Skip to main content

Joint Ordination of Species and Sites: The Unfolding Technique

  • Conference paper
Develoments in Numerical Ecology

Part of the book series: NATO ASI Series ((ASIG,volume 14))

Abstract

Several different methods of gradient analysis, including correspondence analysis and Gaussian ordination, can be characterized as unfolding methods. These techniques are applicable whenever single-peaked response functions are at issue, either with respect to known environmental characteristics or else with respect to data driven reorderings of the sites. Unfolding gives a joint respresentation of the site/species relationships in terms of the distance between two types of points, the location of which can be constrained in various ways. A classification based on loss functions is given, as well as a convergent algorithm for the weighted least squares case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aitchison, J. and J.A.C. Brown, 1957. The Lognormal Distribution. Cambridge University Press, New York, NY.

    Google Scholar 

  • Austin, M.P. 1976. On non-linear species response models in ordination. Vegetatio 33: 33–41.

    Article  Google Scholar 

  • Austin, T.L. jr. 1959. An approximation to the point of minimum aggregate distance. Metron 19: 10–21.

    Google Scholar 

  • Barlow, R.E., D.J. Bartholomew, J.M. Bremner, and H.D. Brunk. 1972. Statistical Inference under Order Restrictions. Wiley, New York, NY.

    Google Scholar 

  • Braun-Blanquet, J. and H. Jenny. 1926. Vegetationsentwicklung und Bodenbildung in der alpinen STUFE der Zentralalpen. Neue Denkschr. Schweiz. Naturforsch. Ges. 63: 175–349.

    Google Scholar 

  • Bray, R.J. and J.T. Curtis. 1957. An ordination of the upland forest communities of Southern Wisconsin. Ecol. Monogr. 27: 325–249.

    Google Scholar 

  • Brown, R.T. and T.T. Curtis. 1952. The upland conifer-hardwood forests of nothern Wisconsin. Ecol. Monogr. 22: 217–234.

    Google Scholar 

  • Browne, M.J. and M.W. Greenacre. 1986. An efficient alternating least squares algorithm to perform multidimensional unfolding. Psychometrika 51: in press.

    Google Scholar 

  • Carroll, J.D. 1969. Polynomial factor analysis. Proc. 77’th Annual Convention of the APA. 4: 103–104.

    Google Scholar 

  • Carroll, J.D. 1972. Individual differences and multidimensional scaling, p. 105–155. In R.N. Shepard et al. [ed.] Multidimensional Scaling, Vol I: Theory. Seminar Press, New York, NY.

    Google Scholar 

  • Coombs, C.H. 1950. Psychological Scaling without a unit of measurement. Psych. Rev. 57: 148–158.

    Google Scholar 

  • Coombs, C.H. 1964. A Theory of Data. Wiley, New York, NY.

    Google Scholar 

  • Coombs, C.H. and J.E.K. Smith. 1973. On the detection of structure in attitudes and developmental processes. Psych. Rev. 80: 337-351.

    Google Scholar 

  • Cottam, G. and J.T. Curtis. 1956. The use of distance measures in phytosociological sampling. Ecology 37: 451–460.

    Article  Google Scholar 

  • Coxon, A.P.M. 1974. The mapping of family-composition preferences: A scaling analysis. Social Science Research 3: 191–210.

    Google Scholar 

  • Curtis, J.T. and R.P. Mcintosh. 1951. An upland continuum in the prairie-forest border region of Wisconsin. Ecology 32: 476–496.

    Article  Google Scholar 

  • Davison, M.L., P.M. King, K.S. Kitchener, and C.A. Parker. 1980. The stage sequence concept in cognitive and social development. Developm. Psych. 16: 121–131.

    Google Scholar 

  • De Leeuw, J. 1977. Applications of convex analysis to multidimensional scaling, p. 133–145. In J.R. Barra et al. [ed.] Recent Developments in Statistics. North-Holland, Amsterdam.

    Google Scholar 

  • De Leeuw, J. 1982. Nonlinear principal component analysis, p. 77–89. In H. Caussinus et al. [ed.] COMPSTAT 1982. Physica Verlag, Vienna.

    Google Scholar 

  • De Leeuw, J. 1984. The Gifi system of nonlinear multivariate analysis, p. 415–424. In E. Diday et al. [ed.] Data Analysis and Informatics, HI. North-Holland, Amsterdam.

    Google Scholar 

  • De Leeuw, J. 1987a. Nonlinear multivariate analysis with optimal scaling. In this volume.

    Google Scholar 

  • De Leeuw, J. 1987b. Nonlinear path analysis with optimal scaling. In this volume.

    Google Scholar 

  • De Leeuw, J. and W.J. Heiser. 1977. Convergence of correction matrix algorithms for multidimensional scaling, p. 735–752. In J. Lingoes [ed.] Geometric representations of relational data. Mathesis Press, Ann Arbor, Mich.

    Google Scholar 

  • De Leeuw, J. and W.J. Heiser. 1980. Multidimensional scaling with restrictions on the configuration, p. 501–522. In P.R. Krishnaiah [ed.] Multivariate Analysis, Vol V. North-Holland, Amsterdam.

    Google Scholar 

  • De Leeuw, J. and W.J. Heiser. 1982. Theory of multidimensional scaling, p. 285–316. In P.R. Krishnaiah and L.N. Kanal [ed.] Handbook of Statistics, Vol 2. North-Holland, Amsterdam.

    Google Scholar 

  • Defays, D. 1978. A short note on a method of seriation. Brit. J. Math. Stat. Psych. 31: 49–53.

    Google Scholar 

  • Derman, C., L.J. Gleser, and I. Olkin. 1973. A Guide to Probability Theory and Application. Holt, Rinehart and Winston, New York, NY.

    Google Scholar 

  • Draper, N.R. and H. Smith. 1966. Applied Regression Analysis. Wiley, New York, NY.

    Google Scholar 

  • Flchet, B. 1986. Distances and Euclidean distances for presence-absence characters and their application to factor analysis. In J. de Leeuw et al. [ed.] Multidimensional Data Analysis. DSWO Press, Leiden, in press.

    Google Scholar 

  • Gabriel, K.R. 1971. The biplot graphic display of matrices with application to principal component analysis. Biometrika 58: 453–467.

    Article  Google Scholar 

  • Gauch, H.G. 1982. Multivariate analysis in community ecology. Cambridge University Press, Cambridge.

    Google Scholar 

  • Gauch, H.G. and G.B. Chase. 1974. Fitting the Gaussian curve to ecological data. Ecology 55: 1377–1381.

    Article  Google Scholar 

  • Gauch, H.G., G.B. Chase, and R.H. Whittaker. 1974. Ordination of vegetation samples by Gaussian species distributions. Ecology 55: 1382–1390.

    Article  Google Scholar 

  • Gause, C.F. 1930. Studies of the ecology of the orthoptera. Ecology 11: 307–325.

    Article  Google Scholar 

  • Gifi, A. 1981. Nonlinear Multivariate Analysis. Department of Data Theory, University of Leiden, Leiden.

    Google Scholar 

  • Gittins, R. 1985. Canonical Analysis: A Review with Applications in Ecology. Physica Verlag, Berlin.

    Book  Google Scholar 

  • Good All, D.W. 1954. Objective methods for the classification of vegetation, IE. An essay in the use of factor analysis. Aust. J. Bot. 2: 304–324.

    Google Scholar 

  • Greenacre, M.J. 1978. Some objective methods of graphical display of a data matrix. Special Report, Dept. of Statistics and Operations Research, University of South-Africa, Pretoria.

    Google Scholar 

  • Greenacre, M.J. 1984. Theory and Applications of Correspondence Analysis. Academic Press, London.

    Google Scholar 

  • Greenacre, M.J. and L.G. Underhill. 1982. Scaling a data matrix in a low-dimensional Euclidean space, p. 183–268. In D.M. Hawkins [ed.] Topics in Applied Multivariate Analysis, Cambridge University Press, Cambridge.

    Google Scholar 

  • Greig-Smith, P. 1983. Quantitative Plant Ecology, 3rd Ed. Blackwell Scient. Publ., London.

    Google Scholar 

  • Grundy, P.M. 1951. The expected frequencies in a sample of an animal population in which the abundances of species are lognormally distributed, I. Biometrika 38: 427–434.

    Google Scholar 

  • Guttman, L. 1950. The principal components of scale analysis. In S.A. Stouffer et al. [ed.] Measurement and Prediction. Princeton University Press, Princeton, NJ.

    Google Scholar 

  • Guttman, L. 1968. A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika 33: 469–506.

    Article  Google Scholar 

  • Hayashi, C. 1952. On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Ann. Inst. Statist. Math. 2: 93–96.

    Google Scholar 

  • Hayashi, C. 1954. Multidimensional quantification - with applications to analysis of social phenomena. Ann. Inst. Stat. Math. 5: 121–143.

    Google Scholar 

  • Hayashi, C. 1956. Theory and example of quantification, II. Proc. Inst. Stat. Math. 4: 19–30.

    Google Scholar 

  • Hayashi, C. 1974. Minimum dimension analysis MDA. Behaviormetrika 1: 1–24.

    Article  Google Scholar 

  • Healy, M.J.R. and H. Goldstein. 1976. An approach to the scaling of categorised attributes. Biometrika 63: 219–229.

    Article  Google Scholar 

  • Heiser, W.J. 1981. Unfolding Analysis of Proximity Data. Ph.D.Thesis, University of Leiden, Leiden, The Netherlands.

    Google Scholar 

  • Heiser, W.J. 1985a. Undesired nonlinearities in nonlinear multivariate analysis. In E. Diday et al. [ed.] Data Analysis and Informatics, IV. North-Holland, Amsterdam, in press.

    Google Scholar 

  • Heiser, W.J. 1985b. Multidimensional scaling by optimizing goodness-of-fit to a smooth hypothesis. Internal Report RR-85-07, Dept of Data Theory, University of Leiden.

    Google Scholar 

  • Heiser, W.J. 1986. Order invariant unfolding analysis under smoothness restrictions. Internal Report RR-86-07, Dept of Data Theory, University of Leiden.

    Google Scholar 

  • Heiser, W.J. and J. De Leeuw. 1979. How to use SMACOF-I ( 2nd edition ). Internal Report, Dept. of Data Theory, University of Leiden.

    Google Scholar 

  • Heiser, W.J. and J. Meulman. 1983a. Analyzing rectangular tables by joint and constrained multidimensional scaling. J. Econometrics 22: 139–167.

    Article  Google Scholar 

  • Heiser, W.J. and J. Meulman. 1983b. Constrained multidimensional scaling, including confirmation. Applied Psych. Meas. 22: 139-167.

    Google Scholar 

  • Hill, M.0. 1974. Correspondence analysis: a neglected multivariate method. Applied Statistics 23: 340–354.

    Google Scholar 

  • Hill, M.O. 1977. Use of simple discriminant functions to classify quantitative phytosociological data, p. 181–199. In E. Diday et al. [ed.] Data Analysis and Informatics, I. INRIA, Le Chesnay, France.

    Google Scholar 

  • Hill, M.O., R.G.H. Bunce, and M.W. Shaw. 1975. Indicator species analysis, a divisive polythetic method of classifcation, and its application to a survey of native pinewoods in Scotland. J. Ecol. 63: 5971–613.

    Google Scholar 

  • Hill, M.O. and H.G. Gauch. 1980. Detrended correspondence analysis: an improved ordination technique. Vegetatio 42: 47–58.

    Article  Google Scholar 

  • Hodson, F.R. et al. [ed.] 1971. Mathematics in the Archaeological and Historical Sciences. Edinburgh University Press, Edinburgh.

    Google Scholar 

  • Hovland, C.I., O.J. Harvey, and M. Sherif. 1957. Assimilation and contrast effects in reactions to communication and attitude change. J. Abnorm. Soc. Psych. 55: 244–252.

    Article  CAS  Google Scholar 

  • Hubert, L. and Ph. Arable. 1986. Unidimensional scaling and combinatorial optimization. In J. de Leeuw et al. [ed.] Multidimensional Data Analysis. DSWO Press, Leiden (in press).

    Google Scholar 

  • Igoshina, K.N. 1927. Die Pflanzengesellschaften der Alluvionen der Flüsse Kama und Tschussowaja (in Russian with German summary). Trav. de l’lnst. Biol, k l’Univ. de Perm 1: 1–117.

    Google Scholar 

  • Ihm, P. and H. Van Groenewoud. 1975. A multivariate ordering of vegetation data based on Gaussian type gradient response curves. J. Ecol. 63: 767–777.

    Article  Google Scholar 

  • Ihm, P. and H. Van Groenewoud. 1984. Correspondence analysis and Gaussian ordination. COMPSTAT Lectures 3. Physica Verlag, Vienna, 5–60.

    Google Scholar 

  • Johnson, R.W. and D.W. Goodall. 1980. A maximum likelihood approach to non-linear ordination. Vegetatio 41: 133–142.

    Article  Google Scholar 

  • Kendall, D.G. 1963. A statistical approach to Flinders Petrie’s sequence dating. Bull. Inst. Statist. Inst. 40: 657–680.

    Google Scholar 

  • Kershaw, K.A. 1968. Classification and ordination of Nigerian savanna vegetation. J. Ecol. 56: 467–482.

    Article  Google Scholar 

  • Kershaw, K.A. and J.H.H. Looney. 1985. Quantitative and Dynamic Plant Ecology, 3rd Ed. Edward Arnold Publ., London.

    Google Scholar 

  • Kooijman, S.A.L.M. 1977. Species abundance with optimum relations to environmental factors. Ann. Systems Res. 6: 123–138.

    Google Scholar 

  • Kruskal, J.B. 1964a. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29: 1–28.

    Article  Google Scholar 

  • Kruskal, J.B. 1964b. Nonmetric multidimensional scaling: a numerical method. Psychometrika 29: 115–129.

    Article  Google Scholar 

  • Kruskal, J.B. 1977. Multidimensional scaling and other methods for discovering structure, p. 296–339. In K. Enslein, A. Ralston and H.S. Wilf [ed.] Statistical Methods for Digital Computers, Vol HI. Wiley, New York, NY.

    Google Scholar 

  • Kruskal, J.B. and J.D. Carroll. 1969. Geometrical models and badness-of-fit functions, p.639–671. In P.R. Krishnaiah [ed.] Multivariate Analysis I I. Academic Press, New York, NY.

    Google Scholar 

  • Kuhn, H.W. 1967. On a pair of dual nonlinear programs, p. 38–54. In J. Abadie [ed.] Methods of nonlinear programming. North-Holland, Amsterdam.

    Google Scholar 

  • Legendre, L. and P. Legendre. 1983. Numerical Ecology. Elsevier Scient. Publ., Amsterdam.

    Google Scholar 

  • McDonald, R.P. 1962. A general approach to nonlinear factor analysis. Psychometrika 27: 397–415.

    Article  Google Scholar 

  • McDonald, R.P. 1967. Nonlinear factor analysis. Psychometric Monograph 15.

    Google Scholar 

  • Meulman, J. and W.J. Heiser. 1984. Constrained multidimensional scaling: more directions than dimensions, p. 137–142. In T. Havrdnek et al. [ed.] COMPSTAT 1984, Proceedings in Computational Statistics. Physica Verlag, Vienna.

    Google Scholar 

  • Nishisato, S. 1980. Analysis of categorical data: dual scaling and its applications. University of Toronto Press, Toronto.

    Google Scholar 

  • Noy-Meir, I. and M.P. Austin. 1970. Principal component ordination and simulated vegetational data. Ecology 51: 551–552.

    Article  Google Scholar 

  • Poole, K.T. 1984. Least squares metric, unidimensional unfolding. Psychometrika 49: 311–323.

    Article  Google Scholar 

  • Ramsay, J.O. 1977. Maximum likelihood estimation in multidimensional scaling. Psychometrika 42: 241–266.

    Article  Google Scholar 

  • Roberts, F.S. 1976. Discrete mathematical models. Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Schriever, B.F. 1985. Order Dependence. Ph.D. Thesis, Amsterdam: Mathematical Centre.

    Google Scholar 

  • Shepard, R.N. 1958. Stimulus and response generalization: deduction of the generalization gradient from a trace model. Psych. Rev. 65: 242–256.

    Google Scholar 

  • Shepard, R.N. 1974. Representation of structure in similarity data: problems and prospects. Psychometrika 39: 373–421.

    Article  Google Scholar 

  • Shepard, R.N. and J.D. Carroll. 1966. Parametric representation of nonlinear data structures, p. 561–592. In P.R. Krishnaiah [ed.] Multivariate Analysis, Vol. I. Academic Press, New York, NY.

    Google Scholar 

  • Swan, J.M.A. 1970. An examination of some ordination problem by use of simulated vegetation data. Ecology 51: 89–102.

    Article  Google Scholar 

  • Takane, Y., F.W. Young, and J. De Leeuw. 1977. Nonmetric individual differences multidimensional scaling: an alternating least squares method with optimal scaling features. Psychometrika 42: 7–67.

    Article  Google Scholar 

  • Ter Braak, C.J.F. 1985. Correspondence analysis of incidence and abundance data: properties in terms of a unimodal response model. Biometrics 41: 859–873.

    Article  Google Scholar 

  • Ter Braak, C.J.F. 1986a. Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology 67: in press.

    Google Scholar 

  • Ter Braak, C.J.F. 1986b. The analysis of vegetation-environment relationships by canonical correspondence analysis. Vegetatio 65: in press.

    Google Scholar 

  • Ter Braak, C.J.F. And L.G. Barendregt. 1986. Weighted averaging of species indicator values: its efficiency in environmental calibration. Math. Biosciences 78: 57–72.

    Article  Google Scholar 

  • Thurstone, L.L. 1927. A law of comparative judgment. Psych. Rev. 34: 278–286.

    Google Scholar 

  • Van Rijckevorsel, J.L.A. 1986. About horseshoes in multiple correspondence analysis, p. 377–388. In W. Gaul and M. Schader [ed.] Classification as a tool of research. North-Holland, Amsterdam.

    Google Scholar 

  • Whittaker, R.H. 1948. A vegetation analysis of the Great Smokey Mountains. Ph.D. Thesis, University of Illinois, Urbana.

    Google Scholar 

  • Whittaker, R.H. 1967. Gradient analysis of vegetation. Biol. Rev. 42: 207–264.

    Google Scholar 

  • Whittaker, R.H. 1978. Ordination of Plant Communities. Dr. W. Junk Publ., The Hague.

    Google Scholar 

  • Whittaker, R.H. and H.G. Gauch. 1978. Evaluation of ordination techniques, p. 277–336. In R.H. Whittaker [ed.] Ordination of Plant Communities. Dr. W. Junk Publ., The Hague.

    Google Scholar 

  • Wilkinson, E.M. 1971. Archaeological seriation and the travelling salesman problem, p. 276–283. In F.R. Hodson et al. [ed.] Mathematics in the Archaeological and Historical Sciences. Edinburgh University Press, Edinburgh.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Heiser, W.J. (1987). Joint Ordination of Species and Sites: The Unfolding Technique. In: Legendre, P., Legendre, L. (eds) Develoments in Numerical Ecology. NATO ASI Series, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-70880-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-70880-0_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-70882-4

  • Online ISBN: 978-3-642-70880-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics