Analysis of non-stationary climate-related extreme events considering climate change scenarios: an application for multi-hazard assessment in the Dar es Salaam region, Tanzania

Abstract

In this paper we have put forward a Bayesian framework for the analysis and testing of possible non-stationarities in extreme events. We use the extreme value theory to model temperature and precipitation data in the Dar es Salaam region, Tanzania. Temporal trends are modeled writing the location parameter of the generalized extreme value distribution in terms of deterministic functions of explanatory covariates. The analyses are performed using synthetic time series derived from a Regional Climate Model. The simulations, performed in an area around the Dar es Salaam city, Tanzania, take into account two Representative Concentration Pathways scenarios from the Intergovernmental Panel on Climate Change. Our main interest is to analyze extremes with high spatial and temporal resolution and to pursue this requirement we have adopted an individual grid box analysis approach. The approach presented in this paper is composed of the following key elements: (1) an advanced Bayesian method for the estimation of model parameters, (2) a rigorous procedure for model selection, and (3) uncertainty assessment and propagation. The results of our analyses are intended to be used for quantitative hazard and risk assessment and are presented in terms of hazard curves and probabilistic hazard maps. In the case study we found that for both the temperature and precipitation data, a linear trend in the location parameter was the only model performing better than the stationary one in the areas where evidence against the stationary model exists.

Appendices

Appendix 1: The GEV family of distributions)

Denoting daily observations by $x_1, x_2, \ldots,$ the classical model for extremes is obtained by analyzing the behavior of $\mathbf{z}_i = \hbox { max}\{x_1, x_2, \ldots , x_i \}$ for large values of $i$. Asymptotic considerations suggest that the distribution of a series of measures of extreme physical events extracted from long sequences of data approaches one of the three families of distributions that are combined into the GEV (e.g., Jenkinson 1955; Martins and Stedinger 2000; Coles 2001). The GEV family of distributions has a distribution function of the form:

$$\begin{aligned} G(z|\mu ,\sigma ,\xi ) = {\mathrm{exp}} \left\{ - \left[ 1 + \xi \left( \frac{z - \mu }{\sigma } \right) \right] ^{-1/\xi } \right\} \end{aligned}$$

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defined on the set $\{z{:} 1+ \xi (z-\mu )/\sigma > 0 \},$ and with parameter space $(\mu , \sigma , \xi ){:} \mu \in \mathbb {R}$ (location parameter), $\sigma \ge 0$ (scale parameter), and $\xi \in \mathbb {R}$ (shape parameter) (e.g., Coles 2001). The Frechét (or Type II) and Weibull (or type III) classes of EV distribution correspond, respectively, to the cases $\xi > 0$ and $\xi < 0$ in this parameterization. These two cases have bounds in the domain of $z$: for instance, $\mu - \sigma /\xi \le z < \infty$ for $\xi > 0$, and $-\infty < z \le \mu - \sigma /\xi$ for $\xi < 0$. The special case of the GEV distribution obtained for $\xi \rightarrow 0$ is the Gumbel (or Type I) class, with distribution function:

$$\begin{aligned} G(z|\mu ,\sigma ,\xi ) = {\mathrm{exp}} \left\{ - {\mathrm{exp}} \left[ - \left( \frac{z - \mu }{\sigma } \right) \right] \right\};\quad z \in \mathbb {R} \end{aligned}$$

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Appendix 2: Bayesian inference of model parameters

In Bayesian data analysis the model parameters are treated as random variables to account for the imperfect knowledge of their exact values. Beyond the information provided by the data, a Bayesian framework allows to incorporate other sources of information that may be available and that may be encoded to construct the prior density function of the model parameters. The prior information may be available from different sources as past studies in the same or similar regions, global or regional information, or subjective information of experts. The important point to be outlined is that the prior distribution must be formulated independently of the data used for the likelihood. The prior is then a probability distribution which should reflect the knowledge (or lack of) about a parameter before seeing the data. The Bayes theorem is then used to update the prior probability density with the information provided by the data to obtain the posterior distribution:

$$\begin{aligned} p(\theta | \mathbf{z}) = \frac{f(\mathbf{z}|\theta )\pi (\theta )}{\int f(\mathbf{z}|\theta )\pi (\theta ) {\mathrm{d}}\theta } \end{aligned}$$

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where $\mathbf{z} = (z_1, z_2, \ldots , z_n)$ is the vector of data, $\pi (\theta )$ the prior density of the model parameters, $f(\mathbf{z}|\theta )$ the likelihood of the data, and $p(\theta | \mathbf{z})$ the posterior probability density, which is the conditional distribution of the parameters given the observed data. All inference about the parameters is based on the posterior distribution.

In most multi-dimensional cases, posterior simulations are needed for Bayesian inference. To get samples from the posterior distribution we use a Markov chain Monte Carlo (MCMC) method. The MCMC is a general method based on drawing values of $\theta$ from approximate distributions and then correcting those draws to better approximate the target posterior distribution, $p(\theta |\mathbf{z})$. The samples are drawn sequentially with the distribution of the sampled draws depending on the last value drawn (hence, the draws form a Markov Chain). The key to MCMC simulations is to create a Markov process whose stationary distribution is the specified $p(\theta |\mathbf{z})$ and run the simulations long enough that the distribution of the current draws is close enough to this stationary distribution (e.g., Gelman et al. 2004). In our application, we construct the Markov chains via the Metropolis–Hastings algorithm (Metropolis and Ulam 1949; Metropolis et al. 1953). After running the Markov chain, we remove the burn-in period and check the convergence of the simulated sequences. Different approaches are available for this procedure (e.g., Geweke 1992; Geyer 1992; Gelman and Rubin 1992); in our analyses we have implemented the method proposed by Geweke (1992) (often referred as the Geweke-z-score).

Appendix 3: Bayes factor: the Laplace–Metropolis estimator

The Bayes factor, $B_{kl}$, for comparing model $M_k$ to Model $M_l$ for observed data $\mathbf{z}$, is the ratio of the posterior odds for $M_k$ against $M_l$ to the prior odds. When the models $M_k$ and $M_l$ are equally probable a priori, then $B_{kl}$ reduces to:

$$\begin{aligned} B_{kl} = \frac{f(\mathbf{z}|M_l)}{f(\mathbf{z}|M_k)} \end{aligned}$$

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It implies computing the integrated likelihoods for model $M_m$ (also called the marginal likelihood, marginal probability of the data, or predictive probability of the data) that has the form

$$\begin{aligned} f(\mathbf{z}|M_m) = \int f(\mathbf{z}|\theta _m, M_m)f(\theta _m|M_m) \hbox {d}\theta _m \end{aligned}$$

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where $\theta _m$ is the vector of parameters in model $M_m$, and $f(\theta _m|M_m)$ is its prior density (for more details see e.g., Kass and Raftery 1995; Lewis and Raftery 1997b).

To estimate the marginal likelihoods for the Bayes factor calculation we have implemented the Laplace–Metropolis estimator (Raftery 1996; Lewis and Raftery 1997a), which uses the posterior simulation output to estimate the integrated likelihoods. Letting $h(\theta ) \equiv \hbox { log}\{ f(\theta )f(\mathbf{z}|\theta )\}$ (the notation showing the conditioning respect to $M_m$—as in Eq. 12—has been dropped for simplicity) and applying the Laplace approximation for an integral, the following approximation for the integrated likelihood is obtained (Raftery 1996; Lewis and Raftery 1997a):

$$\begin{aligned} f(\mathbf{z}) \approx (2\pi )^{P/2}|\mathbf{H^*}|^{1/2}f(\theta ^*)f(\mathbf{z}|\theta ^*) \end{aligned}$$

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where $\theta ^*$ is the value of $\theta$ at which $h$ attains its maximum and $\mathbf{H^*}$ is minus the inverse Hessian of $h$ evaluated at $\theta ^*$ (Lewis and Raftery 1997a). For numerical reasons we use Eq. 13 in a logarithmic scale ($\hbox {log}\{f(\mathbf{z})\}$). To calculate the Laplace–Metropolis estimator we use the posterior samples generated by the Metropolis–Hastings algorithm to estimate both $\theta ^*$ and $\mathbf{H}^*$. In practice, the components of $\theta ^*$ are estimated calculating the component-wise posterior medians from the sample, whereas $\mathbf{H}^*$, being asymptotically equal to the posterior variance matrix, is estimated using the sample covariance matrix of the posterior simulation output.