NRCSE Nonstationary covariance structures II Drawbacks with deformation
NRCSE Nonstationary covariance structures II
Drawbacks with deformation approach Oversmoothing of large areas Local deformations not local part of global fits Covariance shape does not change over space Limited class of nonstationary covariances
Thetford revisited Features depend on spatial location
Kernel averaging Fuentes (2000): Introduce uncorrelated stationary processes Zk(s), k=1, . . . , K, defined on disjoint subregions Sk and construct where wk(s) is a weight function related to dist(s, Sk). Then
Spectral version so where Hence
Estimating spectrum Asymptotically
Details K = 9; h = 687 km Mixture of Matérn spectra
An example Models-3 output, 81 x 87 grid, 36 km x 36 km. Hourly averaged nitric acid concentration week of 950711.
Models-3 fit
A spectral approach to nonstationary processes Spectral representation: s slowly varying square integrable, Y uncorrelated increments Hence varying spectral density is the space- Observe at grid; use FFT to estimate in nbd of s
Testing for nonstationarity U(s, w) = log has approximately mean f(s, w) = log fs(w) and constant variance in s and w. Taking s 1, . . . , sn and w 1, . . . , wm sufficently well separated, we have approximately Uij = U(si, wj) = fij+eij with the eij iid. We can test for interaction between location and frequency (nonstationarity) using ANOVA.
Details The general model has The hypothesis of no interaction ( ij=0) corresponds to additivity on log scale: (uniformly modulated process: Z 0 stationary) Stationarity corresponds to Tests based on approx 2 -distribution (known variance)
Models-3 revisited Source df sum of squares 2 Between spatial points 8 26. 55 663. 75 Between frequencies 8 366. 84 9171 Interaction 64 30. 54 763. 5 Total 80 423. 93 10598. 25
Moving averages A simple way of constructing stationary sequences is to average an iid sequence. A continuous time counterpart is , where x is a random measure which is stationary and assigns independent random variables to disjoint sets, i. e. , has stationary and independent increments.
Lévy-Khinchine n is the Lévy measure, and xt is the Lévy process. W can construct it from a Poisson measure H(du, ds) on R 2 with intensity E(H(du, ds))=n(du)ds and a standard Brownian motion Bt as
Examples Brownian motion with drift: xt~N(mt, s 2 t) n(du)=0. Poisson process: xt~Po(lt) m=s 2=0, n(du)=ld{1}(du) Gamma process: xt~G(at, b) m=s 2=0, n(du)=ae-bu 1(u>0)du/u Cauchy process: m=s 2=0, n(du)=bu-2 du/p
Spatial moving averages We can replace R for t with R 2 (or a general metric space) We can replace R for s with R 2 (or a general metric space) We can replace b(t-s) by bt(s) to relax stationarity We can let the intensity measure for H be an arbitrary measure n(ds, du)
Gaussian moving averages Higdon (1998), Swall (2000): Let x 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:
Details yields an explicit covariance function which is squared exponential with parameter changing with spatial location. The prior distribution describes “local ellipses, ” i. e. , smoothly changing random kernels.
Local ellipses Foci
Prior parametrization Foci chosen independently Gaussian with isotropic squared exponential covariance Another parameter describes the range of influence of a given ellipse. Prior gamma.
Example Piazza Road Superfund cleanup. Dioxin applied to road seeped into groundwater.
Posterior distribution
Estimating nonstationary covariance using wavelets 2 -dimensional wavelet basis obtained from two functions and : detail functions First generation scaled translates of all four; subsequent generations scaled translates of the detail functions. Subsequent generations on finer grids.
W-transform
Karhunen-Loeve expansion and where Ai are iid N(0, 1) Idea: use wavelet basis instead of eigenfunctions, allow for dependent Ai
Covariance expansion For covariance matrix write Useful if D close to diagonal. Enforce by thresholding off-diagonal elements (set all zero on finest scales)
Surface ozone model ROM, daily average ozone 48 x 48 grid of 26 km x 26 km centered on Illinois and Ohio. 79 days summer 1987. 3 x 3 coarsest level (correlation length is about 300 km) Decimate leading 12 x 12 block of D by 90%, retain only diagonal elements for remaining levels.
ROM covariance
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