Some remarks about Lab 1

image(parana.krige,val=sqrt(parana.kri ge$krige.var)) contour(parana.krige,loc=loci,add=T)

Dist 22 22  meth line Gau e3315ols Gau436.8 e3232rob Gau e3250wlsc Cir6.3 e57 e4wlsc Exp e64 e5wlsc

likfit(parana,nugget=470,cov.model="gaussian ",ini.cov.pars=c(5000,250)) kappa not used for the gaussian correlation function likfit: likelihood maximisation using the function optim. likfit: estimated model parameters: beta tausq sigmasq phi " 260.6" " 521.1" "6868.9" " 336.2" Practical Range with cor=0.05 for asymptotic range: likfit: maximised log-likelihood =

Nonstationary variance Let  (x) be a Gaussian process with constant mean , constant variance , and correlation. f is the same deformation as for the covariance modelling. Define the variance process Its distribution at gauged sites is

Moments of the variance process Mean: Variance: Covariance: Correlation:

Priors  ~ N( ,  ) The full conditional distributions are then of the same form (Gibbs sampler). To set the hyperparameters we use an empirical approach: Let S ii be the sample variance at site i.

Method of moments Setting the sample moments (over the sites) equal to the theoretical moments we get and let that be the prior mean. The prior variance is set appropriately diffuse.

French precipitation Constant varianceNonconstant variance

Prediction vs estimation Leave out 8 stations, use remaining 31 for estimation Compute predictive distribution for the 8 stations Plot observed variances (incl. nugget) vs. estimated variances and against predictive distribution

Estimated variance field

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:

Some examples Let a i =  i, o≤  <1. Then Let a i =(2i+1)  i. Then Given C(p,q)

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 Deformation of sphere g=(g 1,g 2 ) latitude def longitude def Avoid crossing of latitudes or longitudes Poles are fixed points Equator can be fixed as well

Simple latitude deformation knot Iterated simple deformations

Two-dimensional deformation Let b and  depend on longitude  Alternating deform longitude and latitude. locationscaleamplitude

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