Fuzzy sets and eigenanalysis in community studies: classification and ordination are “two faces of the same coin”

Abstract

With this paper we want to stress that, on the basis of some matrix algebraic theorems, eigenvectors of similarity matrices are strictly related with clusters that we can obtain with clustering procedures applied to the same similarity matrices and that the fuzzy sets obtained by cluster analysis can be efficiently used as ordination axes and also as tools to measure the diagnostic value (or the indicator value) of attributes (species or other characters) of the ecological systems.

Abbreviations

CA:

Correspondence analysis

FS:

Fuzzy set

OBC:

Ordination based on classification

PCA:

Principal component analysis

References

  1. Amadei, M., R. Bagnaia, L. Laureti, F. Lugeri, N. Lugeri, E. Feoli, M. Dragan and M. Fernetti. 2003. Il progetto Carta della natura alla scala 1:250,000. Metodologia di realizzazione. APAT Manuali e Linee Guida 17/2003

  2. Andreucci, F., E. Biondi, E. Feoli and V. Zuccarello. 2000. Modeling environmental responses of plant associations by fuzzy set theory. Community Ecol. 1: 73–80.

    Article  Google Scholar 

  3. Banykwa, F., E. Feoli and V. Zuccarello. 1990. Fuzzy set system ordination of Serengeti short grasslands, Tanzania. J. Veg. Sci. 1: 97–104.

    Article  Google Scholar 

  4. Biondi, E., E. Feoli and V. Zuccarello. 2004. Modeling environmental responses of plant associations: A review of some critical concepts in vegetation study. Crit. Rev. Plant. Sci. 23: 149–156.

    Article  Google Scholar 

  5. Burba, N., E. Feoli and M. Malaroda. 2008. MATEDIT: A software tool to integrate information in decision making processes. In: R. Neves, J.W. Baretta and M. Mateus (eds.), Perspectives on Integrated Coastal Zone Management in South America. IST PRESS, Lisbon, pp. 123–127.

    Google Scholar 

  6. Ceschin, S., V. Zuccarello and G. Caneva. 2010. Role of macrophyte communities as bioindicators of water quality: application on the Tiber River basin (Italy). Plant Biosyst. 144: 528–536.

    Article  Google Scholar 

  7. Dale, M. B. 1988. Knowing when to stop: cluster concept–concept cluster. Coenoses 1: 11–31.

    Google Scholar 

  8. De Cáceres, M. and P. Legendre. 2009. Associations between species and groups of sites: indices and statistical inference. Ecology 90: 3566–3574.

    Article  Google Scholar 

  9. De Cacéres, M., Legendre, P., Wiser, S.K. and Brotons, L. 2012. Using species combinations in indicator value analysis. Methods Ecol. Evol. 3: 973–982.

    Article  Google Scholar 

  10. Dufrêne, M and P. Legendre. 1997. Species assemblages and indicator species: the need for a flexible assymetrical approach. Ecol. Monogr. 67: 345–366.

    Google Scholar 

  11. Feoli E. 1973a. Un esempio di ordinamento di tipi fitosociologici mediante l’analisi delle componenti principali. Not. Fitosoc. 7:21–27.

    Google Scholar 

  12. Feoli E. 1973b. Un indice che stima il peso dei caratteri per classificazioni monotetiche. Giorn. Bot. Ital. 107:263–268.

    Article  Google Scholar 

  13. Feoli, E. 1976. Correlation between single ecological variables and vegetation by means of cluster analysis. Not. Fitosoc, 12: 77–82.

    Google Scholar 

  14. Feoli, E. 1977a. On the resolving power of principal component analysis in plant community ordination. Vegetatio 33: 119–125.

    Google Scholar 

  15. Feoli, E. 1977b. A criterion for monothetic classification of phytosociological entities based on species ordination. Vegetatio 33:147–152.

    Google Scholar 

  16. Feoli, E. 1984. Some aspects of classification and ordination of vegetation data in perspective. Stud. Geobot. 4: 7–21.

    Google Scholar 

  17. Feoli, E. 2012. Diversity patterns of vegetation systems from the perspective of similarity theory. Plant Biosyst. 146: 797–804.

    Article  Google Scholar 

  18. Feoli, E. and G. Bressan. 1972. Affinità floristica dei tipi di vegetazione bentonica della Cala di Mitigliano (Massa Lubrense, Napoli). Giorn. Bot. Ital. 106:245–256.

    Article  Google Scholar 

  19. Feoli, E. and L. Feoli Chiapella. 1980. Evaluation of ordination methods through simulated coenoclines: some comments. Vegetatio 42: 35–41.

    Article  Google Scholar 

  20. Feoli, E. and D. Lausi. 1980. Hierarchical levels in syntaxonomy based on information functions. Vegetatio 42: 113–115.

    Article  Google Scholar 

  21. Feoli, E. and L. Orlóci. 1979. Analysis of concentration and detection of underlying factors in structured tables. Vegetatio 40: 49–54.

    Article  Google Scholar 

  22. Feoli, E. and L. Orlóci (eds.) 1991. Computer Assisted Vegetation Analysis. Kluwer, Boston.

    Google Scholar 

  23. Feoli, E. and L. Orlóci. 2011. Can similarity theory contribute to the development of a general theory of the plant community? Community Ecol. 12: 135–141.

    Article  Google Scholar 

  24. Feoli, E. and V. Zuccarello. 1986. Ordination based on classification: yet another solution? Abstr. Bot. 10: 203–219.

    Google Scholar 

  25. Feoli, E. and V. Zuccarello. 1988. Syntaxonomy: A source of useful fuzzy sets of environmental analysis? Coenoses 3: 141–147.

    Google Scholar 

  26. Feoli, E. and V. Zuccarello. 1994. Naivete of fuzzy system space in vegetation dinamics? Coenoses 9: 25–3.

    Google Scholar 

  27. Feoli, E. and Zerihun Woldu. 2000. Fuzzy set analysis of the Ethiopian Rift Valley vegetation. Plant Ecol. 147: 219–22.

    Article  Google Scholar 

  28. Feoli, E., G. Ferro and P. Ganis. 2006. Validation of phytosociological classifications based on a fuzzy set approach. Community Ecol. 7: 98–117.

    Article  Google Scholar 

  29. Gauch, H.G. and R.H. Whittaker. 1972. Coenocline simulation. Ecology 53: 446–451.

    Article  Google Scholar 

  30. Gauch, H.G. 1982. Multivariate Analysis in Community Ecology. Cambridge University Press, New York.

    Book  Google Scholar 

  31. Goodall, D.W. 1964. A probabilistic similarity index. Nature 203:1098.

    Article  Google Scholar 

  32. Goodall, D.W. 1966. A new similarity index based on probability. Biometrics 22:882–907.

    Article  Google Scholar 

  33. Goodall, D.W. 1993. Probabilistic indices for classification – Some extensions. Abstr. Bot. 17:125–132.

    Google Scholar 

  34. Gower, J.C. 1971. A general coefficient of similarity and some of its properties. Biometrics 27:857–871.

    Article  Google Scholar 

  35. Hill, M.O., R.G.H. Bunce and M.V. Shaw. 1975. Indicator species analysis, a divisive polythetic method of classification, and its application to survey of native pinewoods in Scotland. J. Ecol. 63: 597–613.

    Article  Google Scholar 

  36. Hill, M. O. 1979. TWINSPAN - A FORTRAN programme for arranging multivariate data in an ordered two-way table by classification of individuals and attributes. Cornell University, Ithaca, New York.

    Google Scholar 

  37. Kenkel, N.C. and L. Orlóci. 1986. Applying metric and nonmetric multidimensional scaling to ecological studies: some new results. Ecology 67: 919–928.

    Article  Google Scholar 

  38. Mantel, N. 1967. The detection of disease clustering and a generalized regression approach. Cancer Res. 27: 209–220.

    CAS  PubMed  PubMed Central  Google Scholar 

  39. Marsili-Libelli, S. 1989. Fuzzy clustering of ecological data. Coenoses 4: 95–106.

    Google Scholar 

  40. Moraczewski, I.R. 1993a. Fuzzy logic for phytosociology. 1 Syntaxa as vague concept. Vegetatio 106:1–11.

    Article  Google Scholar 

  41. Moraczewski, I.R. 1993b. Fuzzy logic for phytosociology. 2 Generalizations and prediction. Vegetatio 106:13–20.

    Article  Google Scholar 

  42. Moraczewski, I.R. 1996. Fuzzy rough set as a tool for ecological data analysis. Coenoses 11:55–67.

    Google Scholar 

  43. Noy-Meir, I. 1973. Data transformation in ecological ordination. I. Some advantages of non-centering. J. Ecol. 61:329–341.

    Article  Google Scholar 

  44. Olano, J.M., J.J. Loidi, A. González and A. Escudero. 1998. Improving the interpretation of fuzzy partitions in vegetation science with constrained ordinations. Plant Ecol. 134:113–118.

    Article  Google Scholar 

  45. Orlóci, L. 1967. Data centering: a review and evaluation with reference to component analysis. Syst. Zool. 16:208–212.

    Article  Google Scholar 

  46. Orlóci, L. 1972. On objective functions of phytosociological resemblance. Am. Midl. Nat. 88:28–55.

    Article  Google Scholar 

  47. Orlóci, L. 1978. Multivariate Analysis in Vegetation Research. 2nd ed. Junk, The Hague.

    Google Scholar 

  48. Pillar, V.D. 1999. How sharp are classifications? Ecology 80: 2508–2516.

    Article  Google Scholar 

  49. Podani, J. 2000. Introduction to the Exploration of Multivariate Biological Data. Backhuys, Leiden.

    Google Scholar 

  50. Podani, J. and B. Csányi. 2010. Detecting indicator species: some extensions of the IndVal measure. Ecol. Indic. 10: 1119–1124.

    Article  Google Scholar 

  51. Podani, J. and E. Feoli. 1991. A general strategy for the simultaneous classification of variables and objects in ecological data tables. J. Veg. Sci. 2:435–444.

    Article  Google Scholar 

  52. Roberts, D.W. 1986. Ordination on the basis of fuzzy set theory. Vegetatio 66: 123–143.

    Article  Google Scholar 

  53. Roberts, D.W. 2008. Statistical analysis of multidimensional fuzzy set ordinations. Ecology 89: 1246–1260.

    Article  PubMed  PubMed Central  Google Scholar 

  54. Roberts, D.W. 2009. Comparison of multidimensional fuzzy set ordination with CCA and DB-RDA. Ecology 90: 2622–2634.

    Article  PubMed  PubMed Central  Google Scholar 

  55. Shu, L., A. Chen, M. Xiong and W. Meng. 2011. Efficient SPectrAl neighborhood blocking for entity resolution. Proceedings IEEE 27th International Conference on Data Engineering. pp. 1063–1078.

  56. von Luxburg, U. 2007. A tutorial on spectral clustering. Statistics and Computing 17:395–416.

    Article  Google Scholar 

  57. van der Maarel, E., L. Orlóci and S. Pignatti. 1980. Data Processing in Phytosociology: Retrospect and Anticipation. Junk, The Hague.

    Book  Google Scholar 

  58. Wilkinson, J.H. 1965. The Algebraic Eigenvalue Problem. Oxford University Press, London.

    Google Scholar 

  59. Wildi, O. 2010. Data Analysis in Vegetation Ecology. John Wiley & Sons, Padstow.

    Book  Google Scholar 

  60. Woldu, Zerihun, Feoli, E. and L. Nigatu. 1989. Partitioning an elevation gradient of vegetation from South-eastern Ethiopia by probabilistic methods. Vegetatio 81: 189–198.

    Article  Google Scholar 

  61. Zimmerman, H.G. 1996. Fuzzy Set Theory and its Applications. 3rd ed. Kluwer Academic Publishers, Dordrecht.

    Book  Google Scholar 

  62. Zuccarello, V., M. Allegrezza, E. Biondi and R. Calandra. 1999. Valenza ecologica di specie e di associazioni prative e modelli di distribuzione lungo gradienti sulla base della teoria degli insiemi sfocati (Fuzzy set Theory). Braun-Blanquetia 16: 121–226.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to E. Feoli.

Rights and permissions

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Cite this article

Feoli, E., Zuccarello, V. Fuzzy sets and eigenanalysis in community studies: classification and ordination are “two faces of the same coin”. COMMUNITY ECOLOGY 14, 164–171 (2013). https://doi.org/10.1556/ComEc.14.2013.2.6

Download citation

Keywords

  • Clusters
  • Data matrices
  • Diagnostic value
  • Indicator value
  • Ordination axes
  • Similarity matrices
  • Vegetation