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True diversities: A comment on Lou Jost’s “Entropy and diversity” Sönke Hoffmann Otto-von-Guericke-University Magdeburg, Faculty of Economics and Management, Dept. of Economic Policy, P.O. 4120, 39016 Magdeburg, Germany. E-mail: [email protected] Andreas Hoffmann University of Applied Sciences Magdeburg-Stendal (FH), Dept. of Water Resource Management, Breitscheidstraße 2, 39114 Magdeburg, Germany. E-mail: [email protected] Abstract Jost (2006) recently discussed Hill’s (1973) effective number of species and concluded by naming it the ‘true’ diversity. Due to the inherent multi-faceted character of diversity we doubt that any known diversity concept can be called the true one. Instead, we may identify ‘good’ and ‘bad’ concepts, or even ‘best’ and ‘worst’ ones, depending on the match-up between properties the context requires and properties the concept provides. Using this terminology, we agree, that Hill’s (1973) effective number is one of the best approaches to quantify community diversity in ecology. Keywords: Diversity measurement, Entropy, Information 1 Introduction In his article “Entropy and diversity” Jost (2006) gives a very unambiguous statement on what diversity is. According to Jost, there is a class of ‘true’ diversities, better known as effective number and usually denoted Na . Jost may aim for the ultimate guidance through Ricotta’s (2004) “Jungle of biodiversity”, but some of his statements are too absolute in their character. In what follows we comment on these aspects and argue that the true diversity is not what the name suggests. 2 What is diversity? At the very beginning of his article Jost (2006) emphasizes: “Diversity [. . . ] has been confounded with the indices used to measure it; a diversity index is not necessarily itself a “diversity”. The radius of a sphere is an index of its volume but is not itself the volume, and using the radius in place of the volume in engineering equations will give dangerously misleading results. This is what biologists have done with diversity indices.” and justifies this statement as follows: “[. . . ] most common diversity measure, the Shannon-Wiener index, is an entropy, giving the uncertainty in the outcome of a sampling process.” Saying that something is not itself ‘a diversity’ requires an unambiguous and commonly accepted understanding of what ‘a diversity’ is. This prerequisite would be easily satisfied if diversity was something naturally given – a physical quantity, like volume, mass or energy. But diversity is not. Instead it is something very multi-faceted and inherently subjective, influenced by at least two sub-questions (cf. Baumgärtner 2006): (1) Diversity for what purpose? (2) Diversity of what? Answers to the first question are manifold but manageable, whereas answers to the second seem to be bounded only by our imagination. Quite apposite to the matter, Magurran (1988) compared diversity with a fata-morgana, that takes very different and blurred shapes from different viewpoints. The often cited ‘plethora’ of diversity concepts is nothing but a natural consequence of the ‘plethora’ of possible answers to questions (1) and (2). In contrast to physical quantities, diversity is nothing naturally given but it is implicitly defined by a formal concept that is assumed to measure the quantity under consideration. This is the common practice in natural and social sciences. As Peet (1974) for example recognized quite early in ecology: “Diversity, in essence, has always been defined by the indices used to measure it [. . . ].” 1 From this perspective it is not possible to say what diversity is ultimately. Diversity in all its dimensions and facets cannot be captured by a single definition or mathematical formalism. Because many different views on diversity exist, many diversites exist, none of them being per se more a diversity than another. On the other hand it is equally is not diversity. Jost argues that the mathematical term Phard to 1say what + S H ≡ c i pi log pi , c ∈ R cannot be a diversity because it is an entropy. Generally seen, H S neither is a diversiy, nor is it an entropy, or any other phenomenological quantity of our real world. Instead, it is nothing but a mathematical expression with certain inherent properties (see Aczél and Daróczy 1975 for details). And these properties may or may not fit to a possible ‘being’ that should be captured by H S . This ‘being’ can be entropy, uncertainty, information, inequality, evenness, diversity or other phenomenons. Entropy is certainly much more objective and physical in its character than diversity, but nevertheless, it is similarly described by astro physicist Tim Thompson1 : “The easist answer to the question, What is entropy?, is [. . . ]: Entropy is what the equations define it to be. You can interpret those equations to come up with a prosey explanation, but remember that the prose and the equations have to match up, because the equations give a firm, mathematical definition [. . . ], that just won’t go away.” Jost is clearly right, saying that radius is not volume and that radius is instead a index variable of volume. However, this does not say anything about the ‘beings’ of diversity and entropy. Using the expression H S to measure community diversity, biologists have certainly not confounded diversity with the indices used to measure it. Instead they have implicitly defined diversity by using this concept, being very aware of the fact that H S is not diversity, but one of many ways to make diversity explicit. The fundamental question arising is not whether H S itself is a diversity or not, – this cannot be answered anyway –, but whether this mathematical expression makes sense in a given context or not. To put it in the words of philosopher Norton (2003): “there is no ‘correct’ [...] definition to be ‘found’, as one might discover a gem under a rock. We are looking for a definition that is useful in deliberative discourse [. . . ]. Proposed definitions will be judged by their usefulness.” For the sake of ontological and etymological consistency we have to reformulate P Jost’s statement. We can either say, ‘Expression c i pi log p1i is neither entropy nor P diversity but simply a mathematical term’, or we say ‘Expression c i pi log p1i can be used to define and measure entropy as it also can be used to define and measure one of many facets of diversity’. How good H S finally performs as a model of the quantity under consideration is a fundamentally different question. Depending P on the context, the number that formula c i pi log p1i finally provides may be a complete nonsense or the most meaningful number of all. In any case, it is an element in the set of available options. 1 See http://www.tim-thompson.com/entropy1.html 2 3 Some proses on H S The legitimation to use a mathematical expression like H S as a measurement concept arises from required properties in a given context on the one hand (What should diversity be?) and the inherent properties of the expression (What is diversity?) on the other. Such interplay between Thompson’s “proses and equations” is finally judged with regard to their match-up. The first, and one of the best matching proses on H S , was given by Ludwig Boltzmann and Josiah Willard Gibbs, who tried to explain classical thermodynamics, and especially its second law, by statistical means of mechanics. Here, H S quantifies a macrostate mean value of energy over possible microstates i (e.g. Beck and Schlögl 1993). As a tribute to Boltzman and Gibbs, H S is usually called the BoltzmannGibbs entropy by physicists. Another substantial and well-matching prose was given by the ‘father of information theory’, Claude E. Shannon, who established H S as a measure of reduced uncertainty (information) in his seminal theory of communication (Shannon 1948). Although the word entropy is, etymologically seen, quite misleading in the context of information theory, most scientists call H S the Shannon entropy. In fact, the creator of the word ‘entropy’ is Rudolf Clausius who proprosed: “[. . . ] to name the quantity S the entropy of the system, after the Greek word trope, the transformation. I have deliberately chosen the word entropy to be as similar as possible to the word energy: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate.” (Clausius 1850, translation by W.F. Magie) The name “Shannon entropy” may be due to the mathematician John von Neumann, who is quoted as having proposed the word ‘entropy’ to Shannon for two reasons (Weinberg 1981): “First, your formula is identical in structure with the entropy of statistical thermodynamics. An second, and more important, no one understands entropy. You will therefore always be at an advantage in an argument.” Other uses of H S were successively made in economics, political sciences, linguistics and many more disciplines. The characterizing properties of H S were also found to be suitable for the ecological measurement of community diversity (Margalef 1958, Pielou 1974, 1975). The most popular assumption in this context is that “[. . . ] diversity, however defined, is a single statistic in which the number of species and evenness are confounded” (Pielou 1969, see also Magurran 2004, p. 17), S S And P this is1exactly what H is able to provide. Alternatively, the expression H = c i pi log pi may also measure average species rarity – another possible facet of community diversity – where pi is the relative abundance of species i and log (1/pi ) is the rarity function of that species (Patil and Taillie 1982). The latter proposal 3 is less popular but not less intuitive: A species with no abundance is, compared to other species, infinitely rare, while a species with abundance pi = 1 is, compared to other species, not rare at all. There are many interpretations of H S . Certainly not all of them have the same objective and unambiguous status as entropy or information have. Shannon (1956) himself noticed a ‘bandwagon effect’ caused by his publication and warned that information theory “[. . . ] has perhaps ballooned to an importance beyond its actual accomplishments.” Using H S as a diversity measure may or may not be admissible. It depends on the interplay between the properties required and the properties provided. The properties required vary with the application contexts and, therefore, H S can neither be generally accepted nor generally rejected. In any case, rejecting H S as diversity measure requires much more than simply observing that H S perfectly fits to measure something else. 4 Effective numbers: The true diversity? An important aspect of diversity concepts is the unit of measurement. Many biologists agree, that the quantity of species diversity should be measured in number of classes (species), corrected by the underlying eveness of the abundance distribution. An effective (species) number N is maximal and equal to the nominal number N = n if all species i = 1 . . . n are equally abundant i.e. pi = 1/n ∀i, and if one species is maximally abundant, i.e. ∃i : pi = 1, then the effective number takes its minimum N = 1. Hill (1973) derived a rather general one-parameter class of diversity indices Na that meets these requirements. This fact made Na very attractive for a variety of scientific disciplines: Economists measure the effective number of firms in an industry (Adelman 1969, Miller 1972, Hannah and Kay 1977), political scientists measure effective number of parties in a parliament (Laakso and Taagepera 1979, Dumont and Caulier 2003) and ecologists measure the effective (evenness-corrected) number of species in a community (Hill 1973, Ricotta 2003b). All three groups may find the same Na matching to their given context. However, economists measure industry or market concentration, political scientists measure fragmentation within the spectrum of political parties, and ecologists measure community evenness or diversification of species. These examples show, once again, and in incomplete analogy to the Shannon entropy, how little a mathematical formalism determines the ontological ‘being’ of what they are able to quantify. Simply observing that H S or some N may be used to measure something different in other contexts does not authorize us to sweepingly reject one of the concepts as a diversity measure in ecology. Nevertheless, Jost (2006) wants Na to be exclusively named the class of “true diversities”. In the first two sections of his paper he gives two reasons: (1) Diversity should be proportional to the number of species if all species are equally abundant. Na is proportional in this sense, H S is not. 4 (2) Let H be a continuous and monotonic P q function. Any entopy-like formula of the class H(f (p)), where f (p) = i pi , q ∈ R can be transformed to Na . The first argument may be plausible in many ecologial contexts, but it is not generally sine qua non. If, for instance, diversity is required to incorporate qualitative differences (disparity) instead of abundance distributions, this assumption becomes all but reasonable. Imagine a hypothetical community consisting of two rather similar species which may be equally abundant, say, one individual of wasp species and one of hornet species. Now we double the community richness by adding one rhino and one albatros. In this context it is hardly plausible to require that diversity has doubled because the disparity of the community has certainly more than doubled. And there are biological contexts that require measuring diversity in terms of disparity. Phylogenetic diversity, for example, is based on differences in genes or traits and not on evenness (Vane-Wright 1991, Faith 1994). Outside biology, similar contexts are even more obvious, e.g. the measurement of economic product diversity (Dixit and Stiglitz 1977, Gans and Hill 1997) or molecular diversity in pharmaceutical chemistry (Chabala 1998). Jost’s (2006) normative requirement is appropriate in large parts of the ecological sciences, but it cannot be used to give Na the status of the ‘true’ diversity. Another aspect of Na ’s fundamentally relative character is the trade-off between typical diversity properties (cf. Hoffmann 2006): The class of effective numbers may be proportional to the nominal number if abundances are equally distributed, but on the other hand this class is neither additive (not even pseudo additive) nor generally concave – two properties that can not be valued as irrelevant per se. Additivity of a diversity measure is needed if the individuals of a community are classifiable in more than one way (Pielou 1975). The measure is required to be concave, if diversity is used as portfolio generating function in financial economics (Fernholz 1999, 2002), if diversity helps finding sparse best-basis selections in applied mathematics (Kreutz-Delgado and Rao 1997, 1998), or, if it needs to be additively decomposable into ‘within’ and ‘along’ community components (Patil and Taillie 1982, Lande 1996, Ricotta 2003a, Keylock 2005). Moreover, in all optimization contexts concavity is desirable because it causes global maxima while non concave measures may not do so (Kapur 1994). Due to the general and far reaching desirability of concavity, this property is sometimes even formulated as an axiom that characterizes diversity functions (e.g. S. Pavoine et al. 2004). What properties are finally more valuable is a matter of individual judgements with respect to the given context and the same holds true for the legitimation to use a mathematical expression as a measure for phenomenological quantities. The second one of Jost’s reasons to call Na the true diversity class is even weaker. He shows that there exists a transformation τ for arbitrary functions of a kind H such that τ (H(f (p)) = Na (Proof 1 in Jost 2006). The generic algorithm needed to obtain τ for any given H is the classical algorithm used to find the equivalent number of equal sized categories (cf. Hannah and Kay 1977, Laakso and Taagepera 1979). Applying this algorithm to classical information and entropy measures, Jost observes that τ R (x) := exp(x) for entropies of order a and τ T (x) := expq (x) for entropies of degree b, where expq (x) is a common q-deformed exponential introduced by Tsallis (1994). Taking into account the logarithmic nature of entropy-like formalisms, this observation is not very surprising, and, more importantly, it does not prove anything 5 in favour of Na being a ‘true’ diversity. To illustrate that point, consider a more obvious analogue: Let L(p) = ap+b a, b ∈ R, a 6= 0 be a sub-class of linear functions. Now, lets take any two elements in L, say L1 (p) = 3p and L2 (p) = 2p + 1. Both functions can be transformed into an arbitrary expression of L, for example into L(p) = p. In this case the transformations would be τ L1 (x) = 13 x and τ L2 (x) = 1 (x − 1) for L1 and L2 respectively. The generic algorithm to obtain τ is: (1) 2 subtract b (2) divide by a. It can be easily proved that for arbitrary L there exists a τ such that τ (L(p)) = L(p), but would such proof be a reason to call L(p) = p the ‘true’ linear function? Certainly not. Instead we call L one of many linear functions; one that is characterized by certain properties, such as L(0) = 0, which is plausible in one context and non-matching to another. 5 Conclusion There are good and bad diversity measurement concepts and the distinction between them is determined by properties, contexts and individual judgements. A major benefit of Hill’s (1973) one-parameter class of effective numbers Na is its straightforward unit of measurement and an easy to comprehend proportionality property. A major drawback is that Na is neither additive nor generally concave. However, the effective number may be called a ‘good’ diversity measure, or even ‘the best’ in contexts, where a straightforward unit is more valuable than additivity or concavity. But giving it the absolute status of the “true” diversity is inadequate and misleading. Na incorporates some but definitely not all desirable properties of diversity measures in general. There are ways to get the ‘plethora’ of diversity models under better control. Information theorists and physicists successfully generalized Shannon entropy along different properties (e.g. Rényi 1961, Havrda and Charvát 1967, Borges and Roditi 1998, Kaniadakis et al. 2005) and some economists have already tried to derive a universal, all-encompassing theory of diversity (e.g. Weitzman 1992, Nehring and Puppe 2002). Hoffmann (2006) recently discussed a general diversity measure that includes all classical entropy-like formalisms as well as the effective number class. In complete analogy to the work of Jost (2006) or Hill (1973) all these models are very helpful to make the general meaning of diversity successively more comprehensible, but regarding their ‘true’ universality they fail. 6 Bibliography Aczél, J. and Daróczy, Z. 1975. On measures of information and their characterizations. - Academic Press. Adelman, M. 1969. 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