Bayesian modelling of extreme wind speed at Cape Town, South Africa

Abstract

In the framework of generalized extreme value (GEV) distribution, the frequentist and Bayesian methods have been used to analyse the extremes of annual maxima wind speed recorded by automatic weather stations in Cape Town, Western Cape, South Africa. In the frequentist approach, the GEV distribution parameters were estimated using maximum likelihood, whereas in the Bayesian method the Markov Chain Monte Carlo technique with the Metropolis–Hastings algorithm was used. The results show that the GEV model with trend in the location parameter appears to be a better model for annual maxima data. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from other weather stations. The results from the Bayesian analysis show that posterior inference might be affected by the choice of priors and hence by the distance between a weather station used to formulate the priors and the point of interest.

Appendices

Appendix 1: The profile likelihood confidence intervals

Let ${\varvec{\theta }}$ = ($\mu$, $\sigma$, $\xi$) be a vector of GEV model parameters and $\ell ({\varvec{\theta }})$ be the log-likelihood for ${\varvec{\theta }}$, given in Eq. (5). Then the profile likelihood intervals for return levels are constructed as follows.

Re-parameterize the GEV model parameters ${\varvec{\theta }}$ = ($\mu$, $\sigma$, $\xi$) so that $y_p$ becomes one of the GEV model parameters. Using Eq. (6), for example, the location parameter $\mu$ can be expressed as

$$\begin{aligned} \mu = y_p + \frac{\sigma }{\xi }\left[ 1 - \{- \log (1 - p)\}^{-\xi } \right] \end{aligned}$$

for $\xi \ne 0$. Let $\ell (y_p, \sigma , \xi )$ be the log-likelihood for ${\varvec{\theta }}$ = ($y_p$, $\sigma$, $\xi$).

Now partition ${\varvec{\theta }}$ = ($y_p$, $\sigma$, $\xi$) into two components (${\varvec{\theta }}_{(1)}$, ${\varvec{\theta }}_{(2)}$) where ${\varvec{\theta }}_{(1)} = y_p$ and ${\varvec{\theta }}_{(2)}$ = ($\sigma$, $\xi$). The profile log-likelihood for ${\varvec{\theta }}_{(1)} = y_p$ is defined as

$$\begin{aligned} \ell _p(y_p) = \max _{{\varvec{\theta }}_{(2)}}\ell (y_p, {\varvec{\theta }}_{(2)}). \end{aligned}$$

That is, for a given value of $y_p$, the profile log-likelihood is the maximized log-likelihood with respect to ${\varvec{\theta }}_{(2)}$ = ($\sigma$, $\xi$).

Let $\hat{{\varvec{\theta }}}$ be the MLE of ${\varvec{\theta }}$. Then under suitable regularity conditions, for large m

$$\begin{aligned} D_p(y_p) = 2\,\{\ell (\hat{{\varvec{\theta }}}) - \ell _p(y_p) \} \sim \chi _1^2. \end{aligned}$$

Thus, the set of values, say $C_{\alpha }$, for which $\{y_p: D_p(y_p) \le c_{\alpha } \}$ provides a ($1 - \alpha$) confidence interval for $y_p$, where $c_{\alpha }$ is the ($1-\alpha$) quantile of the $\chi _1^2$ distribution. The ($1 - \alpha$)100% profile likelihood confidence interval for $y_p$ is the set of values $C_{\alpha }$ given by the points of intersection between the line $\frac{1}{2} \times \chi ^2_1(\alpha )$ and the profile log-likelihood for $y_p$.

Appendix 2: R codes used for analysis