Page 214 - Special Topic Session (STS) - Volume 2

STS486 Tonio D.B. et al.
Thus, some restrictions should be placed on () to ensure the

characteristics of diversity profiles. In the following, we adopt () to indicate
the constrained functional approximation of a generic diversity profile. It may
be represented as follows (Ramsay, 1998):
() = + −(−1) exp (∫ ()) (3)
1
0
where and are arbitrary constants, −1 means taking the indefinite
1
0
integral, and w(x) is a Lebesgue square integrable function. Specifically, ()

is the solution of the differential equation () = () −1 (). It is
straightforward to verify that for = 1 the solution is a positive function, for
= 2 the solution is a monotone function and for = 3 the solution is a
convex function. Since the function () is unconstrained, it can be expanded
in terms of K known basis functions as follows:
() = ∑ () (4)
=1

where is the k-th coefficient which defines the linear combination and

() is the k-th basis function. The main result of the transformation in Eq.

(3) is that we can overcome the problem of finding the constrained functional
() by estimating the unconstrained function () that, in what follows, we
shall term the unconstrained diversity profile. We point out that the function
() has the same information on diversity of () and, whenever it is
required, we can invert the transformation and go back to the constrained
diversity profile functions by putting the coefficients () in Eq. (3).
Once the functional approximation of the profile has been obtained,
different functional tools can be computed to evaluate the biodiversity of a
community (Di Battista et al, 2016). Then, the assessment of uncertainty of an
obtained estimator is a fundamental step in all statistical analysis. In function
estimation problems, simultaneous confidence bands provide a unified set of
graphical and analytical tools to harness such tasks as data exploration, model
specification or validation, assessment of variability in estimation, prediction,
and inference (Di Battista and Fortuna, 2017). In FDA and particularly in an
ecological context, the bootstrap methodology turn out to be often the only
practical alternative to derive the sampling distribution of a functional statistic
(Cuevas et al., 2006). Effectively, the normality assumption could be too strong
for diversity profiles. For this reason, we extend the bootstrap methodology to
the framework of biodiversity assessment to build confidence intervals for the
mean diversity profile.
Let (), (), … , () be the sample of unconstrained functions

1
2
derived from the original diversity profile observed in the i-th sites, i=1,2,...,n,
and = ( (), … , ()) be the sample statistics under consideration. The

1
bootstrap resamples (), … , () are calculated following the following
∗
∗

1
scheme (Febrero-Bande and de la Fuente, 2012):
∗
() = () + () (5)

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