Some remarks about Lab 1

image(parana.krige,val=sqrt(parana.krige$krige.var))

contour(parana.krige,loc=loci,add=T)

Dist 2 2 meth line

Gau 582 6.3 e3 315 ols

Gau 43 6.8 e3 232 rob

Gau 273 5.5 e3 250 wlsc

Cir 6.3 e5 7 e4 wlsc

Exp 437 5.2 e6 4 e5 wlsc

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: 16808.82

likfit: maximised log-likelihood = -669.3

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

ρϑ f(x) − f(y)( )

ν(x) = exp(η(x))

rν[ ] ∝

1

νi∏%Σ

− 12

×exp(− 12 (log(

rν) − μ1)T %Σ−1(log(

rν) − μ1))

%2

Moments of the variance process

Mean:

Variance:

Covariance:

Correlation:

E(ν(x)) =exp( + 12 %

2 )

Var(ν(x)) =exp(2 + %2 ) exp(%2 ) −1( )

Cov(ν(x), ν(y)) =exp(2 + %2 )

× exp %2ρϑ ( f(x) −f(y){ } −1( )

Corr(ν(x), ν(y)) =

exp %2ρϑ ( f(x) −f(y){ } −1

exp(%2 −1)

Priors

~ N(,)

The full conditional distributions are then of the same form (Gibbs sampler).

To set the hyperparameters we use an empirical approach: Let Sii be the sample variance at site i.

%−2 ~ Γ(γ,δ)

ESii =E E(S ii νi )( ) =E(νi ) =exp( + 12 %

2 )

Var(Sii ) =Var E(S ii νi( ) +EVar(S ii νi )

=Exp(2 + %2 )(exp(%2 ) −1)

Method of momentsSetting 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.

=log(S) − 12 log

1n (Sii − S)2

i=1

n

∑ + S2

S2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

1%2 =−log

1n (S ii −S)

2

i=1

n

∑ +S2

S2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

French precipitation

Constant variance Nonconstant

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 R3.

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 Pi the Legendre polynomials:

€

C(p,q) = aii= 0

∞

∑ Pi (cosγpq )

Pn (x) =1

2nn!dn

dxnx2 −1( )

n⎡⎣

⎤⎦

Some examples

Let ai=ρi, o≤ρ<1. Then

Let ai=(2i+1)ρi. Then

Given C(p,q)

€

C(p,q) =1− ρ2

1− 2ρcos γpq + ρ2 − 1

C(p,q) =1

(1−2ρcos pq + ρ2 )

12

ak =2k + 14π

C(p,q)Pk (cos pq )dΩqq∫

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data. Subset with 839 stations with data 1950-1991 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=(g1,g2)

latitude def

longitude def

Avoid crossing of latitudes or longitudes

Poles are fixed points

Equator can be fixed as well

g1 : S2 → [−90,90]

g2 : S2 → [−180,180]

Simple latitude deformation

f(θ) =

(1−b)90 −ξ

(θ −ξ)2 +b(θ −ξ) + ξ, ξ ≤θ ≤90

−(1−b)90 −ξ

(θ −ξ)2 +b(θ −ξ) + ξ, ξ > θ ≥−90

⎧

⎨⎪⎪

⎩⎪⎪

knot

Iterated simpledeformations

Two-dimensional deformation

Let b and ξ depend on longitude

Alternating deform longitude and latitude.

ξ(φ) = α ξ exp(−cos2 (φ− ηξ )

βξ

)

locationscaleamplitude

Three iterations

Resulting isocovariance curves

Comparison

Isotropic Anisotropic

Assessing uncertaintyCov(Z(p),Z(q)) =2r(p,q)

Another current climate problem

General circulation models require accurate historical ocean surface temperature records

Data from buoys, ships, satellites