Some remarks about Lab 1. image(parana.krige,val=sqrt(parana.kri ge$krige.var))...
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Transcript of Some remarks about Lab 1. image(parana.krige,val=sqrt(parana.kri ge$krige.var))...
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