A Functional Approach to Diversity Profiles

Abstract

Diversity plays a central role in ecological theory and its conservation and management are important issues for the wellbeing and stability of ecosystems. The aim of this work is to provide a reliable theoretical framework for performing statistical analysis on ecological diversity by means of the joint use of diversity profiles and functional data analysis. We point out that ecological diversity is a multivariate concept as it is a function of the relative abundances of species in a biological community. For this, several researchers have suggested using parametric families of indices of diversity for obtaining more information from the data. Patil and Taillie introduced the concept of intrinsic diversity ordering which can be determined by using the diversity profile. It may be noted that the diversity profile is a non-negative and convex curve which consists of a sequence of measurements as a function of a given parameter. Thus, diversity profiles can be explained through a process that is described in a functional setting. Recent developments in environmental studies have focused on the opportunity to evaluate community diversity changes over space and/or correlation of diversity with environmental characteristics. For this, we develop an innovative analysis of diversity based on a functional data approach. Whereas conventional statistical methods process data as a sequence of individual observations, functional data analysis is designed to process a collection of functions or curves. Moreover, unconstrained models may lead to negative and/or non-convex estimates for the diversity profiles. To overcome this problem, a transformation is proposed which can be constrained to be non-negative and convex. We focus on some applications showing how functional data analysis provides an alternative way of understanding biological diversity and its interaction with natural and/or human factors. Copyright (c) 2009 Royal Statistical Society.

Diversity and equitability ordering profiles applied to study forest structure

“…As an alternative to always plotting the intrinsic ordering when using a single index, an analytical approach could also be performed, by programmatically assuring that M k consistently increases/decreases for all k (Gove et al, 1992), or testing for Shur concavity (Gattone and Battista, 2009). Gove et al (1995) applied this method to assure diversity of basal area proportions among size classes, though using the right tail-sum method R k ðk; T k Þ instead of majorization.…”

“…This can be carried out by comparing their intrinsic diversity profiles, which are obtained by accumulating the relative dominances of their abundances. Recent research (Gattone and Battista, 2009) has attempted to enhance the use of profiles for describing ecological diversity.…”

Parametric Functional Analysis of Variance for Fish Biodiversity Assessment

“…It is a complete metric space with respect to the metric derived from its norm that is a Cauchy sequence of vectors always converges to a well defined limit in the space. Since diversity profile expresses diversity as a function of the relative abundance vector in a fixed domain, it can be analyzed in a functional context (Gattone and Di Battista, 2009). Thus, in an ecological setting, the functional space S could be the family of diversity profiles, such as the β profile in Equation (1), and for each i-th sites, i = 1, 2, …, N, each relative abundance vector can be assumed as a single parameter, pi = (pi1, …, pis) = θi, so that, p = θ, whereas β represents the functional domain (Di Battista and Fortuna, 2013).…”

“…Furthermore, in the specific case of the beta profile, the smooth function f(β) in Equation (2) must satisfy the non-egativity and the convexity of diversity profile. These restrictions can be obtained arbitrarily in several ways (see for example Barlow et al, 1972;Ramsay, 1988;Gattone and Di Battista, 2009). The way of pre-processing the original data is an important issue due to the impact that this process might have on the subsequent statistical analysis (Pé rez and Vilar, 2013).…”

“…However, the Shannon index is particularly sensitive to the presence of rare species in a community, while the Simpson index is particularly sensitive to changes in the relative abundances of the most dominant species. This highlights how the use of a single indicator greatly reduces the complexity of the ecological systems and hides the multidimensional aspect of biodiversity (Patil and Taillie, 1979;Gattone and Di Battista, 2009;Gove et al, 1994). For these reasons, a universally accepted measure has not been established (Ricotta, 2005;Di Battista et al, 2016).…”