Asymmetric Dependence in Hydrological Extremes

Extremal dependence describes the strength of correlation between the largest observations of two variables. It is usually measured with symmetric dependence coefficients that do not depend on the order of the variables. In many cases, there is a natural asymmetry between extreme observations that cannot be captured by such coefficients. An example for such asymmetry is large discharges at an upstream and a downstream stations on a river network: an extreme discharge at the upstream station will directly influence the discharge at the downstream station, but not vice versa. Simple measures for asymmetric dependence in extreme events have not yet been investigated. We propose the asymmetric tail Kendall's tau as a measure for extremal dependence that is sensitive to asymmetric behavior in the largest observations. It essentially computes the classical Kendall's tau but conditioned on the extreme observations of one of the two variables. We show theoretical properties of this new coefficient and derive a formula to compute it for existing copula models. We further study its effectiveness and connections to causality in simulation experiments. We apply our methodology to a case study on river networks in the United Kingdom to illustrate the importance of measuring asymmetric extremal dependence in hydrology. Our results show that there is important structural information in the asymmetry that would have been missed by a symmetric measure. Our methodology is an easy but effective tool that can be applied in exploratory analysis for understanding the connections among variables and to detect possible asymmetric dependencies. Compound events describe situations where the simultaneous behavior of two or more variables lead to severe impacts. For instance, the dependence between climate or hydrological variables can lead to particular conditions that result in extreme events at the same time in different locations. Since the physical processes behind these phenomena are very complex, there can be a stronger influence from one variable on another than the other way around. In such cases, there is asymmetry in the dependence between the extreme observations of the two variables. The traditional measures of dependence are symmetric and cannot detect any asymmetries. We propose a new measure that is sensitive to asymmetric behavior in extremes. It is based on an extension of the Kendall's tau coefficient, a classical dependence measure. We derive evidence from theory and simulation experiments for the effectiveness of our new methodology. We then apply it to a case study on river networks in the United Kingdom where we show that our measure detects asymmetric behavior of extreme discharges with a preferred direction from upstream to downstream stations. Our work points out the importance of considering proper tools for analyzing the connections between different variables in particular in the presence of asymmetry in extreme observations. Coefficients that can detect asymmetry in dependence among extremes are crucial since traditional methods rely on assumption of symmetryWe propose a conditional version of Kendall's tau that allows to detect asymmetries between extremes, conditioning on one variable at a timeThis new measure can be used for exploratory analysis, model assessment, and to detect directional asymmetries or causal structures in extremes

Automated summary

This article introduces the concept of extremal dependence and the limitations of traditional symmetric dependence measures. It proposes a new measure called asymmetric tail Kendall's tau to capture asymmetry in extreme events. The effectiveness of this measure is demonstrated through theoretical analysis, simulation experiments, and a case study on river networks in the United Kingdom. The study highlights the importance of measuring asymmetric extremal dependence in hydrology.