Copulas and their potential for ecology

Abstract

<jats:title>Abstract</jats:title><jats:p>All branches of ecology study relationships among and between environmental and biological variables. However, standard approaches to studying such relationships, based on correlation and regression, provide only a small slice of the complex information contained in the relationships. Other statistical approaches exist that provide a complete description of relationships between variables, based on the concept of the <jats:italic>copula</jats:italic>; they are applied in finance, neuroscience and other fields, but rarely in ecology. We here explore the concepts that underpin copulas and examine the potential for those concepts to improve our understanding of ecology. We find that informative copula structure in dependencies between variables is common across all the environmental, species-trait, phenological, population, community, and ecosystem functioning datasets we considered. Many datasets exhibited asymmetric tail associations, whereby two variables were more strongly related in their left compared to right tails, or <jats:italic>vice versa</jats:italic>. We describe mechanisms by which observed copula structure and asymmetric tail associations can arise in ecological data, including a Moran-like effect whereby dependence structures between environmental variables are inherited by ecological variables; and asymmetric or nonlinear influences of environments on ecological variables, such as under Liebig’s law of the minimum. We also describe consequences of copula structure for ecological phenomena, including impacts on extinction risk, Taylor’s law, and the stability through time of ecosystem services. By documenting the importance of a complete description of dependence between variables, advancing conceptual frameworks, and demonstrating a powerful approach, we aim to encourage widespread use of copulas in ecology, which we believe can benefit the discipline.</jats:p>

Tail associations in ecological variables and their impact on extinction risk

“…How does the nature of the density dependence mediate this influence? Will density dependence reverse or reproduce the earlier result (Ghosh et al 2020), using non‐density‐dependent models, that left‐tail dependence in environmental variables accentuates and right‐tail dependence mitigates extinction risks? For Q1, we simulate a simple 2‐stage model.…”

“…Greater right‐tail associations would probably not have the same effect, even if overall correlations between locations were the same, because in that case, it would be very good years for the component populations that would occur at the same time in many patches. Ghosh et al (2020) substantiated this intuition by showing that it held true for a modeling setup using a spatial version of a very simple non‐density‐dependent population growth model (the Lewontin‐Cohen model). In essence, if environmental "catastrophes" (i.e., extremely bad years for a population) are widely spatially synchronized it should create much greater metapopulation extinction risk than if "bonanzas" (i.e., extremely good years) are widely synchronized, even if overall (non‐tail‐specific) levels of environmental synchrony are the same in both these scenarios (Fig.…”

“…Asymmetric tail associations have been documented between environmental variables, and between the same environmental variable measured in different places (Serinaldi 2008, Li et al 2013, Goswami et al 2018, She and Xia 2018). Asymmetric tail associations have also been documented between numerous ecological variables and between environmental and ecological variables (Ghosh et al 2020).…”

“…1 shows that two variables can have a wide range of patterns of tail association, from extreme left‐ to extreme right‐tail association, with the same overall correlation. So tail association is a distinct concept from correlation itself and may have distinct ecological ramifications (Ghosh et al 2020). Asymmetric tail associations have been documented between environmental variables, and between the same environmental variable measured in different places (Serinaldi 2008, Li et al 2013, Goswami et al 2018, She and Xia 2018).…”

“…By so doing, we hope to open these potentially practically important questions to wider examination by other researchers, using a variety of models and perspectives. Though tail associations are used to assess risks in fields such as finance and hydrology (Alexander 2008, Chen et al 2011, Li et al 2013, Goswami et al 2018), to our knowledge asymmetric tail associations have rarely been assessed for their importance for extinction risks of populations and species (but see Ghosh et al 2020). Our essential goal is to advance the idea that tail associations may also be important in this context.…”