Multivariate return period calculation via survival functions

Abstract

[1] The concept of return period is fundamental for the design and the assessment of many engineering works. In a multivariate framework, several approaches are available to its definition, each one yielding different solutions. In this paper, we outline a theoretical framework for the calculation of return periods in a multidimensional environment, based on survival copulas and the corresponding survival Kendall's measures. The present approach solves the problems raised in previous publications concerning the coherent foundation of the notion of return period in a multivariate setting. As an illustration, a practical hydrological application is presented.

Copulas Based on Marshall–Olkin Machinery

“…Notice that, as consequence of [27,Proposition 2], when the number of parameters is equal to the number of pairs d(d − 1)/2, then the estimator given by (11) does not depend on the weights. These previous estimates help to quantify the critical levels and return periods corresponding to this dataset (see, e.g., [30,31]). In hydrology, a critical level p corresponding to a return period T is defined through the relationship T = 1 1 − P(C (F 1 (Y 1 ) Figure 6 shows the estimated critical levels, along with confidence intervals, associated to the fitted dataset.…”

Estimation procedures for exchangeable Marshall copulas with hydrological application

“…Here we present an illustration of the estimation procedures for EM copulas in an hydrological application. The data we are considering are collected at the Ceppo Morelli dam, and are essentially the same as those investigated in De Michele et al (2005) (see also Salvadori et al ( , 2013a), to which we make reference for further details. The dam is located in the valley of Anza catchment, a sub-basin of the Toce river (Northern Italy), and was built to produce hydroelectric energy.…”

Probability Analysis of the Water Table and Driving Factors Using a Multidimensional Copula Function

“…In recent years, it has been widely applied to various aspects of hydrological studies [13]. The main applications of the Copula Function include the analysis of: precipitation characteristics [12], the correlation between flood peak and flood volume [14,15], the frequency and recurrence interval of floods [16][17][18], characteristics of storms [19,20], the frequency and recurrence intervals of droughts [21], drought assessment [22], risk assessment [23], and an assessment of environmental hydrological model performance [24]. Due to the specialty of the water table, the establishment of the Copula Function for this purpose is relatively difficult; therefore, no research that uses the Copula Function to analyze the water table has been reported.…”