Nonstationary covariance structures II 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

Nonstationary spatial covariance: Basic idea: the parameters of a local variogram model–nugget, range, sill, and anisotropy–vary spatially. For deformation, which do? Not nugget nor sill. I’m citing the published methodologies with visualizations, choosing the order of (1) the Piecewise Gaussian model as the simplest conceptually, then (2) Nott & Dunsmuire, which is hard to explain, although conceptually related to moving window kriging, (3) Pintore & Holmes approach, which uses Karhunen-Loeve or Fourier expansions, (4) Process convolution, (5) Deformation. Montse’s kernel averaging doesn’t have a good visualization. I then have slides with equations for Process convolution, and Kernel averaging, before going into the deformation material.

Major approaches: Haas, 1990, Moving window kriging Kim, Mallock & Holmes, 2005, Piecewise Gaussian modeling Nott & Dunsmuir, 2002, Biometrika, Average of locally stationary processes Fuentes, 2002, Kernel averaging of orthogonal, locally stationary processes. Pintore & Holmes, 2005, Fourier and Karhunen-Loeve expansions Higdon & Swall, 1998, 2000, Gaussian moving averages or “process convolution” model Nychka, Wikle & Royle, 2002. Wavelet expansion. Need to provide an actual reference list, not this numbered list.

Thetford revisited: Spectrum of canopy heights Ridge in neg quadrat 46.4% of var Low freq ridge in pos quadrat 19.7% of var High freq ridge in pos quadrat 6.1% of var

Looking at several adjacent plots Features depend on spatial location HF q<0 LF q>0 LF q<0 HF q>0 other

Moving window approach For kriging at s, consider only sites “near” s Near defined small enough to make stationarity/isotropy reasonable

The moving window approach In purely spatial context, pick the ns nearest sites (for spacetime, use a window in space and time and grow until get ns sites) Fit a model based only on the chosen sites and perform appropriate kriging Difficulty: not a single model, so can be discontinuous Overall covariance may not be positive definite

Kernel averaging Fuentes (2000): Introduce uncorrelated stationary processes Zk(s), k=1,...,K, defined on 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, 81x87 grid, 36km x 36km. 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 is the space-varying spectral density 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 s1,...,sn and w1,...,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: Z0 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 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.

Covariance In the squared exponential kernel case:

Lévy-Khinchine General representation for psii: n is the Lévy measure, and xt is the Lévy process. We can construct it from a Poisson measure H(du,ds) on R2 with intensity E(H(du,ds))=n(du)ds and a standard Brownian motion Bt as

Examples Brownian motion with drift: xt~N(mt,s2t) n(du)=0. Poisson process: xt~Po(lt) m=s2=0, n(du)=ld{1}(du) Gamma process: xt~G(at,b) m=s2=0, n(du)=ae-bu1(u>0)du/u Cauchy process: m=s2=0, n(du)=bu-2du/p

Spatial moving averages We can replace R for t or s with R2 (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 correlation 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.

Prior parametrization Bivariate normal covariance can be described by an ellipse (level curve), determined by area, center and one focus. Focus chosen independently Gaussian with isotropic squared exponential covariance. Another parameter describes the range of influence (scale) of a given ellipse. Prior gamma.

Local ellipses Focus Ellipse determined by focus, center, and fixed area Focus components independent Gaussian time series

Posterior Compute posterior on a grid Mostly Metropolis-Hastings algorithm Full model allows the size of the kernels to vary across locations

Example Piazza Road Superfund cleanup. Dioxin applied to road to prevent dust storms seeped into groundwater.

Posterior distribution