z(s1),...,Z(sn) estimate Z(s0) (the process at an unobserved location) (an average of the process) In the environmental context often time series of observations at the locations. or

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

15. A Gaussian formula If then

16. Simple kriging Let X = (Z(s1),...,Z(sn))T, Y = Z(s0), so that X=1n, Y=, XX=[C(si-sj)], YY=C(0), and YX=[C(si-s0)]. Then This is the best unbiased linear predictor when and C are known (simple kriging). The prediction variance is

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

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

19. Some other kriging variants Indicator kriging Block kriging Co-kriging Using a covariate to improve kriging Disjunctive kriging A nonlinear version of kriging: expand the field into CONS and co-krige these

20. 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.

21. Ordinary kriging where and kriging variance

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

23. Fitted variogram

24. Kriging surface

25. Kriging standard error

26. A better combination

27. 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 m2() ?

28. Some results 1. Under Gaussianity, m2 ≥ m1( with equality iff p2(X)=p(X;) a.s. 2. Under Gaussianity, if is sufficient, and if the covariance is linear in  then 3. An unbiased estimator of m2(is where is an unbiased estimator of m1().

29. 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)