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.