short course on space-time modeling instructors: peter guttorp johan lindström paul sampson
TRANSCRIPT
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Short course on space-time modeling
Instructors:Peter GuttorpJohan LindströmPaul Sampson
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Schedule9:10 – 9:50 Lecture 1: Kriging9:50 – 10:30 Lab 110:30 – 11:00 Coffee break11:00 – 11:45 Lecture 2:
Nonstationary covariances11:45 – 12:30 Lecture 3: Gaussian
Markov random fields12:30 – 13:30 Lunch break13:30 – 14:20 Lab 214:20 – 15:05 Lecture 4: Space-
time modeling15:05 – 15:30 Lecture 5: A case
study15:30 – 15:45 Coffee break15:45 – 16:45 Lab 3
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Kriging
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The geostatistical model
Gaussian processμ(s)=EZ(s) Var Z(s) < ∞Z is strictly stationary if
Z is weakly stationary if
Z is isotropic if weakly stationary and
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The problem
Given observations at n locationsZ(s1),...,Z(sn)
estimateZ(s0) (the process at an unobserved
location)
(an average of the process)
In the environmental context often time series of observations at the locations.
or
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Some history
Regression (Bravais, Galton, Bartlett)Mining engineers (Krige 1951, Matheron, 60s)Spatial models (Whittle, 1954)Forestry (Matérn, 1960)Objective analysis (Gandin, 1961)More recent work Cressie (1993), Stein (1999)
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A Gaussian formula
If
then
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Simple krigingLet X = (Z(s1),...,Z(sn))T, Y = Z(s0), so that
μX=μ1n, μY=μ, ΣXX=[C(si-sj)], ΣYY=C(0), and
ΣYX=[C(si-s0)].
Then
This is the best unbiased linear predictor when μ and C are known (simple kriging).
The prediction variance is
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Some variants
Ordinary kriging (unknown μ)
where
Universal kriging (μ(s)=A(s)βfor some spatial variable A)
where Still optimal for known C.
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Universal kriging variance
simple kriging variance
variability due to estimating μ
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The (semi)variogram
Intrinsic stationarityWeaker assumption (C(0) needs not exist)Kriging predictions can be expressed in terms of the variogram instead of the covariance.
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The exponential variogram
A commonly used variogram function is γ(h) = σ2 (1 – e–h/ϕ. Corresponds to a Gaussian process with continuous but not differentiable sample paths.More generally,
has a nugget τ2, corresponding to measurement error and spatial correlation at small distances.
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Nugget Effective range
Sill
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Ordinary kriging
where
and kriging variance
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An example
Precipitation data from Parana state in Brazil (May-June, averaged over years)
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Variogram plots
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Kriging surface
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Bayesian kriging
Instead of estimating the parameters, we put a prior distribution on them, and update the distribution using the data.Model:
Matrix withi,j-elementC(si-sj; φ)(correlation)
measurementerror
θβσφτT
(Z(s1)...Z(sn))T
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Prior/posterior of φ
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Estimated variogram
ml
Bayes
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Prediction sites
1
2
3
4
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Predictive distribution
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References
A. Gelfand, P. Diggle, M. Fuentes and P. Guttorp, eds. (2010): Handbook of Spatial Statistics. Section 2, Continuous Spatial Variation. Chapman & Hall/CRC Press.
P.J. Diggle and Paulo Justiniano Ribeiro (2010): Model-based Geostatistics. Springer.