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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 (a) µ µ µ (b) µ µ µ µ µ µ µ µ µ µ µ µ 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. 318 Amici, V. 2011. Dealing with vagueness in complex forest landscapes: a soft classification approach through a niche-based distribution model. Ecological Informatics 6: 371–383. Bezdek, J.C., Ehrlich, R. & Full, W. 1984. FCM: the fuzzy cmeans clustering algorithm. Computers & Geosciences 10: 191– 203. Duff, T.J., Bell, T.L. & York, A. 2014. Recognizing fuzzy vegetation pattern: the spatial prediction of floristically defined fuzzy communities using species distribution modelling methods. Journal of Vegetation Science 25: 323–337. Feoli, E. & Zuccarello, V. 1988. Syntaxonomy: a source of useful fuzzy sets for environmental analysis? Coenoses 3: 141–147. Fisher, P., Cheng, T. & Wood, J. 2007a. Higher order vagueness in geographical information: empirical geographical population of type n fuzzy sets. Geoinformatica 11: 311–330. Fisher, P., Wood, J. & Cheng, T. 2007b. Higher order vagueness in a dynamic landscape: multi-resolution morphometric analysis of a coastal dunefield. 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Journal of Vegetation Science 23: 395–405. Sorensen, R.A. 1985. An argument for the vagueness of the ‘vague’. Analysis 45: 134–137. Swanson, A.K., Dobrowski, S.Z., Finley, A.O., Thorne, J.H. & Schwartz, M.K. 2013. Spatial regression methods capture prediction uncertainty in species distribution model projections through time. Global Ecology and Biogeography 22: 242– 251. Williamson, T. 1999. On the structure of higher-order vagueness. Mind 108: 127–143. Zadeh, L. 1965. Fuzzy sets. Information and Control 8: 338–353. Journal of Vegetation Science Doi: 10.1111/jvs.12152 © 2014 International Association for Vegetation Science