Nonstationary covariance structure I: Deformations

Evidence of anisotropy 15 o red 60 o green 105 o blue 150 o brown

Another view of anisotropy

General setup Z(x,t) =  (x,t) + (x) 1/2 E(x,t) +  (x,t) trend + smooth + error We shall assume that  is known or constant t = 1,...,T indexes temporal replications E is L 2 -continuous, mean 0, variance 1, independent of the error  C(x,y) = Cor(E(x,t),E(y,t)) D(x,y) = Var(E(x,t)-E(y,t)) (dispersion)

Geometric anisotropy Recall that if we have an isotropic covariance (circular isocorrelation curves). If for a linear transformation A, we have geometric anisotropy (elliptical isocorrelation curves). General nonstationary correlation structures are typically locally geometrically anisotropic.

The deformation idea In the geometric anisotropic case, write where f(x) = Ax. This suggests using a general nonlinear transformation. Usually d=2 or 3. G-plane D-space We do not want f to fold.

Implementation Consider observations at sites x 1,...,x n. Let be the empirical covariance between sites x i and x j. Minimize where J(f) is a penalty for non-smooth transformations, such as the bending energy

SARMAP An ozone monitoring exercise in California, summer of 1990, collected data on some 130 sites.

Transformation This is for hr. 16 in the afternoon

Identifiability Perrin and Meiring (1999): Let If (1) f and f -1 are differentiable in R n (2)  (u) is differentiable for u>0 then (f,  ) is unique up to a scaling for  and a homothetic transformation for f (scaling, rotation, reflection)

Richness Perrin & Senoussi (2000): Let f and f -1 be differentiable, and let r(x,y) be continuously differentiable. Then (stationarity) iff Let f(0)=0,c i i th column of. Then (isotropy) iff and

The Brownian sheet Let and Then So the Brownian sheet can be thought of as a stationary deformation. It is however not an istropic deformation.

Estimating variability Resample time slices with replacement from the original data (to maintain spatial structure). Re-estimate deformation based on each bootstrap sample. Kriging estimates can be made based on each of the bootstrap estimates, to get a better sense of the variability.

French rainfall data Altitude-adjusted 10-day aggregated rainfall data Nov-Dec for 39 sites from Languedoc-Rousillon region of France.

Estimated deformation

G-plane equicorrelation contours

D-plane Equicorrelation Contours

Uncertainty in deformation Model refitted using 24 of the 36 sites.

The smoothing parameter Cross-validation: Leave out sampling station i, estimate(θ,f) from remaining n-1 stations. Minimize the prediction error for site i, summed over i. Together with bootstrap estimate of variability, very computer intensive.

Thin-plate splines

A Bayesian implementation Likelihood: Prior: Linear part: fix two points in the G-D mapping put a (proper) prior on the remaining two parameters smoothing parameter

Computation Metropolis-Hastings algorithm for sampling from highly multidimensional posterior. Given estimates of D-plane locations, f(x i ), the transformation is extrapolated to the whole domain using thin plate splines. Predictive distributions for (a) temporal variance at unobserved sites, (b) the spatial covariance for pairs of observed and/or unobserved sites, (c) the observation process at unobserved sites.

California ozone

Posterior samples

Other applications Point process deformation (Jensen & Nielsen, Bernoulli, 2000) Deformation of brain images (Worseley et al., 1999)

Global processes Problems such as global warming require modeling of processes that take place on the globe (an oriented sphere). Optimal prediction of quantities such as global mean temperature need models for global covariances. Note: spherical covariances can take values in [-1,1]–not just imbedded in R 3. Also, stationarity and isotropy are identical concepts on the sphere.

Isotropic covariances on the sphere Isotropic covariances on a sphere are of the form where p and q are directions,  pq the angle between them, and P i the Legendre polynomials. Example: a i =(2i+1)  i

Global temperature Global Historical Climatology Network 7280 stations with at least 10 years of data. Subset with 839 stations with data selected.

Isotropic correlations

Spherical deformation Need isotropic covariance model on transformation of sphere/globe Covariance structure on convex manifolds Simple option: deform globe into another globe Alternative: MRF approach

A class of global transformations Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude). (Das, 2000)

Three iterations

Resulting isocovariance curves

Comparison IsotropicAnisotropic

Assessing uncertainty

Another current climate problem General circulation models require accurate historical ocean surface temperature recordsocean surface temperature Data from buoys, ships, satellites