The tales that the distribution tails of non-Gaussian autocorrelated processes tell: efficient methods for the estimation of the k-length block-maxima distribution

ABSTRACT

The focal point of this work is the estimation of the distribution of maxima without the use of classic extreme value theory and asymptotic properties, which may not be ideal for hydrological processes. The problem is revisited from the perspective of non-asymptotic conditions, and regards the so-called exact distribution of block-maxima of finite-sized k-length blocks. First, we review existing non-asymptotic approaches/models, and introduce an alternative and fast model. Next, through simulations and comparisons (using asymptotic and non-asymptotic models), involving intermittent processes (e.g. rainfall), we highlight the capability of non-asymptotic approaches to model the distribution of maxima with reduced uncertainty and variability. Finally, we discuss an alternative use of such models that concerns the theoretical estimation of the multi-scale probability of obtaining a zero value. This is a useful finding when the scope is the multi-scale modelling of intermittent hydrological processes (e.g. intensity–duration–frequency models). The work also provides step-by-step recipes and an R package.

KEYWORDS:

• non-asymptotic distribution of k-length block maxima
• non-Gaussian marginal distribution
• autocorrelated processes
• intermittent processes
• Gaussian copula

Editor A. Castellarin Associate editor I. Prosdocimi

Editor A. Castellarin Associate editor I. Prosdocimi

Acknowledgements

I thank the Associate Editor and the three reviewers for their constructive comments and suggestions that significantly improved the content and quality of the manuscript. Also, I acknowledge the motivational and always fruitful discussions on the broader topic of stochastics with my NTUA colleagues, Panagiotis Kossieris, Andreas Efstratiadis, Demetris Koutsoyiannis and Christos Makropoulos.

Disclosure statement

No potential conflict of interest was reported by the author.

Data and software availability

The analysis, simulations, and visualization were performed in R language (R Core Team 2017). The source code for the anySim R package (Tsoukalas et al. 2020) is available at the GitHub repository: https://github.com/itsoukal/anySim. The source code for the bBextremes R package (introduced in this work) is available at the GitHub repository: https://github.com/itsoukal/bBextremes. Also, all the figures/plots presented herein were produced via ggplot2 R package (Wickham 2016).

Supplementary material

Supplemental data for this article can be accessed here

Notes

¹ The simulations were performed using the anySim R package (Tsoukalas et al. 2020), which is capable of simulating Nataf-based (i.e. Gaussian copula) processes with any marginal distribution (with finite variance) and (valid; i.e. positive definite) correlation structure (temporal, spatial or combination of them).

² The distribution fitting was performed using the lmom R package (Hosking 2019). In particular, for the $\mathcal{B}rXII$ distribution we employed the pelp function which, as described, provides “Parameter estimation for a general distribution by the method of Lmoments.” It is noted that for computational efficiency, and optimization-related reasons, in the case of $\mathcal{B}rXII$ distribution (as well as for other models comprising a scale and multiple shape parameters), the argument type should be set to “s” (for more details see p. 37 of the package manual).

Additional information

Funding

This research is co-financed by Greece and the European Union (European Social Fund – ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning” in the context of the project “Reinforcement of Postdoctoral Researchers – 2nd Cycle” (MIS-5033021), implemented by the State Scholarships Foundation (ΙΚΥ).