Abstract
Correspondence analysis has found widespread application in analysing vegetation gradients. However, it is not clear how it is robust to situations where structures other than a simple gradient exist. The introduction of instrumental variables in canonical correspondence analysis does not avoid these difficulties. In this paper I propose to examine some simple methods based on the notion of the plexus (sensu McIntosh) where graphs or networks are used to display some of the structure of the data so that an informed choice of models is possible. I show that two different classes of plexus model are available. These classes are distinguished by the use in one case of a global Euclidean model to obtain well-separated pair decomposition (WSPD) of a set of points which implicitly involves all dissimilarities, while in the other a Riemannian view is taken and emphasis is placed locally, i.e., on small dissimilarities. I show an example of each of these classes applied to vegetation data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- MST:
-
Minimal Spanning Tree
- CA:
-
Correspondence Analysis
- w-b-c:
-
Williams, Bunt and Clay (1991).
References
Agarwal, P. K., J. Matousek and S. Suri. 1992. Farthest neighbors, maximum spanning trees, and related problems in higher dimensions. Comput. Geom.: Theory and Appl. 4:189–201.
Al Ayouti, B. 1992. New forms of graphical representation in data analysis: additive forests. Proc Conf. Distancia, Rennes.
Allison, L. & C. S. Wallace. 1994. The posterior probability distribution of alignments and its application to parameter estimation of evolutionary trees and to optimisation of multiple alignments. J. Molecular Evolution 39:418–430.
Althöfer, I., G. Das, D. Dobkin, D. Joseph and J. Soares. 1993 On sparse spanners of weighted graphs. Discrete Comput. Geom. 9:81–100.
Ash, P. F. and E. D. Bolker. 1986. Generalized Dirichlet tessellation. Geometriae Dedicata 20: 209–243.
Aurenhammer, F. 1991. Voronoi diagrams - a survey of a fundamental geometric data structure. A. C. M. Computing Surveys 23: 345–405.
Austin, M. P. 1976. On nonlinear species response models in ordination. Vegetatio 33: 33–41.
Austin, M. P. 1990. Community theory and competition in vegetation. In: J. B. Grace and D. Tilman (eds.), Perspectives on Plant Competition. Academic Press, San Diego, pp. 215–238.
Babad, Y. M. & J. A. Hoffer. 1984. Even no data has value. Commun. Assoc. Comput. Mach. 27: 748–756.
Bandelt, H-J. and A. W. M. Dress. 1992. A canonical decomposition theory for metrics on a finite set. Adv. Math. 92: 47–105.
Barkman, J. J. 1965. Die Kryptogamenflora einiger Vegetationstypen in Drente und ihr Zusammenhang mit Boden und Mikroklima. In: R. Tuxen (ed.), Biosoziologie, Ber Symp. Int. Ver. Vegetskunde. Stolzenau/Weser 1960, pp. 157–171.
Beals, E. W. 1973. Ordination: mathematical elegance and ecological naivete. J. Ecol. 61: 23–35.
Birks, H. L. B., S. M. Peglar and H. A. Austin. 1996. An annotated bibliography of canonical correspondence analysis and related constrained ordination methods 1986–1993. Abstracta Botanica 20: 17–36.
Bradfield, G. E. and N. C. Kenkel. 1987. Nonlinear ordination using flexible shortest path adjustment of ecological distances. Ecology 68: 750–753.
Cai, L. 1994. NP-completeness of minimum spanner problems. Discrete Appl. Math. 48: 187–194.
Camerini, P. M. 1978. The min-max spanningtree problem and some extensions. Inform. Proc. Lett. 7: 10–14.
Camerini, P. M, F Maffioli, S. Martello and P. Toth. 1986. Most and least uniform spanning trees. Discrete Applied Math. 15: 81–187.
Chatterjee, S. and A. Narayanan. 1992. A new approach to discrimination and classification using a Hausdorff type metric. Austral. J. Statist. 34:391–406.
Critchlow, D. 1985. Metric Methods for Analyzing Partially Ranked Data. Springer-Verlag, New York.
Culik II, K. and H. A. Maurer. 1978. String representations of graphs. Internt. J. Computer Math. Sect. A 6: 272–301.
Dale, M. B. 1975. On the objectives of ordination. Vegetatio 30: 15–32.
Dale, M. B. 1994. Straightening the horseshoe: a Riemannian resolution? Coenoses 9: 43–53
De’ath, G. 1999. Principal curves: a new technique for indirect and direct gradient analysis. Ecology 80: 2237–2253.
de Soete, G. 1988. Tree representations of proximity data by least squares methods. In: H. H. Bock (ed.), Classification and Related Methods of Data Analysis. North Holland, Amsterdam, pp. 147–156.
de Vries 1952. Objective combination of species. Acta Bot. Neerl. 1: 497–499.
de Vries, D. M., J. P. Baretta and G. Haming. 1954. Constellation of frequent herbage plants based on their correlation in occurrence. Vegetatio 5/6: 105–111.
Deichsel, G. 1980. Random walk clustering in large data sets. COMPSTAT1980. Physica-Verlag, Vienna pp. 454–459.
Diday, E. and P. Bertrand. 1986. An extension to hierarchical clustering: the pyramidal presentation. In: E. S. Gelsema and L. N. Kanak (eds.), Pattern Recognition in Practice. Elsevier Science, Amsterdam, pp. 411–424.
Dobkin, D., S. J. Friedman, and K. J. Supowit. 1990. Delaunay graphs are almost as good as complete graphs. Discrete Comput. Geom. 5: 399–407.
Dress, A. W. M, D. H. Huson and V. Moulton. 1996. Analyzing and visualizing sequence and distance data using “SplitsTree”. Discrete Applied Math. 71: 95–109.
Duckworth, J. C., R. G. H. Bunce and A. J. C. Malloch. 2000. Vegetation-environment relationships in Atlantic European calcareous grasslands. J. Veg. Sci. 11:15–22.
Edgoose, T. and L. Allison. 1999. MML Markov classification of sequential data. Statistics and Computing 9: 269–278.
Eilertson, O., R. H. Økland, T. Økland and O. Pederson. 1989. The effects of scale range, species removal and downweighting of rare species on eigenvalue and gradient length in DCA ordination. J. Veg. Sci. 1:261–270.
Eppstein, D. 1992. The farthest point Delaunay triangulation minimizes angles. Computational Geometry Theory and Applications 1: 143–148.
Escofier, B., H. Benali and K. Bachar. 1990. Comment introduire la contiguité en analyse des correspondances? Application en segmentation d’image. Rapport de recherche de l’INRIA - Rennes, RR-1191, 22 pages - Mars 1990.
Falinski, J. 1960. Zastosowanie taksonomii wroclawskiej do fitosocjologii. Acta Soc. bot. Pol. 29: 333–361.
Famili, A. and P. Turney. 1991. Intelligently Helping Human Planner in Industrial Process Planning. AIEDAM 5: 109–124.
Fayyad, U., G. Piatetsky-Shapiro and P. Smyth. 1996. From Data Mining to Knowledge Discovery. In: U. Fayyad et al. (eds.), Advances in Knowledge Discovery and Data Mining. AAAI/MIT Press, Menlo Park, CA, pp. 1–34.
Friedman, J. H. and L. C. Rafsky. 1979. Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7: 697–717.
Friedman, J. H. and L. C. Rafsky. 1983 Graph-theoretic measures of multivariate association and prediction. Ann. Statist. 11: 377–391.
Gabriel, K. R. and C. L. Odoroff. 1984. Resistant lower rank approximation of matrices. In: E. Diday, M. Jambu, L. Lebart, J. Pagés, and R. Tomassone (eds.), Data Analysis and Informatics III North Holland, Amsterdam. pp. 23–30.
Gabriel, K. R., C. L. Odoroff and S. Choi. 1988. Fitting lower dimensional ordinations to incomplete similarity data. In: H. H. Bock (ed.), Classification and Related Methods of Data Analysis. Elsevier- North Holland, pp. 445–454.
Gabriel, K. R. and R. R. Sokal. 1969. A new statistical approach to geographical analysis. Syst. Zool. 18: 54–64.
Gimingham, C. H. 1961. North European heath communities: a network of variation. J. Ecol. 49: 655–694.
Godehardt, E. and H. Herrmann. 1988. Multigraphs as atool for numerical classification. In: H. Bock (ed.), Classification and related methods of Data Analysis. Elsevier, North Holland, pp. 219–229.
Goodall, D. W. and R. W. Johnson. 1982. Non-linear ordination in several dimensions: a maximum likelihood approach. Vegetatio 48: 197–208.
Goodall, D. W. and R. W. Johnson. 1987. Maximum likelihood ordination: some improvements. Vegetatio 73: 3–13.
Hill, M. O. 1973. Reciprocal averaging: an eigenvector method of ordination. J. Ecol. 61: 237–249.
Hubert, L. and P. Arabic 1992. Correspondence analysis and optimal structural representations. Psychometrika 56: 119–140.
Hubert, L. and P. Arabie. 1994. The analysis of proximity matrices through sums of matrices having (anti-)Robinson forms. Brit. J. Math. Statist. Psychol. 47: 1–40.
Hubert, L. and J. Schultz. 1975. Hierarchical clustering and the concept of space distortion. Brit. J. Math. Statist. Psychol. 28:121–133.
Hubert, L. and J. Schultz. 1976. Quadratic assignment as a general data analysis strategy. Brit. J. Math. Statist. Psychol. 29: 190–241.
Huisman, J., H. Olff and L. F. M. Fresco. 1993. A hierarchical set of models for species response analysis. J. Veg. Sci. 4: 37–46.
Ihm, P and H. van Groenewoud. 1975A multivariate ordering of vegetation data based on Gaussian type gradient response curves J. Ecol. 63: 767–777.
Karadžić, B. and R. Popović. 1994. A generalized standardization procedure in ecological ordination: test with Principal Components Analysis. J. Veg. Sci. 5: 259–262.
Keil, J. M. and C. A. Gutwin. 1992. Classes of graphs which approximate the complete Euclidean graph. Computational Geometry 7: 13–28.
Kendall, D. G. 1971. Seriation from abundance matrices. In: F. R. Hodson, D. G. Kendall and P. Tautu (eds.), Mathematics in the Archaeological and Historical Sciences. Edinburgh Univ. Press. pp. 215–252.
Klauer, K. C. 1989. Ordinal network representation: representing proximities by graphs. Psychometrika 54: 737–750.
Levcopoulos, C. and A. Lingas. 1989. There are planar graphs almost as good as complete graphs and about as cheap as the minimal spanning tree. Proc. Internatl. Symp. Optimal Algorithms. Lecture Notes in Computer Science 401, Springer, Berlin, pp. 9–13.
Maa, J-F, D. K. Pearl and R. Bartoszyński. 1996. Reducing multidimensional two-sample data to one-dimensional interpoint comparisons. Annals Statistics 24: 1069–1074.
Mcintosh, R. P. 1973. Matrix and plexus techniques. In: R. H. Whittaker (ed.), Ordination and Classification of Communities. Dr. W. Junk, den Haag. pp. 157–191.
Minchin, P. R. 1987. An evaluation of the relative robustness of techniques for ecological ordination. Vegetatio 69: 89–107.
Mulder, H. M. and A. Schrijver. 1979. Median graphs and Helly hypergraphs. Discrete Mathematics 25: 41–50.
Murtagh, F. 1983. A probability theory of hierarchic clustering using random dendrograms. J. Statist. Comput. Simul. 18: 145–157.
Naga, R. A. and G. Antille. 1990. Stability of robust and non-robust principal components analysis. Comput. Statist. Data Anal. 10: 169–174.
Naouri, J-C. 1970. Analyse factorielle des correspondances continues. Publ. l’Inst. Statist. Univ. Paris 19:1–100.
O’Callaghan, J. 1974. An alternative definition for the neighbourhood of a point. I. E. E. E. Trans. Comput. C-24: 1121–1125.
Oksanen, J. and R. R. Minchin. 1997. Instability of ordination results under changes in input data order: explanations and remedies. J. Veg. Sci. 8: 447–454.
Orth, B. 1988. Representing similarities by distance graphs: monotone network analysis (MONA). In: H. H. Bock (ed.), Classification and Related methods of Data Analysis. North Holland, Amsterdam. pp. 489–496.
Posse, C. 1995. Tools for two-dimensional exploratory projection pursuit. J. Computer Graphics Statist. 4: 83–100.
Taguri, M., M. Hiramatsu, T. Kittaka and K. Wakimoto. 1976. Graphical representation of correlation analysis of ordered data by linked vector pattern. J. Jap. Statist. Soc. 6: 17–25.
Tamassia, R. and I. G. Tollis. 1995. Graph Drawing. DIMACS Internatl. Workshop, Princeton 1994. Lecture Notes in Computer Science 894, Springer, Berlin.
Tausch, R. J., D. A. Charlet, D. A. Weixelman and D. C. Zamudio. 1995. Patterns of ordination and classification instability resulting from changes in input data order. J. Veg. Sci. 6: 897–902.
ter Braak, C. J. F. 1986. Canonical correspondence analysis a new eigenvector technique for multivariate direct gradient analysis. Ecology 67: 1167–1179.
Toussaint, G. T. 1980. The relative neighbourhood graph of a finite planar set. Patt. Recog. 12: 261–268.
Vaidya, P. M. 1991. A sparse graph almost as good as the complete graph on points in K dimensions. Discrete Comput. Geom. 6: 369–381.
Van Groenewoud, H. 1992. The robustness of Correspondence, Detrended Correspondence and TWINSPAN analysis. J. Veg. Sci. 3: 239–246.
Vasilevich, V. I. 1967. A continuum in the coniferous and parvifoliate forest of the Karelian isthmus. Bot. Zhur SSSR 52: 45–53 (in Russian).
Veltkamp, R. C. 1992. The γ-neighbourhood graph. Computational Geometry: Theory and Applications 1: 227–246.
Wallace, C. S. 1995. Multiple factor analysis by MML estimation Tech. Rep. 95/218, Dept Computer Science, Monash University, Clayton Victoria3168, Australia21 pp.
Wallace, C. S. & D. L. Dowe. 2000. MML clustering of multi-state, Poisson, von Mises circular and Gaussian distributions. Statistics and Computing 10: 73–83.
Williams, W. T. 1973. Partition of information: the CENTPERC problem. Austral. J. Bot. 21: 277–281.
Williams, W. T. 1980. TWONET: A new program for the computation of a two-neighbour network. Austral. Comput. J. 12: 70.
Williams, W. T., J. S. Bunt and H. J. Clay. 1991. Yet another method of species-sequencing. Marine Ecol. Prog. Ser. 72: 283–287
Wishart, D. 1969. Mode analysis: A generalisation of nearest neighbour which reduces chaining effects. In: A. J. Cole (ed.), Numerical Taxonomy. Academic Press, New York. pp. 282–308.
Yanai, H. 1988. Partial correspondence analysis and its properties. In: E. Diday, C. Hayashi, M. Jambu and N. Ohsumi (eds.), Recent Developments in Clustering and Data Analysis. Academic Press, New York and London. pp. 259–266.
Author information
Authors and Affiliations
Corresponding author
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.
About this article
Cite this article
Dale, M.B. On plexus representation of dissimilarities. COMMUNITY ECOLOGY 1, 43–56 (2000). https://doi.org/10.1556/ComEc.1.2000.1.7
Published:
Issue Date:
DOI: https://doi.org/10.1556/ComEc.1.2000.1.7