New Measures for Comparing the Species Diversity Found in Two or More Habitats

Both the weighted entropy, which generalizes the Shannon entropy, and the weighted quadratic index, which generalizes the Gini-Simpson index, are used for getting a unified treatment of some diversity measures proposed recently in ecology. The weights may reflect the ecological importance, rarity, or economic value of the species from a given habitat. The weighted measures, being concave functions, may be used in the additive partition of diversity. The weighted quadratic index has a special advantage over the weighted entropy because its maximum value has a simple analytical formula which allows us to introduce a normed measure of dissimilarity between habitats. A special case of weighted quadratic index is the Rich-Gini-Simpson index which, unlike the Shannon entropy and the classic Gini-Simpson index, behaves well when the number of species is very large. The weighted entropy and the weighted quadratic index may also be used to measure the global diversity among the subsets of species. In this context, Rao's quadratic index of diversity between the pairs of species, based on the phylogenetic distance between species, is obtained as a particular case and is generalized to measure the diversity among the triads of species as well.

[1]  E. C. Pielou,et al.  Species-diversity and pattern-diversity in the study of ecological succession. , 1966, Journal of theoretical biology.

[2]  G. Patil,et al.  Diversity as a Concept and its Measurement , 1982 .

[3]  Zoltán Botta-Dukát,et al.  Rao's quadratic entropy as a measure of functional diversity based on multiple traits , 2005 .

[4]  Mark Kon,et al.  A New Phylogenetic Diversity Measure Generalizing the Shannon Index and Its Application to Phyllostomid Bats , 2009, The American Naturalist.

[5]  K. Shimatani,et al.  On the measurement of species diversity incorporating species differences , 2001 .

[6]  L. Jost.  Entropy and diversity , 2006 .

[7]  János Izsák,et al.  A link between ecological diversity indices and measures of biodiversity. , 2000 .

[8]  E. C. Pielou.  Shannon's Formula as a Measure of Specific Diversity: Its Use and Misuse , 1966, The American Naturalist.

[9]  Dorion Sagan,et al.  Microcosmos: Four Billion Years of Evolution from Our Microbial Ancestors , 1986 .

[10]  H. Tuomisto.  A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity , 2010 .

[11]  Calyampudi R. Rao.  Diversity and dissimilarity coefficients: A unified approach☆ , 1982 .

[12]  L. Jost.  Partitioning diversity into independent alpha and beta components. , 2007, Ecology.

[13]  C. R. Rao,et al.  Cross entropy, dissimilarity measures, and characterizations of quadratic entropy , 1985, IEEE Trans. Inf. Theory.

[14]  Antonio Di Crescenzo,et al.  On Weighted Residual and Past Entropies , 2006 .

[15]  C. Ricotta,et al.  Towards a unifying approach to diversity measures: bridging the gap between the Shannon entropy and Rao's quadratic index. , 2006, Theoretical population biology.

[16]  P. Couteron,et al.  An operational, additive framework for species diversity partitioning and beta‐diversity analysis , 2007 .

[17]  D. Chessel,et al.  From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis. , 2004, Journal of theoretical biology.

[18]  E. H. Simpson.  Measurement of Diversity , 1949, Nature.

[19]  A. Chao,et al.  Partitioning diversity for conservation analyses , 2010 .

[20]  R. Routledge.  On Whittaker's Components of Diversity , 1977 .

[21]  Silviu Guiasu,et al.  A quantitative-qualitative measure of information in cybernetic systems (Corresp.) , 1968, IEEE Trans. Inf. Theory.

[22]  Silviu Guiasu,et al.  Conditional and Weighted Measures of Ecological Diversity , 2003, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[23]  O. Hardy,et al.  Characterizing the phylogenetic structure of communities by an additive partitioning of phylogenetic diversity , 2007 .

[24]  Silviu Guiaşu,et al.  Information theory with applications , 1977 .

[25]  L. Jost,et al.  Interpreting and estimating measures of community phylogenetic structuring , 2008 .

[26]  Carlo Ricotta,et al.  A parametric diversity measure combining the relative abundances and taxonomic distinctiveness of species , 2004 .

[27]  T. O. Crist,et al.  The additive partitioning of species diversity: recent revival of an old idea , 2002 .

[28]  Otto Wildi,et al.  Additive partitioning of plant species diversity in an agricultural mosaic landscape , 2000, Landscape Ecology.

[29]  Hanna Tuomisto,et al.  A diversity of beta diversities: straightening up a concept gone awry. Part 2. Quantifying beta diversity and related phenomena , 2010 .

[30]  Carlo Ricotta,et al.  Bridging the gap between ecological diversity indices and measures of biodiversity with Shannon's entropy: comment to Izsák and Papp , 2002 .

[31]  L. Jost.  Mismeasuring biological diversity: : Response to Hoffmann and Hoffmann (2008) , 2009 .

[32]  S. Pavoine,et al.  Measuring diversity from dissimilarities with Rao's quadratic entropy: are any dissimilarities suitable? , 2005, Theoretical population biology.

[33]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[34]  Bernadette Bouchon.  Useful information and questionnaires , 1976 .

[35]  A. Rényi.  On Measures of Entropy and Information , 1961 .

[36]  R. Macarthur.  PATTERNS OF SPECIES DIVERSITY , 1965 .

[37]  R. Whittaker.  Evolution and measurement of species diversity , 1972 .

[38]  C. Ricotta,et al.  Diversity partitioning of Rao's quadratic entropy. , 2009, Theoretical population biology.

[39]  Harry F. Recher,et al.  On the Relation between Habitat Selection and Species Diversity , 1966, The American Naturalist.

[40]  A. Magurran,et al.  Measuring Biological Diversity , 2004 .

[41]  Sergi Villagrasa,et al.  Statistical approach of user’s experience in the visualization of architectural images in different environments , 2010 .

[42]  Hans-Peter Weikard,et al.  Diversity measurement combining relative abundances and taxonomic distinctiveness of species , 2006 .

[43]  Sönke Hoffmann,et al.  Is there a “true” diversity? , 2008 .

[44]  M. Hill.  Diversity and Evenness: A Unifying Notation and Its Consequences , 1973 .

[45]  R. Lande.  Statistics and partitioning of species diversity, and similarity among multiple communities , 1996 .

[46]  Silviu Guiasu,et al.  Grouping data by using the weighted entropy , 1986 .

[47]  P. Couteron,et al.  Additive apportioning of species diversity: towards more sophisticated models and analyses , 2004 .