Ecological Modelling 152 (2002) 1–3 www.elsevier.com/locate/ecolmodel Bridging the gap between ecological diversity indices and measures of biodiversity with Shannon’s entropy: comment to Izsa´k and Papp Carlo Ricotta * Department of Plant Biology, Uni6ersity of Rome ‘‘La Sapienza’’, Piazzale Aldo Moro 5, 00185 Rome, Italy Received 31 May 2001; accepted 11 October 2001 Abstract Most ecological diversity indices summarize the information about the relative abundances of community species without reflecting taxonomic differences between species. Nevertheless, in the environmental conservation practice, data on species abundances are generally unknown. In such cases, to summarize the conservation value of a given site, so-called ‘biological diversity’ measures need to be used. Most of these measures are based on taxonomic relations among species and ignore species relative abundances. In a recent paper, Izsa´k and Papp suggest that the quadratic entropy index (Q) is the only diversity index used to date in the ecological practice that incorporates both species relative abundances and a measure of the pairwise taxonomic differences between species in the analyzed data set. I show here that a number of traditional ecological diversity measures can be generalized to take into account a taxonomic weighting factor. Since these new indices violate part of the mathematical properties that an index should meet to be termed an ecological diversity index, I defined this new family of indices ‘weak diversity indices’. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Biodiversity measures; Ecological diversity; Quadratic entropy; Shannon’s entropy; Species richness 1. Introduction data on species abundances are generally un- known. Often the only available data is the num- Traditional ecological diversity indices, such as ber of species. For instance, focusing on the Shannon index and the Simpson index, sum- conservation problems, species abundances are marize the information about the relative abun- mostly irrelevant. In addition, the comparison of dances of species within a species sample or species abundances is largely meaningless between community without regard to species names or systematically remote organisms, such as oaks differences between species. Nevertheless, for and orchids (Izsa´k and Papp, 2000). large-scale environmental protection purposes, In these circumstances, to summarize the con- servation value of a given site, so-called ‘biologi- * Tel.: +39-06-4991-2408; fax: +39-06-4457-540. cal diversity’ measures are generally preferred. E-mail address: [email protected] (C. Ricotta). Most of these measures are based on taxonomic 0304-3800/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0304-3800(01)00468-9 2 C. Ricotta / Ecological Modelling 152 (2002) 1–3 relations among species and are not linked to number of edges) between the i-th and the j-th traditional diversity indices. To our knowledge, species in the taxonomic tree. An illustration of an Vane-Wright et al. (1991) were the first to suggest artificial taxonomic tree along with its species that, for conservation purposes, we should quantify distance matrix is given in Fig. 1. For example, in the ‘taxonomic distinctiveness’ of different species. Fig. 1, the taxonomic distance dac from species a to For a given data set, Vane-Wright et al. (1991) species c is 4. proposed a measure of the degree of ‘independent If the species abundance values are unknown (i.e. evolutionary history’ that takes into account the if the only available data is species richness), we can topology of the corresponding taxonomic tree, but attribute the same abundance value 1/N to all ignores species relative abundances. Successively, species present. In this case, Eq. (1) expresses the various refinements of this basic idea have been mean taxonomic distance between two randomly actively pursued (Faith, 1992, 1995). In this view, selected species and can be written in the form: Izsa´k and Papp (2000) suggest that the quadratic N N 1 % % entropy index (Rao, 1982) is the only ecological J= 2 dij. (2) diversity index used to date in the ecological N i=1j=1 practice that incorporates both species relative Izsa´k and Papp (2000) note that J does not satisfy abundances and a measure of the pairwise taxo- set monotonicity, a property generally required for nomic differences between species in the analyzed biodiversity measures. This property ensures that data set. the index value will increase by adding a new Consider an N-species community characterized species to a species set A. Formally, denoting a @ \ by the relative abundance vector P=(p1, p2,…, pN) biodiversity index by I, I(A {x}) I(N). Con- 0 0 N such that such that 0 pi 1 and i=1pi =1. Rao versely, the extensive counterpart of J: (1982) defined the quadratic diversity index Q or N N 2 % % quadratic entropy as: F=N J= dij, (3) i=1j=1 N N % % Q= dij pi pj, (1) where F is the sum of the taxonomic differences of i=1j=1 species in the analyzed data set satisfy set where dij is the difference between the i-th and the monotonicity. F is therefore an adequate measure j-th species (dij =dji and dii =0) and Q expresses the of biodiversity if data on species relative abun- average difference between two randomly selected dances are unknown (Izsa´k and Papp, 2000). No- individuals. It is easily shown that for dij =constant tice that F is not simply a (species) richness index " for all i j, Q reduces to dij(1−D), where D is the insofar as it is not a monotone increasing function N 2 Simpson index i=1p i . To apply Q in practice, the of the number of species in the data set. Instead, differences dij need to be specified. Izsa´k and Papp its values are jointly determined by the number of (2000) propose to generate species differences based species and the (taxonomic) differences between the on the taxonomic distance between two species, let species (Izsa´k and Papp, 2000). dij stand for the topological distance (i.e. the 2. Linking ecological diversity and biodiversity with Shannon’s entropy In a similar manner, it is also possible to bridge the gap between traditional ecological diversity indices based on species relative abundances and conservation-oriented biodiversity indices starting N Fig. 1. Illustration of an artificial taxonomic tree along with its from Shannon’s entropy H=− i=1pi log pi. species distance matrix to clarify how to measure taxonomic Shannon’s entropy H of a given community is a distances. information-theoretic measure of uncertainty in C. Ricotta / Ecological Modelling 152 (2002) 1–3 3 % % % predicting the relative abundance of species. Since p2,…, pN andtoap 1, p 2,…, p N permutation of those uncertainty is maximal when entropy is highest, are identical. Therefore, I propose to define these information theory forms one of the basic founda- indices that derive from traditional ecological di- tions of ecological diversity theory (Orlo´ci, 1991). versity indices but that have their greatest value for Within this framework, Casquilho et al. (1998) non-completely even communities ‘weak diversity proposed a generalized version of Shannon’s en- indices’. Notice that the weak diversity formulation tropy of Shannon’s entropy reported in Eq. (4) can be N easily extended to parametric diversity indices such % D HG =− wi pi log pi, (4) as the i diversity family (Patil and Taillie, 1982) i=1 1 D N i i = i=1wi pi(1−p i )ortoAcze´l and Daro´czy where wi is a weighting factor that embodies the i ‘quality’ of the i-th species in the data set. Based h 1 (1975) generalized entropy H = on the species distance matrix of Fig. 1, a simple 21−h −1 N h way to compute a taxonomic weight wi associated i=1wi(p i −pi) to obtain a vector representation of to the i-th species in the data set is to add all dij community diversity as suggested by Patil and elements along row i or column i of the species Taillie (1982). Will weak diversity indices be able matrix. The result of this summation represents the to effectively summarize the important information total taxonomic distance from the i-th species to all contained in the analyzed species assemblages? This other species in the data set. For example, in Fig. is a critical question and their answer will provide 1, the taxonomic weight wa associated to species a valuable insights into the practice of nature conser- is 18. vation. If the species abundance values are unknown, two expression analogous to J and F can be derived (see Eqs. (2) and (3)): References N 1 % Acze´l, J., Daro´czy, Z., 1975. On Measures of Information and JG = wi log N (5) their Characterizations. Academic Press, New York. Ni=1 Casquilho, J., Neves, M., Rego, F., 1998. Extenso˜es da func¸a˜o and de Shannon e equil´ıbrios de proporc¸o˜es—uma aplicac¸a˜oao mosaico de paisagem. Anais do Instituto Superior de Agro- N % nomia 48, 77–99. FG =NJG =log N wi, (6) i=1 Faith, D.P., 1992. Conservation evaluation and phylogenic diversity. Biol. Conserv. 61, 1–10. N N N where i=1wi = i=1 j=1dij and log N is the en- Faith, D.P., 1995. Phylogenic pattern and the quantification of tropy introduced by Hartley (1928), who wanted organismal biodiversity. In: Hawksworth, D.L. (Ed.), Biodi- the entropy to depend only upon the number of versity Measurement and Estimation. Chapman & Hall, species, not upon their relative abundances. There- London, pp. 45–58. Hartley, R.V., 1928. Transmission of information. Bell Syst. fore, since FG takes into account species richness Techn.