Page 212 - Special Topic Session (STS) - Volume 2

STS486 Tonio D.B. et al.
issue is given by diversity profiles. They are non-negative and convex curves,
which express diversity as a function of the relative abundance vector. They
provide a comfortable representation of diversity because they consider its
multivariate nature; return a graphical representation of diversity; and allow to
compare different communities when the profiles do not intersect. Thus, the
behaviour of these curves gives important information about biodiversity in a
community.
In the literature, different diversity profiles have been proposed; the main
ones are the β diversity profile (Patil and Taillie, 1979), the intrinsic diversity
profile (Patil and Taillie, 1979), and the Hill's number Hill, 1973), among the
others. The formulation of a diversity profile can be generalized as follows:
{∆ : ∈ } (1)

where ∆ are various diversity measures obtained by varying x in the domain

,which can be finite or infinite. The curve which joins the (, ∆ ) pairs for ∈

is termed a diversity profile, and depicts in a single picture simultaneous
values of diversity measures with varying sensitivities to the rare and abundant
species as a function of the parameter x. Hence, a diversity profile measures
diversity through a curve rather than a scalar as in the case of diversity indexes.
This emphasizes the importance of using such an approach in environmental
studies as it does not collapse the information of a multidimensional set (the
biological community) into a single number (Gattone and Di Battista, 2009).
Due to these characteristics, Gattone and Di Battista (2009) proposed to
analyze them through the functional data analysis (FDA) approach (Ramsay
and Silverman, 2005; Ferraty and View, 2006). The latter allows to obtain
several advantages in an ecological context. Indeed, we can analyze the shape
of the profile through functional tools (Di Battista et al., 2016, Maturo e Di
Battista, 2018) and evaluate the behaviour of the profile throughout the
reference domain. The functional approach is particularly helpful when an
inferential approach for biodiversity is required. Indeed, making inference on
diversity profiles starting from the abundance vector with standard
multivariate prodedures, involves many unresolved issues. The solutions
proposed in the literature mainly concern the use of independent replications
of a sampling design (Barabesi and Fattorini, 1998) and the use of jackknife to
build confidence intervals for the diversity estimator (Fattorini and Marcheselli,
1999). However, in practice, replications of paths for a given sampling design
could be quite expensive and time consuming (Di Battista and Gattone, 2004),
and the jackknife requires that the elements of the frequencies vector are all
different from each other. Moreover, in some cases, the jackknife procedure
may fail to return a convex diversity profile. On the other hand, the FDA
approach analyses the profile as a function; thus, for each sample unit, a single
observation is observed, overcoming problems of simultaneous multivariate
inference (Di Battista and Fortuna, 2017).

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