Short course on space-time modeling
Instructors:Peter GuttorpJohan LindströmPaul Sampson
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
Kriging
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
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
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)
A Gaussian formula
If
then
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
Some variants
Ordinary kriging (unknown μ)
where
Universal kriging (μ(s)=A(s)βfor some spatial variable A)
where Still optimal for known C.
Universal kriging variance
simple kriging variance
variability due to estimating μ
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.
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.
Nugget Effective range
Sill
Ordinary kriging
where
and kriging variance
An example
Precipitation data from Parana state in Brazil (May-June, averaged over years)
Variogram plots
Kriging surface
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
Prior/posterior of φ
Estimated variogram
ml
Bayes
Prediction sites
1
2
3
4
Predictive distribution
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.