Estimating Design discharge for the river Rhine using Bayesian analysis

This case study is based on a research paper J.M. van Noortwijk, H.J. Kalk and E.H. Chbab: ”Bayesian estimation of design loads”. ( PDF). In this case study the paper is referred to as vNK&C.

Introduction

This case study presents an example of parameter estimation and model comparison within a Bayesian framework. The following step provide a walk-through of the problem:

• Flooding of the river Rhine (this case study relates to the Lobith region) presents a considerable risk.

• Flood defences are required; however:

  • Building excessive defences wastes valuable resources and money; but

  • Building inadequate defences exposes the region to flood risk.

Therefore, a “design discharge” i.e., the maximum discharge for which the defences remain effective, is defined. In the Netherlands, the design discharge is the discharge that should statistically occur only once every 1250 years. This design discharge is chosen to balance the expense of defence construction against the risk of flood inundation.

However, in vNK&C, only 97 years of data (1901 - 1998) are available for defining an exceedance probability distribution from which to extrapolate the discharge associated with a 1 in 1250 year event.

vNK&C describe how a number of candidate probability density functions fit the observed data reasonably well; however, each function generates a different value when extrapolated to the 1 in 1250 discharge.

In the present study, vNK&C apply modern Bayesian methods to estimate the parameters for 9 candidate probability density functions and also to compare the functions using Bayes weights (Bayes factors): these provides the odds that the observed data was generated by each of the prospective distributions.

The numerical methods used by vNK&C for Bayesian estimation and bayes weights is involved, an overwiew of the methods is available in Kass and Raftery (1995) ( postscript). The homepage for Bayesian Inference for the Physical Sciences (BIPS) provides many useful links for anyone interested in finding out more about Bayesian methods.

Methods

vNK&C tackled the problem in two stages:

• (1) The parameters of the prospective density functions were estimated using Bayesian methods.

• (2) Bayes weights were determined for each density function. The density functions considered were:

  • Exponential,

  • Rayleigh,

  • Normal,

  • Lognormal,

  • Gamma,

  • Weibull,

  • Gumbel,

  • generalised gamma,

  • generalised extreme value,

Parameter estimation:

The inputs required for Bayes theorem, in order to perform parameter estimation were:

• A likelihood function. This represents the inherent uncertainty in the observations assuming the model that generated them is correct.

• A prior density function for the model parameters. This represents prior knowledge of the model parameter uncertainty. vNK&C chose “non-informative priors” i.e., the prior assumptions about the values for the parameters describing the “shape” of the prospective density functions is as weak as possible, allowing the observational data to “speak for itself”. Rather than uniform priors which can be shown to be unsatisfactory (see for example Ibrahim and Laud (1991) or the definitive text book: Box and Tiao (1973)), vNK&C define appropriate Jeffreys priors for each distribution.

The result of applying Bayes theorem to the above likelihhood function and model parameter prior density function is to generate a posterior model parameter density function that incorporates observational data and model uncertainty.

From the resulting posterior parameter density function, the posterior predictive probability of exceedance for a given discharge could then be found.

Figure 1 (reproduced with kind permission of the authors) shows the results.

Figure 1: Predictive exceedance probability of annual maximum river Rhine discharge.

Bayes Weights:

Having calculated the parameters and exceedance probabilities using Bayes theorem, the next task carried out by vNK&C was to calculate the odds that each of the prospective distribution was responsible for the observed data using the following steps:

• An assumption is made that the data were generated by one of m prospective models.

• Assume an equal prior probability that any of the models produced the data.

• Bayes theorem was used to computer the posterior probability that model i produced the data (where i = 1,…,m).

Table 1 shows the results, also shown is the estimated 1/1250 quantile for each distribution (with a maximum likelihood estimate included for comparison).

Comment

vNK&C have shown that Bayes estimates and Bayes weights can be used to estimate a design discharge in a way that takes into account both parameter uncertainty and distribution-type uncertainty.

The Rayleigh distribution provided the highest Bayes weight. vNK&C suggest this may partly be due to the low number of parameters describing this model as Bayesian analysis automatically penalises models with many parameters i.e., if two models fit the data equally well, the model with the lowest dimensional parameter space will generate a higher Bayes weight.

The design discharge is higher using Bayesian estimation compared to maximum likelihood. This result demonstrates how a realistic consideration of model parameter uncertainty calls for a higher margin of error in the design discharge.

References

• Box, G. E. P. and G. C. Tiao (1973). Bayesian inference in statistical analysis. Reading [Mass.]? ; London, Addison-Wesley.

• Ibrahim, J. G. and P. W. Laud (1991). "On Bayesian Analysis of Generalized Linear Models Using Jeffrey's Prior." Journal of the American Statistical Association 86(416): 981-986.

• Kass, R. E. and A. E. Raftery (1995). "Bayes Factors." Journal of the American Statistical Association 90(430): 23.

• van Noortwijk, J. M., H. J. Kalk and E. H. Chbab (2004). "Bayesian estimation of design loads." HERON 49(2): 189-205.