Meet the professor Friday January 23 at SFU
- Slides: 39
Meet the professor Friday, January 23 at SFU 4: 30 Beer and snacks reception
NRCSE Spatial Covariance
Valid covariance functions Bochner’s theorem: The class of covariance functions is the class of positive definite functions C: Why?
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
Isotropic correlation Jv(u) is a Bessel function of the first kind and order v. Hence and in the case d=2 (Hankel transform)
The Bessel function J 0 – 0. 403
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.
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,
The spherical Corresponding variogram nugget sill range
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.
Some other covariance/variogram families Name Covariance Wave Rational quadratic Linear None Power law None Variogram
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
A robust empirical variogram estimator (Z(x)-Z(y))2 is chi-squared for Gaussian data Fourth root is variance stabilizing Cressie and Hawkins:
Least squares Minimize Alternatives: • fourth root transformation • weighting by 1/ 2 • generalized least squares
Maximum likelihood Z~Nn( , ) = [ (si-sj; )] = V( ) Maximize and maximizes the profile likelihood
A peculiar ml fit
Some more fits
All together now. . .
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
Stein’s result Covariance functions C 0 and C 1 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 V 0 the variance under C 0 of some random variable. If limn. V 0(e 0(n))=0, then
The Fourier transform
Properties of Fourier transforms Convolution Scaling Translation
Parceval’s theorem Relates space integration to frequency integration. Decomposes variability.
Aliasing Observe field at lattice . Since of spacing the frequencies and ’= +2 m/ are aliases of each other, and indistinguishable. The highest distinguishable frequency is , the Nyquist frequency.
Illustration of aliasing Aliasing applet
Spectral representation Stationary processes Spectral process Y has stationary increments If F has a density f, it is called the spectral density.
Estimating the spectrum For process observed on nxn grid, estimate spectrum by periodogram Equivalent to DFT of sample covariance
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
Some common isotropic spectra Squared exponential Matérn
A simulated process
Thetford canopy heights 39 -year thinned commercial plantation of Scots pine in Thetford Forest, UK Density 1000 trees/ha 36 m x 120 m area surveyed for crown height Focus on 32 x 32 subset
Spectrum of canopy heights
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 log. N) operations to calculate instead of O(N 3).
Using non-gridded data Consider where Then Y is stationary with spectral density Viewing Y as a lattice process, it has spectral density
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
A simulated example
Estimated variogram
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