Expert elicitation of the variogram

Phuong N. TruongGerard B.M. Heuvelink

John Paul Gosling

Wageningen University (NL)Food and Environmental Research Agency (UK)

Pedometrics 2011

• It is a measure of spatial variability of soil properties

• It is required for kriging and spatial stochastic simulation

The variogram is a key tool in pedometrics

But it is not always available:

• At least 200 observations are required to estimate it reliably (perhaps 60 will do)

• In many cases this may be too expensive or otherwise impossible

ASK EXPERTS!

What else can we do to estimate the variogram?

But how should expert information be derived?

• We often read “...was derived using expert knowledge...”, without further explanation

• But much can go wrong when consulting experts• Must therefore make use of formal and

established procedures: expert elicitation

• Aims to construct a probability distribution that properly represents the expert’s knowledge

• Scientific field in its own right, many text books, conferences and journals

• Involves contributions from statistics and psychology (understanding human judgement)

Expert elicitation

1. Background and preparation2. Identify and recruit expert(s)3. Motivating and training the expert(s), e.g. to

avoid overconfidence and anchoring4. Structuring and decomposition (ensure that

expert agrees with how the problem is structured)

5. The elicitation itself6. Assess the adequacy of the elicitation

Expert elicitation procedure involves six steps

1. Elicitation of quantiles, i.e. ask experts for a value such that for several values for

2. Start with median (, next use bisection method and move to quartiles, and so on

3. Fit a suitable distribution

4. Show result to expert and allow modification

5. Aggregate results from multiple experts (two main modes: ‘behavioural’ and ‘mathematical’)

Closer look at step 5 in case of elicitation of a univariate

continuous distribution by multiple experts:

1. Choose lags for which the semivariance is to be elicited

2. Elicit semivariance for each lag using Cressie’s robust estimator:

Our approach to expert elicitation of the variogram

2 γ̂ (|h|)=(median(|𝑍 (𝑥 )−𝑍 (𝑥+h )|))2/0.457

3. Fit variogram using least squares

4. Show simulated realisations along transect and allow expert to modify their judgement

5. Pool variograms of multiple experts using mathematical aggregation

Expert Geostatistician

Cannot ask:What are the nugget, sill, range and shape of the variogram?

Can ask:Could you determine a value , such that the absolute difference of the variable of interest at two locations a distance apart, is equally likely to be less than or greater than this value ?

Summary of variogram elicitation procedure

Individual expertPooled pdf Feedback

Not satisfy

Elicitation of the variogram

Satisfy

Pooled variogram

SimulationIndividual expert Feed back

Not satisfy

End whole elicitation procedure Satisfy

End elicitation procedure for marginal pdf

Elicitation of marginal

probability distribution

Let’s try a live demo (but screendumps prepared as a

back-up)

Web-based implementation at

www.variogramelicitation.org

• Expert elicitation of variograms is possible but takes a lot of effort

• Web-based tool ready to be used, feel free to try it out at www.variogramelicitation.org

• No real case study yet, but we plan to elicit the variogram of soil pH for a region in the UK

• Variogram elicitation useful in case of too few or no data, and also for:– Spatial sample design optimization for minimization of

the average (universal) kriging variance (Brus & Heuvelink, Geoderma 138, pp. 86-95)

– Use as a prior in Bayesian estimation of the variogram

Conclusions

Thank you!

Starting page

Round 1: marginal distribution

Feedback round 1

First page round 2

Elicitation of semivariance at seven lags

Feedback by simulated reality along transect