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STS486 Tonio D.B. et al.
Starting from these considerations, simultaneous confidence bands for the
mean diversity profile function are obtained using a bootstrap procedure.
Since diversity profiles are constrained functions, our proposal consists in pre-
processing diversity profiles via a differential equation method (Ramsay, 1998)
and bootstrapping the unconstrained functions. This allows us to respect the
characteristics of the diversity profile and to work in an appropriate functional
2
space, that is the space.
The paper is organized as follows: Section 2 provides a brief review of the
diversity profile evaluation in a functional context focusing on the estimation
of constrained curves. The section continues with the construction of
bootstrap simultaneous confidence bands for the functional mean estimator.
Section 4 deals with an application to a real dataset concerning fish
biodiversity in Lazio rivers (Italy) and Section 5 concludes the paper.

2. Methodology: FDA approach for evaluating diversity profiles
Diversity profile are functions of the abundance vector, whose knowledge
requires a census of the population under study, which is unfeasible in most
cases (Barabesi and Fattorini, 1998).
Let us suppose that an ecological population is composed of N units and
T
is partitioned into s species j=1,2,...,s. Let N=(N1,,...,Ns) be the species
abundance vector whose generic element Nj represents the number of
individuals belonging to the j-th species, and let p=(p1,...,ps) be the relative
T
s
s
abundance vector with pj=Pj/∑ j=1Nj such that 0≤pj≤ 1 and ∑ j=1pj=1. The
abundances must be estimated by means of a sample survey, following a
model-based or a design-based approach. The latter is widely applied in an
ecological context, because it considers the values of a variable of interest as
fixed quantities and the selection probabilities, introduced with the design, are
used in defining the properties of the estimators, without making any
assumptions about the population (Thompson, 1992).
Let us suppose that abundance data have been collected from a biological
community. Then, for each i-th sample unit (habitat, environmental site, etc.),
i=1,2,...,n, a diversity profile ∆ix can be obtained. Since diversity profiles are
presented as curves, they may be represented in a functional framework as
follows (Gattone and Di Battista, 2009):
∆ () = () + () ∈ , = 1,2, … , (2)

where () is an arbitrary smooth function; and () denotes an unknown

independent zero-mean error term. Usually, the functional form of the
diversity profile, (),can be reconstructed from the observed raw sampled

data points {∆ : ∈ } using basis function expansion and smoothing

(Ramsay and Silverman, 2005). However, diversity profiles are a special case of
functional data in that they are non-negative, convex and decreasing curves.

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