Groundwater permeability Easy to solve the forward problem: flow of groundwater given permeability of aquifer Inverse problem: determine permeability from flow (usually of tracers With some models enough to look at first arrival of tracer at each well (breakthrough times Notation ψ is permeability b is breakthrough times ˆb(ψ expected breakthrough times L(ψ b exp( 1 (b 2σ 2 k ˆb k (ψ 2 k Illconditioned problems: different permeabilities can yield same flow Use regularization by prior on log(ψ MRF Gaussian Convolution with MRF (discretized
MRF prior π(x λ λ m/ 2 exp( 1 2 λxt Wx where 1 if j ~ k W = n j if j = k 0 otherwise and n j =#{i:i~j}
Kim, Mallock & Holmes, JASA 2005 Analyzing Nonstationary Spatial Data Using Piecewise Gaussian Processes Studying oil permeability Voronoi tesselation (choose M centers from a grid Separate power exponential in each regions
Nott & Dunsmuir, 2002, Biometrika Consider a stationary process W(s, correlation R, observed at sites s 1,..,s n. Write W(s = [ R(s s i ] T R(s i s j W(si 1 [ ] + δ(s kriging predictor = λ(s T [W(s i ] + δ(s δ(s has covariance function R δ (s,t = R(t s [ ] T R(s i s j R(s s i [ ] 1 R(t si kriging error
More generally Consider k independent stationary spatial fields W i (s and a random vector Z. Write µ i (s = λ i (sz and create a nonstationary process by Z(s = w i (sµ i (s + w i (s 1 2 δ (s i Its covariance (with Γ=Cov(Z is R(s,t = i,j w i (sw i (tλ i (s T Γλ j (t + w i (s 1 2 w i (t 1 2 R δ i (s,t i Fig. 2. Sydney wind pattern data. Contours of equal estimated correlation with two different fixed sites, shown by open squares: (a location 33 85 S, 151 22 E, and (b location 33 74 S, 149 88 E. The sites marked by dots show locations of the 45 monitored sites.
Karhunen-Loéve expansion There is a unique representation of stochastic processes with uncorrelated coefficients: Z(s = α k φ k (s Var(α k = λ k where the φ k (s solve C(s,tφ k (tdt = λ k φ k (s and are orthogonal eigenfunctions. Example: temporal Brownian motion C(s,t=min(s,t φ k (s=2 1/2 sin((k-1/2πt/((k-1/2π Conversely, C(s,t = λ k φ k (sφ k (t
Discrete case Eigenexpansion of covariance matrix Empirically SVD of sample covariance Example: squared exponential k=1 5 20 Tempering Stationary case: write Z(s j = α k φ k (s j + ε(s j with covariance C(s i,s j = λ k φ k (s i φ k (s j + σ 2 1(i = j To generalize this to a nonstationary case, use spatial powers of the λ k : η(s/ 2 η(t/ C(s,t = λ k λ 2 k φ k (sφ k (t Large η corresponds to smoother field
A simulated example η(s = 0.01 s 2 3 Estimating η(s Regression spline log η(s = β 0 + β 1 s + β j+2 ψ(s;u j ψ(s;u = s u 2 log s u Knots u i picked using clustering techniques Multivariate normal prior on the βʼs. r j=1
Piazza Road revisited Tempering More smoothness More fins structure
Covariances A B C D Karhunen-Loeve expansion revisited Cov(Z(s 1,Z(s 2 = C(s 1,s 2 and Z(s = i=1 = λ i φ i (s 1 φ i (s 2 where α ι are iid N(0,λ i Idea: use wavelet basis instead of eigenfunctions, allow for dependent α i i=1 α i φ i (s
Spatial wavelet basis Separates out differences of averages at different scales Scaled and translated basic wavelet functions Estimating nonstationary covariance using wavelets 2-dimensional wavelet basis obtained from two functions φ and ψ: S(x 1,x 2 = φ(x 1 φ(x 2 H(x 1,x 2 = ψ(x 1 φ(x 2 V(x 1,x 2 = φ(x 1 ψ(x 2 D(x 1,x 2 = ψ(x 1 ψ(x 2 detail functions First generation scaled translates of all four; subsequent generations scaled translates of the detail functions. Subsequent generations on finer grids.
W-transform Covariance expansion For covariance matrix Σ write Σ = ΨDΨ T ; D = Ψ 1 Σ(Ψ T 1 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. 3x3 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
Some open questions Multivariate Kronecker structure Nonstationarity Covariates causing nonstationarity (or deterministic models Comparison of models of nonstationarity Mean structure