On the Exact Distribution of Correlated Extremes in Hydrology

Abstract

The analysis of hydrological hazards usually relies on asymptotic results of extreme value theory, which commonly deals with block maxima or peaks over threshold (POT) data series. However, data quality and quantity of block maxima and POT hydrological records do not usually fulfill the basic requirements of extreme value theory, thus making its application questionable and results prone to high uncertainty and low reliability. An alternative approach to better exploit the available information of continuous time series and nonextreme records is to build the exact distribution of maxima (i.e., nonasymptotic extreme value distributions) from a sequence of low-threshold POT. Practical closed-form results for this approach do exist only for independent high-threshold POT series with Poisson occurrences. This study introduces new closed-form equations of the exact distribution of maxima taken from low-threshold POT with magnitudes characterized by an arbitrary marginal distribution and first-order Markovian dependence, and negative binomial occurrences. The proposed model encompasses and generalizes the independent-Poisson model and allows for analyses relying on significantly larger samples of low-threshold POT values exhibiting dependence, temporal clustering, and overdispersion. To check the analytical results, we also introduce a new generator (called Gen2Mp) of proper first-order Markov chains with arbitrary marginal distributions. An illustrative application to long-term rainfall and streamflow data series shows that our model for the distribution of extreme maxima under dependence takes a step forward in developing more reliable data-rich-based analyses of extreme values.

Towards Generic Simulation for Demanding Stochastic Processes

“…For a process with exponential distribution, which is a subcase of the gamma distribution, there exist generation algorithms for the case of short-range (Markov) dependence (e.g., [46]). As already mentioned, a more general algorithm for generation of any type of marginal distribution has recently been proposed by Lombardo et al [28], but again under the condition of the Markov dependence. However, the method proposed here can generate such a process irrespective of the type of the dependence, whether SRD or LRD.…”

“…A more general algorithm for generation of any type of marginal distribution was recently proposed by Lombardo et al [28], but only under the condition of Markov dependence, thus leaving out problems with more complex dependence, including LRD. Recent advances include the use of machine learning methods in stochastic simulation, e.g., [29], which, however, have the disadvantages of being implicit in their mathematical structure, and non-parsimonious.…”

Towards Generic Simulation for Demanding Stochastic Processes

“…[47]). As already mentioned, a more general algorithm for generation of any type of marginal distribution has recently been proposed by Lombardo et al [28] but again under the condition of Markov dependence. However, the method proposed here can generate such a process irrespective of the type of the dependence, whether SRD or LRD.…”

“…A more general algorithm for generation of any type of marginal distribution has recently been proposed by Lombardo et al [28] but only under the condition of Markov dependence, thus leaving out problems with more complex dependence, including LRD.…”

Stochastic investigation of hydrological extremes

“…De Michele (2019) provides a review of approaches to derive the exact distribution of maxima without assuming 'identically distributed' extremes. With a similar rationale but focused on relaxing the independence assumption, Lombardo et al (2019) derive the exact distribution of maxima taken from low threshold POT with magnitudes characterized by an arbitrary marginal distribution and first-order Markovian dependence, and negative binomial occurrences. Volpi et al (2015) derive the distribution function of the waiting time for processes with Markovian dependence, while Serinaldi and Lombardo (2020) derive the probability distribution of the waiting time till the th extreme also under longrange dependence.…”