A simplified vine copula-based probabilistic method for quantifying multi-dimensional ecological niches and niche overlap: take a three-dimensional case as an example

Abstract

For quantifying m-dimensional ($m \ge 3$) niche regions and niche overlaps using a copula-based approach, commonly used copulas, including Archimedean and elliptical copula families, are unsatisfactory alternatives in characterizing a complex dependence structure among multiple variables, especially when bi-variate copulas characterizing dependency structures of two-dimensional sub-variables differ. To solve the problem, we improve the copula-based niche space modeling approach using simplified vine copulas, a powerful tool containing various bi-variate dependence structures in one multivariate copula. Using four simulated data sets, we then check the performance of simplified vine copula approximation when the simplifying assumption is invalid. Finally, we apply the improved copula-based approach to quantifying a three-dimensional niche space of a real case of Swanson et al. (Ecology 96(2):318–324, 2015. https://doi.org/10.1890/14-0235.1) and discover that among various simplified vine and other flexible multi-dimensional copulas, non-parametric simplified vine copula approximation performs best in fitting the data set. In the discussion, to analyze differences in calculating niche overlaps caused by using different copulas, we compare non-parametric simplified vine copula approximation with non-parametric and parametric simplified vine copula approximation, elliptical copula, Hierarchical Archimedean copula estimation, and empirical beta copula and give some comments on the results.

Appendices

Appendix 1

It contains Figs. 9, 10, and 11.

Fig. 9

Normalized bi-variate copula contour plots for BDWF. The red solid contours in each column are marked by the three-dimensional copulas above the first row; the black dashed contours are empirical

Fig. 10

Normalized bi-variate copula contour plots for LKWF. The red solid contours in each column are marked by the three-dimensional copulas above the first row; the black dashed contours are empirical

Fig. 11

Normalized bi-variate copula contour plots for LSCS. The red solid contours in each column are marked by the three-dimensional copulas above the first row; the black dashed contours are empirical

Appendix 2

It contains Figs. 12, 13, and 14.

Fig. 12

Normalized bi-variate ($X_{1}{-}X_{2}$) copula contour plots based on HAC2. The blue contours represent NP-SVCA; the red contours are empirical; the black contours represent HAC2 (the true model). n is sample size used for estimation

Fig. 13

Normalized bi-variate ($X_{1}{-}X_{2}$) copula contour plots based on HAC3. The blue contours represent NP-SVCA; the red contours are empirical; the black contours represent HAC3 (the true model). n is sample size used for estimation

Fig. 14

Normalized bi-variate ($X_{1}{-}X_{2}$) copula contour plots based on HAC4. The blue contours represent NP-SVCA; the red contours are empirical; the black contours represent HAC4 (the true model). n is sample size used for estimation