Download Fuzzy species distribution models: a way to represent plant

Journal of Vegetation Science 25 (2014) 317–318
COMMENTARY
Fuzzy species distribution models: a way to represent
plant communities spatially
Duccio Rocchini
Rocchini, D. ([email protected]):
Department of Biodiversity and Molecular
Ecology, GIS and Remote Sensing Unit,
Fondazione Edmund Mach, Research and
Innovation Centre, Via E. Mach 1, 38010,
S. Michele all’Adige, Trento, TN, IT
Abstract
Fuzzy set theory has generally been applied to smooth classification cut-offs,
with an unavoidable loss of information. In this commentary, I rely on both
advantages and disadvantages of the methods proposed in Duff et al., in this
issue of the Journal of Vegetation Science, to map the variability over space of vegetation classes based on fuzzy sets and species distribution models.
Vegetation composition is fuzzy by its very nature. Classification approaches, despite their robustness, will
always lead to a degradation of the original information
(Palmer et al. 2002). Palmer (2007) explicitly states that:
‘we have no comprehensive theory governing the geometry of environmental variation. Some environmental
gradients vary smoothly and linearly […]. Other gradients may be more spatially unpredictable.’ Such unpredictability is related to the uncertainty of representing
nature using pre-defined deterministic laws. In this case,
uncertainty is particular and important information that
deserves to be maintained and spatially represented,
instead of being abruptly cut off.
Thomas J. Duff and colleagues (Duff et al. 2014)
attempt to describe vegetation classes relying on fuzzy set
theory, leading to what I call ‘fuzzy species distribution
models’, to represent their continuous variation over
space. For the sake of clarity, mathematically speaking, let
U denote a universe of entities u; hence, the fuzzy set F is
F ¼ ðu; lf ðuÞÞju 2 U
where the membership function lf associates for each
entity u the degree of membership in the set F (see Zadeh
1965). The degree of membership ranges through the continuous interval [0, 1].
Two major assumptions suggest fuzzy sets as a powerful
tool for maintaining uncertainty information when mapping vegetation classes: (i) the membership of ecological
entities (spatial units) in classes is not forced to occur
within the integer range [0, 1] as in Boolean logic; (ii) considering different classes [A, B,…, N], the sum of membership values does not necessarily equal 1 for each entity u.
Thus, different classes may overlap to different extents,
overcoming the traditional restriction on the mutually
exclusive nature of vegetation classes (Rocchini & Ricotta
2007).
Strictly, every single spatial unit may belong to several
different classes at the same time, with different degrees of
membership for each class. Duff et al. (2014) applied such
concepts by relying on the following straightforward
scheme:
1. They identified groups (vegetation classes) based on the
fuzzy c-means algorithm (Bezdek et al. 1984) applied to
species/sites matrices;
2. They related the fuzzy membership of each site for each
group to spatial environmental predictors (climate, topography, biomass, fire regimes, etc.) using boosted regression
trees (BRTs);
3. They created predictive surfaces (species distribution
models) of the fuzzy membership for each group (vegetation class).
Figure 1 helps to illustrate differences between discrete
and fuzzy classification. The method of Duff et al. (2014) is
straightforward, robust and allows visualization of the
uncertainty related to each defined vegetation class. And
overall, it is novel.
While pioneering studies have used different fuzzy-set
methods to softly classify plant communities (e.g., Feoli &
Zuccarello 1988; Roberts 1989; Podani 1990), few efforts
have been applied in the past to spatially represent (map)
fuzzy memberships in the different vegetation classes
(groups).
On the one hand, previous attempts have been made to
map the variability of land-use classes using fuzzy sets,
relying on species distribution model rules applied to satellite image classification (Amici 2011). On the other hand,
species distribution models coupled with uncertaintybased models have been used to show the variability in
space of single species. An example of this is provided in
Swanson et al. (2013), mapping the uncertainty in the
distribution of Salvia mellifera in California, US.
Journal of Vegetation Science
Doi: 10.1111/jvs.12152 © 2014 International Association for Vegetation Science
317
Commentary
D. Rocchini
References
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Fig. 1. In discrete classification (a), a single map is delivered with all the
classes following a Boolean membership rule (Fig. 5 in Duff et al. 2014). In
fuzzy classification (b), each vegetation class (group) is represented as a
map of memberships (Fig. 4 in Duff et al. 2014). Reproduced from Rocchini
et al. (2013).
In general, a conceptual drawback in applying fuzzy set
theory to the uncertain description of vegetation classes is
the deterministic relationship between each species and
the related community class (Zadeh 1965). Such a deterministic relationship is a paradox since, in this case, the
description of uncertainty is made with class membership
values that are a-priori suspected to be certain. In other
words, fuzzy classification still assumes that classes exist
and require definition of classes that is not necessarily possible (Schmidtlein et al. 2012). A statement describing a
vague phenomenon must be necessarily vague (Sorensen
1985); this has led to the possibility of assuming, for each
level of the fuzzy membership function, a fuzzy set membership. This is also known as a type 2 fuzzy set, or secondorder vagueness, which has been extended to a higherorder vagueness concept by Williamson (1999). Refer to
Fisher et al. (2007a,b) for an example of higher-order
vagueness and type 2 fuzzy sets applied to detection of a
mountain peak (2007a) and a coastal dune (2007b).
Moreover, the intrinsic difficulty of managing higherorder vagueness in a straightforward manner is the only
(conceptual) drawback I see in the application of fuzzy set
theory to vegetation science, which is actually more philosophical than empirical.
Hence, I personally advocate strengthening methods
such those developed in Duff et al. (2014) to allow a representation of the variability of plant community (fuzzy)
classes over space.
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Journal of Vegetation Science
Doi: 10.1111/jvs.12152 © 2014 International Association for Vegetation Science