Meet the professor

1. Meet the professor • Friday, January 23 at SFU • 4:30 Beer and snacks reception

2. Spatial Covariance NRCSE

3. Valid covariance functions • Bochner’s theorem: The class of covariance functions is the class of positive definite functions C: • Why?

4. Spectral representation • By the spectral representation any isotropic continuous correlation on Rd is of the form • By isotropy, the expectation depends only on the distribution G of . • Let Y be uniform on the unit sphere. Then

5. Isotropic correlation • Jv(u) is a Bessel function of the first kind and order v. • Hence • and in the case d=2 • (Hankel transform)

6. The Bessel function J0 –0.403

7. The exponential correlation • A commonly used correlation function is (v) = e–v/. Corresponds to a Gaussian process with continuous but not differentiable sample paths. • More generally, (v) = c(v=0) + (1-c)e–v/ has a nugget c, corresponding to measurement error and spatial correlation at small distances. • All isotropic correlations are a mixture of a nugget and a continuous isotropic correlation.

8. The squared exponential • Using yields • corresponding to an underlying Gaussian field with analytic paths. • This is sometimes called the Gaussian covariance, for no really good reason. • A generalization is the power(ed) exponential correlation function,

9. The spherical • Corresponding variogram nugget sill range

11. The Matérn class • where is a modified Bessel function of the third kind and order . It corresponds to a spatial field with [–1] continuous derivatives • =1/2 is exponential; • =1 is Whittle’s spatial correlation; • yields squared exponential.

13. Some other covariance/variogram families

14. Estimation of variograms • Recall • Method of moments: square of all pairwise differences, smoothed over lag bins • Problems: Not necessarily a valid variogram • Not very robust

15. A robust empirical variogram estimator • (Z(x)-Z(y))2 is chi-squared for Gaussian data • Fourth root is variance stabilizing • Cressie and Hawkins:

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

17. Maximum likelihood • Z~Nn(,) = [(si-sj;)] =  V() • Maximize • and q maximizes the profile likelihood

18. A peculiar ml fit

19. Some more fits

20. All together now...

21. Asymptotics • Increasing domain asymptotics: let region of interest grow. Station density stays the same • Bad estimation at short distances, but effectively independent blocks far apart • Infill asymptotics: let station density grow, keeping region fixed. • Good estimates at short distances. No effectively independent blocks, so technically trickier

22. Stein’s result • Covariance functions C0 and C1 are compatible if their Gaussian measures are mutually absolutely continuous. Sample at {si, i=1,...,n}, predict at s (limit point of sampling points). Let ei(n) be kriging prediction error at s for Ci, and V0 the variance under C0 of some random variable. • If limnV0(e0(n))=0, then

23. The Fourier transform

24. Properties of Fourier transforms • Convolution • Scaling • Translation

25. Parceval’s theorem • Relates space integration to frequency integration. Decomposes variability.

26. Aliasing • Observe field at lattice of spacing . Since • the frequencies  and ’=+2m/are aliases of each other, and indistinguishable. • The highest distinguishable frequency is , the Nyquist frequency.

27. Illustration of aliasing • Aliasing applet

28. Spectral representation • Stationary processes • Spectral process Y has stationary increments • If F has a density f, it is called the spectral density.

29. Estimating the spectrum • For process observed on nxn grid, estimate spectrum by periodogram • Equivalent to DFT of sample covariance

30. Properties of the periodogram • Periodogram values at Fourier frequencies (j,k)are • uncorrelated • asymptotically unbiased • not consistent • To get a consistent estimate of the spectrum, smooth over nearby frequencies

31. Some common isotropic spectra • Squared exponential • Matérn

32. A simulated process

33. Thetford canopy heights • 39-year thinned commercial plantation of Scots pine in Thetford Forest, UK • Density 1000 trees/ha • 36m x 120m area surveyed for crown height • Focus on 32 x 32 subset

34. Spectrum of canopy heights

35. Whittle likelihood • Approximation to Gaussian likelihood using periodogram: • where the sum is over Fourier frequencies, avoiding 0, and f is the spectral density • Takes O(N logN) operations to calculate • instead of O(N3).

36. Using non-gridded data • Consider • where • Then Y is stationary with spectral density • Viewing Y as a lattice process, it has spectral density

37. Estimation • Let • where Jx is the grid square with center x and nx is the number of sites in the square. Define the tapered periodogram • where . The Whittle likelihood is approximately

38. A simulated example

39. Estimated variogram