Abstract
Cover-abundance estimates are commonly employed in phytosociological investigations to record the performance of species. Because the coded values are on an ordinal scale of measure, various authors have suggested that some transformation is necessary before such values can be used for classification and ordination. However, it is not clear that transformation is a sufficient treatment, and it would seem preferable to use ordinal data directly. In this paper we examine such direct use of partial rankings and show that several dissimilarity measures can be defined for this case without invoking any transformations. They include dissimilarity measures associated with various rank correlation measures and with distances between strings; all the measure are variant forms of Hausdorf's interset distance. Certain other kinds of data, such as those employing dominant and subdominant species and the dry-weight-rank estimation of biomass, are also on an ordinal scale and could be analysed using similar techniques.
To illustrate the approach, a string dissimilarity measure is used to analyse a set of data from Slovakian grasslands which appear to reflect a simple gradient. The original data were recorded with 10 classes of performance and are analysed using hierarchical and nondeterministic, overlapping, classifications.
Similar content being viewed by others
References
Arabie, P. & Carroll, J. D. 1980. MAPCLUS: a mathematical programming approach to fitting the ADCLUS mode. Psychometrika 45: 211–235.
Austin, M. P. & Belbin, L. 1982. A new approach to the species classification problem in floristic analysis. Aust. J. Ecol. 7: 75–89.
Bannister, P. 1966. The use of subjective cover-abundance as a basis for ordination. J. Ecol. 54: 665–674.
Bednarek, A. R. & Ulam, S. M. 1979. An integer valued metric for patterns. Fundamentals of computation theory, pp. 52–57. Academic-Verlag, Berlin.
Ben-Bassat, M. & Zaidenberg, L. 1984. Contextual template matching: a distance measure for patterns with hierarchically dependent features. IEEE Trans. Patt. Anal. Mach. Intell. PAMI 6: 201–211.
Berlekamp, E. R. 1968. Algebraic coding theory. McGraw Hill, New York.
Beyer, W. A., Stein, M. L., Smith, T. F. & Ulam, S. M. 1974. A molecular sequence metric and evolutionary trees. Math. Biosci. 19: 9–25.
Bückenholt, I. & Gaul, W. 1988. Probabilistic multidimensional scaling of paired comparison data. In: Bock, H. H. (ed.), Classification and other related methods of data analysis, pp. 405–412. Elsevier, Amsterdam.
Cayley, A. 1849. A note on the theory of permutations. Phil. Mag. 34: 527–529.
Clifford, H. T. & Goodall, D. W. 1967. A numerical contribution to the classification of the Poaceae. Aust. J. Bot. 15: 499–519.
Critchlow, D. 1985. Metric methods for analyzing partially ranked data. Springer-Verlag, New York.
Currall, J. E. P. 1987. A transformation of the Domin scale. Vegetatio 72: 81–87.
Dale, M. B. 1968. On property structure, numerical taxonomy and data handling. In: Heywood, V. H. (ed.), Modern methods in plant taxonomy, pp. 185–197. Academic Press, London.
Dale, M. B. 1970. Information analysis of quantitative data. In: Patil, G., Pielou, E. & Waters, W. (eds), Stat. Ecol. 3: 1433–140.
Dale, M. B. 1988. Knowing when to stop: cluster concept-concept cluster. Coenoses 3: 11–32.
Dale, M. B. in press. Mutational and nonmutational similarity measures: a preliminary examination. Coenoses.
Dale, M. B. 1988. Fuzzy and nondeterministic classification in vegetation studies: Ideals and instances. Folia Geobot. Phytotax. Praha.
Dale, M. B. & Anderson, D. J. 1972. Qualitative and quantitative information analysis. J. Ecol. 60: 639–653.
Dale, M. B. & Dale, P. E. R. 1986. Similarity and structure attributes in ecological classification. Abstr. Botan. 10: 17–34.
Dale, M. B., Feoli, E. & Ganis, P. 1988. Incorporation of information from the taxonomic hierarchy in comparing vegetation types. Taxon.
Dale, M. B., Ferrari, C., Beatrice, M. & Venanzoni, R. 1986. A comparison of some methods of species selection. Coenoses 1: 35–51.
Day, W. E. & Faith, D. P. 1986. A model in partial orders for comparing objects by dualistic measures. Math. Biosci. 78: 179–192.
deLeeuw, J. & vanRijckevorsel, J. J. L. 1980. HOMALS & PRINCALS, some generalizations of principal component analysis. In: Diday, E., Lebart, L., Pagès, J. P. & Tomassone, R. (eds), Data analysis and informatics. North Holland, Amsterdam.
Estabrook, G. F. 1972. Cladistic methodology: a discussion of the theoretical basis for the induction of evolutionary history. Ann. Rev. Ecol. Syst. 9: 335–341.
Estabrook, G. F. & Meacham, C. A. 1979. How to determine the compatibility of undirected character state trees. Math. Biosci. 46: 251–256.
Faith, D. P. 1987. Robust multidimensional scaling when measured variables have general unimodal relationships to the underlying space. First Conf. International Federation Classification Soc., Aachen.
Feoli, E. & Lagonegro, M. 1983. A resemblance function based on probability: application to field and simulated data. Vegetatio 53: 3–9.
Friedman, J. H. & Rafsky, L. C. 1979. Multivariate generalization of the Wald-Wolfowitz and Smirnov two-sample test. Ann. Stat. 7: 697–717.
Gini, C., Boldrini, M., galvani, L. & Venere, A. 1933. Sui centri della populazione e sulle loro applicazioni. Metron 11: 3–102.
Good, I. J. & Smith, E. P. 1986. An additive algorithm analogous to the singular decomposition; or a comparison of polarization and multiplicative models: An example of qualitative robustness. Comm. Statist.-Simulation Commput. 15: 545–569.
Goodall, D. W. 1964. A probabilistic similarity index. nature 203: 1098.
Gower, J. C. 1966. Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53: 325–338.
Gower, J. C. 1974. The mediancentre. Appl. Statist. 23: 466–470.
Gunlaugsdóttir, E. 1985. Composition and dynamic status of heathland communities in Iceland in relation to recovery measures. Acta Phytogeog. Suec. 75: 1–84.
Hausdorf, F. 1927. Mengenlehre. de Gruyter, Berlin.
Hill, M. O., Bunce, R. G. H. & Shaw, M. W. 1975. Indicator species analysis, a divisive polythetic method of classification and its application to a survey of native pine-woods in Scotland. J. Ecol. 63: 597–613.
Hohn, M. E. & Nuhfer, E. B. 1980. Asymmetric measures of association, classed data and multivariate analysis. Math. Geol. 12: 235–246.
Holgate, P. 1971. Notes on the Marczewski-Steinhaus coefficient of similarity. In: Patil, G., Pielou, E. & Waters, W. (eds). Stat. Ecol. 3: 181–193.
Ito, T., Kodama, Y. & Toyoda, J. 1984. A similarity measure between patterns with nonindependent attributes. IEEE Trans. Patt. Anal. Mach. Intell. PAMI 6: 111–115.
Jensén, S. 1978. Influences of transformation of cover values on classification and ordination of lake vegetation. Vegetatio 37: 19–31.
Jensén, S. & van derMaarel, E. 1974. Numerical approaches to lake classification with special reference to macrophyte communities. Vegetatio 42: 117–128.
Kendall, M. G. 1938. A new measure of rank correlation. Biometrika 30: 81–93.
Kendall, M. G. 1950. Rank correlation methods. Hafner Publishing Co., New York.
Lefkovitch, L. P. 1987. Clustering from ordination. Math. Biosci. 87, 17–30.
Lemone, K. A. 1982. Similarity measures extended to sets of strings. IEEE Trans. Patt. Anal. Mach. Intell. PAMI 4: 345–347.
Levenshtein, V. I. 1965. Binary codes capable of correcting deletions, insertions and reversals. Dokl. Akad. Nauk SSR. 163: 825–828.
Lowrance, R. & Wagner, R. A. 1975. An extension to the string-to-string correction problem. J. Assoc. Comput. Mach. 22: 177–183.
Lu, S-Y. 1984. A tree matching algorithm based on node splitting and merging. IEEE Trans. Patt. Anal. Mach. Intell. PAMI 6: 249–256.
Mallows, C. 1957. Non-Null ranking models. Biometrika 44: 114–130.
't Mannetje, L. & Haydock, K. P. 1963. The dry weight rank method for the botanical analysis of pasture. J. Brit. Grassland Soc. 18: 268–275.
McCullagh, P. 1980. Regression models for ordinal data. J. Royal Statist. Soc. B 42: 109–142.
Meulman, J. 1986. A distance approach to nonlinear multivariate analysis. DSWO Press, Leiden.
Meulman, J. 1988. Nonlinear redundancy analysis via distances. In: Bock, H. H. (ed.), Classification and related methods of data analysis. pp. 515–522. Elsevier, Amsterdam.
Mueller-Dombois, D. R. & Ellenberg, H. 1974. Aims and methods in vegetation ecology, Wiley, New York.
Orlowska, E. & Pawlak, Z. 1984. Representation of nondeterministic information. Theor. Comput. Sci. 29: 27–39.
Ozawa, K. 1983. CLASSIC: a hierarchical clustering algorithm based on asymmetric similarities. Patt. Recog. 16: 201–211.
Pettitt, A. N. 1984. Tied, grouped continuous and ordered categorical data: a comparison of 2 models. Biometrika 71: 35–42.
Plackett, R. L. 1975. The analysis of permutations. Appl. Stat. 24: 193–202.
Prim, R. C. 1957. Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36: 1389–1401.
Sankoff, D. & Kruskal, J. B. 1983. Time warps, string edits and macromolecules: the theory and practice of sequence comparison. pp. 382. Addison Wesley, London.
Sattath, S. & Tversky, A. 1977. Additive similarity trees. Psychometrika 42: 319–345.
Schouten, H. J. A. 1982. Measuring pairwise agreement among many observers. II. Some improvements and additions. Biom. J. 24: 431–435.
Spearman, C. 1904. The proof and measurement of association between two things. Amer. J. Psych. 15: 72–101.
Spearman, C. 1906. A footrule for measuring correlation. Brit. J. Psych. 2: 89–108.
Srivastavan, R. & Srivastavan, A. K. 1985. On fuzzy Hausdorfness concepts. Fuzzy Sets and Systems 17: 67–71.
Tomek, I. 1976. Two modifications of CNN. IEEE Trans. Syst. Man. Cyber. SMC 6: 769–772.
van derBurg, A. & deLeeuw, J. 1983. Non-linear canonical correlation. Brit. J. Math. Statist. Psych. 36: 54–80.
van derMaarel, E. 1979. Transformation of cover-abundance values in phytosociology and its effect on community similarity. Vegetatio 39: 97–114.
van derMaarel, E., Espejel, I. & Moreno-Casasola, P. 1986. Two-step vegetation analysis based on very large data sets. Vegetatio 68: 139–143.
Verhelst, N. D., Koppen, M. G. M. & vanEssen, E. P. 1985. The exact distribution of an index of agreement between partitions. Brit. J. Math. Stat. Psych. 38: 44–57.
Werman, M., Pelg, S. & Rosenfeld, A. 1985. A distance metric for multidimensional histograms. Comput. Vision, Graphics and Image Processing 32: 328–336.
Whitfield, J. W. 1953. The distribution of total rank values for one particular object in m rankings of n objects. Brit. J. Statist. Psych. 6: 35–40.
Zadeh, C. T. 1971. Similarity relations and fuzzy sets. Inform. Sci. 3: 177–200.
Zinnes, J. L. & Griggs, R. A. 1974. Probabilistic, multidimensional unfolding analysis. Psychometrika 39: 327–350.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dale, M.B. Dissimilarity for partially ranked data and its application to cover-abundance data. Vegetatio 82, 1–12 (1989). https://doi.org/10.1007/BF00217977
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00217977