Statistical analysis of extreme events in a non-stationary context via a Bayesian framework: case study with peak-over-threshold data

Abstract

Statistical analysis of extremes currently assumes that data arise from a stationary process, although such an hypothesis is not easily assessable and should therefore be considered as an uncertainty. The aim of this paper is to describe a Bayesian framework for this purpose, considering several probabilistic models (stationary, step-change and linear trend models) and four extreme values distributions (exponential, generalized Pareto, Gumbel and GEV). Prior distributions are specified by using regional prior knowledge about quantiles. Posterior distributions are used to estimate parameters, quantify the probability of models and derive a realistic frequency analysis, which takes into account estimation, distribution and stationarity uncertainties. MCMC methods are needed for this purpose, and are described in the article. Finally, an application to a POT discharge series is presented, with an analysis of both occurrence process and peak distribution.

Appendices

Annex 1: Griddy Gibbs sampling

The aim of this method is to simulate a sample from the posterior distribution $p (\varvec{\theta} | \user2{X}).$ The Gibbs sampling algorithm can be written as follow:

• Choose a starting value ${\varvec{\theta}}^{{(0)}} = {\left({\theta ^{{(0)}}_{1},\ldots,\theta ^{{(0)}}_{k}} \right)},$ and set j = 0.

• Repeat N times:

  • j=j+1;

  • Sample θ ^(j)₁ ∼ p(θ₁ | θ ^{(j −1)}₂ ,...,θ^{(j − 1)}, X),

  • Sample θ ^(j)₂ ∼ p(θ₂ | θ ^(j)₁ ,θ ^{(j − 1)}₃ ,...,θ ^{(j − 1)} _k , X),

  • ...

  • Sample θ ^(j) _q ∼ p(θ _q | θ ^(j)₁ ,...,θ ^(j)_{q − 1} ,θ ^{(j − 1)}_q+1 ,...,θ ^{(j − 1)} _k , X),

  • ... ...

  • Sample θ ^(j) _k ∼ p(θ _k | θ ^(j)₁ ,...,θ ^(j)_{k − 1} , X)

The vectors series $(\varvec{\theta}^{(j)})$ converges to the target posterior distribution as j tends to infinity. In order to decrease the influence of the starting point, the first m iterations are usually deleted, and inference is made with the last N−m iterations. Sensitivity analysis may be necessary to determine acceptable values for m and N (usually, at least a few thousands of iterations are used for m and N). More generally, this method presents the same drawbacks as other iterative simulation techniques. Convergence thus has to be monitored, by choosing several starting points or computing convergence indices. Gelman et al. (1995) or Tanner (1996) provide some guidelines for improving the numerical simulations efficiency.

Unfortunately, the Gibbs sampling algorithm can usually not be used in this raw version, because it involves being able to sample from the full conditional densities p(θ _q | θ ^(j)₁ ,...,θ ^(j)_{q − 1} ,θ ^{(j − 1)}_q+1 ,...,θ ^{(j − 1)} _k ,X ). Ritter and Tanner (1992) proposed the use of a discrete approximation of the cumulative density functions (cdf) of these distributions:

• Choose a grid of points y ₁, ..., y _p .

• Evaluate p (θ _i |θ_1,...,θ_i−1, θ_i+1,...,θ _q , X) on this grid, to obtain w ₁,..., w _p .

• Compute the cumulative sums of w ₁,...,w _p to obtain an approximation of the CDF.

• Sample u from a uniform distribution on [0, 1].

• Transform u by the inverse of the approximate CDF.

This algorithm must be added at each step of the Gibbs iteration. If the full conditional densities are known only up to proportionality, p(θ _q | θ ^(j)₁ ,...,θ ^(j)_{q − 1} ,θ ^{(j - 1)}_q+1 ,...,θ ^{(j − 1)} _k ,X ) can be replaced by f(θ ^(j)₁ ,...,θ ^(j)_{q − 1} ,y,θ ^{(j − 1)}_q+1 ,...,θ ^{(j − 1)} _k ), the product of prior and likelihood, and the cumulative sums of w ₁, ..., w _p must then be divided by the total sum to provide the CDF approximation. Inversion of the CDF can be made with a linear interpolation between two grid points. The grid choice is the most important issue of this technique: it has to be broad enough to cover the range of the distributions, and fine enough to ensure a sufficient accuracy, keeping in mind that this supplementary step is computing-time expensive. Some improvements of the method are described in Ritter and Tanner (1992).

Annex 2: Chib method

The aim of this method is to compute the marginal distribution of the observations, which is the normalizing constant of the Bayes theorem:

$$p({\user2{X}}| M) = \frac{{\pi ({\varvec{\theta }})p({\user2{X}}| {\varvec{\theta }})}}{{p({\varvec{\theta}}| {\user2{X}})}} = \frac{{f({\varvec{\theta}})}}{{p({\varvec{\theta}}| {\user2{X}})}}.$$

(35)

This relationship being true for any vector $\varvec{\theta},$ let us consider a particular ${\varvec{\theta}}^{*} = (\theta ^{*}_{1},\ldots,\theta ^{*}_{k}).$ $f (\varvec{\theta}^{*})$ is directly computable, which is not the case of the denominator. Consider the following relationship:

$$\begin{aligned} p(\theta ^{*}_{1},\ldots,\theta ^{*}_{k} | {\user2{X}}) &= p(\theta ^{*}_{1} | {\user2{X}})p(\theta ^{*}_{2} | \theta ^{*}_{1}, {\user2{X}}) \cdots p(\theta ^{*}_{q} | \theta ^{*}_{1}, \theta ^{*}_{2},\ldots,\theta ^{*}_{{q - 1}}, {\user2{X}}) \\ &\quad \cdots p(\theta ^{*}_{k} | \theta ^{*}_{1},\ldots,\theta ^{*}_{{k - 1}},{\user2{X}}).\\ \end{aligned}$$

(36)

The first term can be evaluated thanks to the sample of the first marginal distribution, by using a Gaussian kernel as an example. The last term can be computed by 1-D numerical integration:

$$p(\theta ^{*}_{k} | \theta ^{*}_{1},\ldots,\theta ^{*}_{{k - 1}},{\user2{X}}) = \frac{{f({\varvec{\theta}}^{*})}}{{{\int {f(\theta ^{*}_{1},\ldots,\theta ^{*}_{{k - 1}},\theta _{k}){\rm d}}}\theta _{k}}}.$$

(37)

The griddy Gibbs sampling can be used to compute intermediary terms. p(θ ^* _q | θ ^*₁ ,θ ^*₂ ,...,θ ^*_{q − 1} ,X) is indeed the first marginal of the distribution p(θ _q ,θ_q+1,...,θ _k | θ ^*₁ , θ ^*₂ ,...,θ ^*_{q − 1} ,X), evaluated at θ ^* _q . The griddy Gibbs algorithm can thus be applied to the non-normalized posterior density with the first q − 1 components being fixed, that is f(θ ^*₁ , θ ^*₂ ,...,θ ^*_{q − 1} , θ _q ,θ _q+1,...,θ _k ).

Although this approach is theoretically valid for any value $\varvec{\theta}^{*}$ with non-zero posterior probability, Chib recommends the use of a high-density point to increase the method accuracy.