skip to main content

Title: Copulas and their potential for ecology
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 some 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 *copula*; they are applied in finance, neuroscience and elsewhere, but rarely in ecology. We explore the concepts that underpin copulas and 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 *vice versa*. We describe mechanisms by which observed copula structure and tail associations can arise in ecological data, including a Moran-like effect whereby dependence structures are inherited from environmental 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 temporal stability of ecosystem services. By documenting the importance of a complete description of dependence between variables, advancing conceptual frameworks, and demonstrating a powerful approach, we encourage widespread use of copulas in ecology, which we believe can benefit the discipline.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ; ; ; ; ;
Date Published:
Journal Name:
Advances in ecological research
Page Range / eLocation ID:
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Species coexistence attracts wide interest in ecology. Modern coexistence theory (MCT) identifies coexistence mechanisms, one of which, storage effects, hinges on relationships between fluctuations in environmental and competitive pressures. However, such relationships are typically measured using covariance, which does not account for the possibility that environment and competition may be more related to each other when they are strong than when weak, or vice versa. Recent work showed that such ‘asymmetric tail associations’ (ATAs) are common between ecological variables, and are important for extinction risk, ecosystem stability, and other phenomena. We extend MCT, decomposing storage effects to show the influence of ATAs. Analysis of a simple model and an empirical example using diatoms illustrate that ATA influences can be comparable in magnitude to other mechanisms of coexistence and that ATAs can make the difference between species coexistence and competitive exclusion. ATA influences may be an important new mechanism of coexistence.

    more » « less
  2. Normalizing flows—a popular class of deep generative models—often fail to represent extreme phenomena observed in real-world processes. In particular, existing normalizing flow architectures struggle to model multivariate extremes, characterized by heavy-tailed marginal distributions and asymmetric tail dependence among variables. In light of this shortcoming, we propose COMET (COpula Multivariate ExTreme) Flows, which decompose the process of modeling a joint distribution into two parts: (i) modeling its marginal distributions, and (ii) modeling its copula distribution. COMET Flows capture heavy-tailed marginal distributions by combining a parametric tail belief at extreme quantiles of the marginals with an empirical kernel density function at mid-quantiles. In addition, COMET Flows capture asymmetric tail dependence among multivariate extremes by viewing such dependence as inducing a low-dimensional manifold structure in feature space. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of COMET flows in capturing both heavy-tailed marginals and asymmetric tail dependence compared to other state-of-the-art baseline architectures. All code is available at

    more » « less
  3. Abstract

    Although our observing capabilities of solar‐induced chlorophyll fluorescence (SIF) have been growing rapidly, the quality and consistency of SIF datasets are still in an active stage of research and development. As a result, there are considerable inconsistencies among diverse SIF datasets at all scales and the widespread applications of them have led to contradictory findings. The present review is the second of the two companion reviews, and data oriented. It aims to (1) synthesize the variety, scale, and uncertainty of existing SIF datasets, (2) synthesize the diverse applications in the sector of ecology, agriculture, hydrology, climate, and socioeconomics, and (3) clarify how such data inconsistency superimposed with the theoretical complexities laid out in (Sun et al., 2023) may impact process interpretation of various applications and contribute to inconsistent findings. We emphasize that accurate interpretation of the functional relationships between SIF and other ecological indicators is contingent upon complete understanding of SIF data quality and uncertainty. Biases and uncertainties in SIF observations can significantly confound interpretation of their relationships and how such relationships respond to environmental variations. Built upon our syntheses, we summarize existing gaps and uncertainties in current SIF observations. Further, we offer our perspectives on innovations needed to help improve informing ecosystem structure, function, and service under climate change, including enhancing in‐situ SIF observing capability especially in “data desert” regions, improving cross‐instrument data standardization and network coordination, and advancing applications by fully harnessing theory and data.

    more » « less
  4. Abstract Aim

    We aimed to measure the dominant spatial patterns in ecosystem properties (such as nutrients and measures of primary production) and the multi‐scaled geographical driver variables of these properties and to quantify how the spatial structure of pattern in all of these variables influences the strength of relationships among them.

    Location and time period

    We studied > 8,500 lakes in a 1.8 million km2area of Northeast U.S.A. Data comprised 10‐year medians (2002–2011) for measured ecosystem properties, long‐term climate averages and recent land use/land cover variables.

    Major taxa studied

    We focused on ecosystem properties at the base of aquatic food webs, including concentrations of nutrients and algal pigments that are proxies of primary productivity.


    We quantified spatial structure in ecosystem properties and their geographical driver variables using distance‐based Moran eigenvector maps (dbMEMs). We then compared the similarity in spatial structure for all pairs of variables with the correlation between variables to illustrate how spatial structure constrains relationships among ecosystem properties.


    The strength of spatial structure decreased in order for climate, land cover/use, lake ecosystem properties and lake and landscape morphometry. Having a comparable spatial structure is a necessary condition to observe a strong relationship between a pair of variables, but not a sufficient one; variables with very different spatial structure are never strongly correlated. Lake ecosystem properties tended to have an intermediary spatial structure compared with that of their main drivers, probably because climate and landscape variables with known ecological links induce spatial patterns.

    Main conclusion

    Our empirical results describe inherent spatial constraints that dictate the expected relationships between ecosystem properties and their geographical drivers at macroscales. Our results also suggest that understanding the spatial scales at which ecological processes operate is necessary to predict the effects of multi‐scaled environmental changes on ecosystem properties.

    more » « less
  5. null (Ed.)
    Synchrony among population fluctuations of multiple coexisting species has a major impact on community stability, i.e. on the relative temporal constancy of aggregate properties such as total community biomass. However, synchrony and its impacts are usually measured using covariance methods, which do not account for whether species abundances may be more correlated when species are relatively common than when they are scarce, or vice versa. Recent work showed that species commonly exhibit such ‘asymmetric tail associations’. We here consider the influence of asymmetric tail associations on community stability. We develop a ‘skewness ratio’ which quantifies how much species relationships and tail associations modify stability. The skewness ratio complements the classic variance ratio and related metrics. Using multi-decadal grassland datasets, we show that accounting for tail associations gives new viewpoints on synchrony and stability; e.g. species associations can alter community stability differentially for community crashes or explosions to high values, a fact not previously detectable. Species associations can mitigate explosions of community abundance to high values, increasing one aspect of stability, while simultaneously exacerbating crashes to low values, decreasing another aspect of stability; or vice versa. Our work initiates a new, more flexible paradigm for exploring species relationships and community stability. This article is part of the theme issue ‘Synchrony and rhythm interaction: from the brain to behavioural ecology’. 
    more » « less