Aquatic Sciences

, Volume 57, Issue 3, pp 255–289 | Cite as

Canonical correspondence analysis and related multivariate methods in aquatic ecology

  • Cajo J. F. ter Braak
  • Piet F. M. Verdonschot
Article

Abstract

Canonical correspondence analysis (CCA) is a multivariate method to elucidate the relationships between biological assemblages of species and their environment. The method is designed to extract synthetic environmental gradients from ecological data-sets. The gradients are the basis for succinctly describing and visualizing the differential habitat preferences (niches) of taxavia an ordination diagram. Linear multivariate methods for relating two set of variables, such as two-block Partial Least Squares (PLS2), canonical correlation analysis and redundancy analysis, are less suited for this purpose because habitat preferences are often unimodal functions of habitat variables. After pointing out the key assumptions underlying CCA, the paper focuses on the interpretation of CCA ordination diagrams. Subsequently, some advanced uses, such as ranking environmental variables in importance and the statistical testing of effects are illustrated on a typical macroinvertebrate data-set. The paper closes with comparisons with correspondence analysis, discriminant analysis, PLS2 and co-inertia analysis. In an appendix a new method, named CCA-PLS, is proposed that combines the strong features of CCA and PLS2.

Key words

Multivariate response data compositional data unimodal model community ecology partial least squares 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, N. J., 1993. Natural versus anthropogenic change in lakes: the role of the sediment record. TREE 8:356–361.Google Scholar
  2. Anderson, N. J., T. Korsman and I. Renberg, 1994. Spatial heterogeneity of diatom stratigraphy in varved and non-varved sediments of a small, boreal-forest lake. Aquat. Sci. 56:40–58.Google Scholar
  3. Anderson, N. J., B. Rippey and C. E. Gibson, 1992. A comparison of sedimentary and diatominferred phosphorus profiles: implications for defining pre-disturbance nutrient conditions. Hydrobiologia 253:357–366.Google Scholar
  4. Austin, M. P. and M. J. Gaywood, 1994. Current problems of environmental gradients and species response curves in relation to continuum theory. J. Veg. Sci. 5:473–482.Google Scholar
  5. Austin, M. P., A. O. Nicholls, M. D. Doherty and J. A. Meyers, 1994. Determining species response functions to an environmental gradient by means of aβ-function. J. Veg. Sci. 5:215–228.Google Scholar
  6. Bakker, C., P. M. J. Herman and M. Vink, 1990. Changes in seasonal succession of phytoplankton induced by the storm-surge barrier in the Oosterschelde (S. W. Netherlands). J. Plankton Res. 12:947–972.Google Scholar
  7. Barker, P., 1994. Book review of “H. van Dam (Editor). Twelfth International Diatom Symposium. Kluwer, Academic Publ. Dordrecht”. Eur. J. Phycol. 29:281–283.Google Scholar
  8. Birks, H. J. B., S. Juggins and J. M. Line, 1990a. Lake surface-water chemistry reconstructions from palaeolimnological data. In: Mason B. J. (ed.), The Surface Waters Acidification Programme, Cambridge University Press, Cambridge, pp. 301–313.Google Scholar
  9. Birks H. J. B., J. M. Line, S. Juggins, A. C. Stevenson and C. J. F. ter Braak, 1990b. Diatoms and pH reconstruction. Phil. Trans. Roy. Soc. London, Ser B 327:263–278.Google Scholar
  10. Birks, H. J. B., S. M. Peglar and H. A. Austin, 1994. An annotated bibliography of canonical correspondence analysis and related constrained ordination methods 1986–1993, Botanical Institute, Bergen, Norway, 58 pp.Google Scholar
  11. Borcard, D., P. Legendre and P. Drapeau, 1992. Partialling out the spatial component of ecological variation. Ecology 73:1045–1055.Google Scholar
  12. Carnes, B. A. and N. A. Slade, 1982. Some comments on niche analysis in canonical space. Ecology 63:888–893.Google Scholar
  13. Carpenter, S. R., T. M. Frost and D. K. T. K. Heisey, 1989. Randomized intervention analysis and the interpretation of whole-ecosystem experiments. Ecology 70:1142–1152.Google Scholar
  14. Charles, D. F. and J. P. Smol, 1994. Long-term chemical changes in lakes: quantitative inferences from biotic remains in the sediment record. In: Baker L. (ed.), Environmental Chemistry of Lakes and Reservoirs, American Chemical Society, Washington, pp. 3–31.Google Scholar
  15. Chessel, D., J.-D. Lebreton and N. Yoccoz, 1987. Propriétés de l'analyse canonique des correspondances; une illustration en hydrobiologie. Rev. Statist. Appl. 35:55–72.Google Scholar
  16. Copp, G. H., 1992. An empirical model for predicting microhabitat of 0+ juvenile fishes in a lowland river catchment. Oecologia 91:338–345.Google Scholar
  17. Cumming, B. F., J. P. Smol and H. J. B. Birks, 1992. Scaled chrysophytes (Chrysophyceae and Synurophyceae) from Adirondack drainage lakes and their relationship to environmental variables. J. Phycol. 28:162–178.Google Scholar
  18. de Jong, S. and R. W. Farebrother, 1994. Extending the relationship between ridge regression and continuum regression. Chemometrics Intell. Lab. Syst. 25:179–181.Google Scholar
  19. de Jong, S. and C. J. F. ter Braak, 1994. Comments on the PLS kernel algorithm. J. Chemometrics 8:169–174.Google Scholar
  20. Descy, J. P., 1979. A new approach to water quality estimation using diatoms. Nova Hedwigia, Beiheft 64:305–323.Google Scholar
  21. Dolédec, S. and D. Chessel, 1994. Co-inertia analysis: an alternative method for studying speciesenvironment relationships. Freshwater Biol. 31:277–294.Google Scholar
  22. Ellenberg, H., 1948. Unkrautgesellschaften als Mass für den Säuregrad, die Verdichtung und andere Eigenschaften des Ackerbodens. Ber. Landtech. 4:130–146.Google Scholar
  23. Eriksson, L., J. L. M. Hermens, E. Johansson, H. J. M. Verhaar and S. Wold, 1995. Multivariate analysis of aquatic toxicity data. Aquat. Sci. this volume.Google Scholar
  24. Escoufier, Y. and P. Roberts, 1979. Choosing variables and metrics by optimizing the RV-coefficient. In: Rustagi J. S. (ed.), Optimizing methods in Statistics, Academic Press, New York, pp. 205–219.Google Scholar
  25. Fairchild, G. W. and J. W. Sherman, 1993. Algal periphyton response to acidity and nutrients in softwater lakes: lake comparison vs. nutrient enrichment approaches. J. N. Am. Benthol. Soc. 12:157–167.Google Scholar
  26. Frank, I.E. and J. H. Friedman, 1993. A statistical view of some chemometric regression tools (with discussion). Technometrics 35:109–148.Google Scholar
  27. Fritz, S. C., S. Juggins and R. W. Batterbee, 1993. Diatom assemblages and ionic characterization of lakes of the northern great plains, North America — a tool for reconstructing past salinity and climate fluctuations. Can. J. Fish. Aquat. Sci. 50:1844–1856.Google Scholar
  28. Gabriel, K. R., 1982. Biplot. In: Kotz S. and N. L. Johnson (eds.), Encyclopedia of Statistical Sciences, Vol. 1, Wiley, New York, pp. 263–271.Google Scholar
  29. Gabriel, K. R. and C. L. Odoroff, 1990. Biplots in biomedical research. Statist. Med. 9:469–485.Google Scholar
  30. Gauch, H. G., 1982. Multivariate analysis in community ecology. Cambridge University Press, Cambridge, 298 pp.Google Scholar
  31. Gause, G. F., 1930. Studies on the ecology of the Orthoptera. Ecology 11:307–325.Google Scholar
  32. Geladi, P., 1988. Notes on the history and nature of partial least squares (PLS) modelling. J. Chemometrics 2:231–246.Google Scholar
  33. Gower, A. M., G. Myers, M. Kent and M. E. Foulkes, 1994. Relationships between macroinvertebrate communities and environmental variables in metal-contaminated streams in south-west England. Freshwater Biol. 32:119–221.Google Scholar
  34. Grantham, B. A. and B. J. Hann, 1994. Leeches (Annelida, Hirundinea) in the experimental lakes area, Northwestern Ontario, Canada — Patterns of species composition in relation to environment. Can. J. Fish. Aquat. Sci. 51:1600–1607.Google Scholar
  35. Green, R. H., 1971. A multivariate statistical approach to the Hutchinsonian niche: bivalve mollucs of central Canada. Ecology 52:543–556.Google Scholar
  36. Green, R. H., 1974. Multivariate niche analysis with temporally varying environmental factors. Ecology 55:73–83.Google Scholar
  37. Green, R. H., 1979. Sampling design and statistical methods for environmental biologists. Wiley, New York, 257 pp.Google Scholar
  38. Greenacre, M. J., 1984. Theory and applications of correspondence analysis, Academic Press, London, 364 pp.Google Scholar
  39. Greenacre, M. J., 1989. The Carroll-Green-Schaffer scaling in correspondence analysis: a theoretical and empirical appraisal. J. Marketing Research 26:358–365.Google Scholar
  40. Greenace, M. J., 1993. Biplots in correspondence analysis. J. Appl. Statist. 20:251–269.Google Scholar
  41. Hawkes, H. A., 1975. River zonation and classification. In: Whitton B. A. (ed.), River Ecology. Studies in ecology, vol. 2, Univ. Calif. Press, pp. 312–374.Google Scholar
  42. Heiser, W. J., 1987. Joint ordination of species and sites: the unfolding technique. In: Legendre P. and L. Legendre (eds.), Developments in numerical ecology. Springer-Verlag, Berlin, pp. 189–224.Google Scholar
  43. Higler, L. W. G. and F. Repko, 1981. The effects of pollution in the drainage area of a Dutch lowland stream on fish and macro-invertebrates. Verh. Int. Verein. Limnol. 21:1077–1082.Google Scholar
  44. Higler, L. W. G. and P.F.M. Verdonschot, in prep. The relation between macro-invertebrates, hydraulics and soil fertilization in two man-made tributaries of a Dutch lowland stream.Google Scholar
  45. Hill M. O., 1973. Reciprocal averaging: an eigenvector method of ordination. J. Ecol. 61:237–249.Google Scholar
  46. Hill, M. O., 1974. Correspondence analysis: a neglected multivariate method. Appl. Statist. 23:340–354.Google Scholar
  47. Hill, M. O., 1979. DECORANA — A FORTRAN program for detrended correspondence analysis and reciprocal averaging. Ecology and Systematics, Cornell University, Ithaca, New York, 52 pp.Google Scholar
  48. Hill, M. O. and H. G. Gauch, 1980. Detrended correspondence analysis, an improved ordination technique. Vegetatio 42:47–58.Google Scholar
  49. Höskuldsson, A., 1988. PLS regression methods. J. Chemometrics 2:211–228.Google Scholar
  50. Hutchinson, G. E., 1968. When are species necessary? In: Lewontin E. (ed.), Population biology and evolution, Syracuse Univ. Press, Syracuse, N. Y., pp. 177–186.Google Scholar
  51. Iwatsubo, S., 1984. The analytical solutions of eigenvalue problem in the case of applying optimal scoring method to some types of data. In: Diday E. (ed.), Data Analysis and Informations III, North Holland, Amsterdam, pp. 31–40.Google Scholar
  52. James, F. C. and C. E. McCullach, 1990. Multivariate analysis in ecology and systematics: panacea or Pandora's box. Ann. Rev. Ecol. Syst. 21:129–166.Google Scholar
  53. Jones, V. J., S. Juggins and J. C. Ellis-Evans, 1993. The relationship between water chemistry and surface sediment diatom assemblages in maritime Antarctic lakes. Ant. Sci. 5:339–348.Google Scholar
  54. Jongman, R. H. G., C. J. F. ter Braak and O. F. R. van Tongeren, 1995. Data analysis in community and landscape ecology, Cambridge Univesity Press, Cambridge, 299 pp.Google Scholar
  55. Kautsky, H. and E. van der Maarel, 1990. Multivariate approaches to the variation in phytobenthic communities and environmental vectors in the Baltic Sea. Mar. Ecol. Progr. Ser. 60:169–184.Google Scholar
  56. Kingston, J. C., H. J. B. Birks, A. J. Uutala, B. F. Cumming and J. P. Smol, 1992. Assessing trends in fishery resources and lake water aluminium from paleolimnological analyses of siliceous algae. Can. J. Fish. Aquat. Sci. 49:116–127.Google Scholar
  57. Krzanowski, W J., 1988. Principles of Multivariate Analysis, Clarendon Press, Oxford.Google Scholar
  58. Lebreton, J.-D., D. Chessel, R. Prodon and N. Yoccoz, 1988. L'analyse des relations espèces-milieu par l'analyses canonique des correspondances. I. Variables de milieu quantitatives. Acta Oecol. Gen. 9:53–67.Google Scholar
  59. Lebreton, J.-D., D. Chessel, M. Richardot-Coulet and N. Yoccoz, 1988. L'analyse des relations espèces-milieu par l'analyse canonique des correspondances. II. Variables de milieu qualitatives. Acta Oecol. Gen. 9:137–151.Google Scholar
  60. Lebreton, J.-D., R. Sabatier, G. Banco and A. M. Bacou, 1991. Principal component and correspondence analysis with respect to instrumental variables: an overview of their role in studies of structure-activity and species-environment relationships. In: Devillers J. and W. Karcher (eds.), Applied Multivariate Analysis in SAR and Environmental Studies, Kluwer, Dordrecht, pp. 85–114.Google Scholar
  61. Line, J. M., C. J. F. ter Braak and H. J. B. Birks, 1994. WACALIB version 3.3 — a computer program to reconstruct environmental variables from fossil assemblages by weighted averaging and to derive sample-specific errors of prediction. J. Palaeolimnol. 10:147–152.Google Scholar
  62. Lohmuller, J.-B., 1988. The PLS program system: latent variables path analysis with partial least squares estimation. Mult. Beh. R. 23:125–127.Google Scholar
  63. Malmqvist, B. and M. Maki, 1994. Benthic macroinvertebrate assemblages in North Swedish streams — environmentla relationships. Ecography 17:9–16.Google Scholar
  64. Manly, B. F. J., 1991. Randomization and Monte Carlo methods in biology, Chapman and Hall, London, 281 pp.Google Scholar
  65. Martens, H. and T. Naes, 1989. Multivariate calibration, Wiley, Chichester, 419 pp.Google Scholar
  66. McLachlan, G. J., 1992. Discriminant Analysis and Statistical Pattern Recognition, Wiley, New York.Google Scholar
  67. Miller, A. J., 1990. Subset Selection in Regression, Champan and Hall, London, 229 pp.Google Scholar
  68. Odum, E. P., 1971. Fundamentals of Ecology 3rd Edition, W. B. Saunders Company, Philadelphia.Google Scholar
  69. Økland, R. H. and O. Eilertsen, 1994. Canonical correspondence analysis with variation partitioning: some comments and an applications. J Veg. Sci. 5:117–126.Google Scholar
  70. Oksanen, J., 1987. Problems of joint display of species and site scores in correspondence analysis. Vegetatio 72:51–57.Google Scholar
  71. Palmer, M. W., 1993. Putting things in even better order: the advantages of canonical correspondence analysis. Ecology 74:2215–2230.Google Scholar
  72. Pantle, R. and H. Buck, 1955. Die biologische Überwachung der Gewässer und die Darstellung der Ergebnisse. Gas- und Wasserfach 96:604.Google Scholar
  73. Rao, C. R., 1952. Advanced Statistical Methods in Biometric Research, Wiley, New York.Google Scholar
  74. Rao, C. R., 1964. The use and interpretation of principal component analysis in applied research. Sankhya A 26:329–358.Google Scholar
  75. Reilly, S. B. and P. C. Fiedler, 1994. Interannual variability of dolphin habitats in the eastern tropical Pacific. 1. Research vessel surveys. Fish. Bull. 92:434–450.Google Scholar
  76. Ruse, L. P., 1994. Chironomid microdistribution in gravel of an English chalk river. Freshwater Biol. 32:533–551.Google Scholar
  77. Sabatier, R., J.-D. Lebreton and D. Chessel, 1989. Multivariate analysis of composition data accompanied by qualitative variables describing a structure. In: Coppi R. and S. Bolasco (eds.), Multiway data tables, North-Holland, Amsterdam, pp. 341–352.Google Scholar
  78. Saris, W. E. and L. H. Stronkhorst, 1984. Causal modelling in nonexperimental research. An introduction to the LISREL approach, Sociometric Research Foundation, Amsterdam.Google Scholar
  79. Shelford, V. E., 1911. Ecological succession: stream fishes and the method of physiographic analysis. Biol. Bull. (Woods Hole) 21:9–34.Google Scholar
  80. Sládecek, V. E., 1986. Diatoms as indicators of organic pollution. Acta hydrochim. hydrobiol. 14:555–566.Google Scholar
  81. Smilauer, P., 1992. CanoDraw User's Guide v. 3.0, Microcomputer Power, Ithaca, NY USA, 118 pp.Google Scholar
  82. Smilauer, P., 1994. Exploratory analysis of palaeoecological data using the program CanoDraw. J. Paleolimnol. 12:163–169.Google Scholar
  83. Snoeijs, P. J. M., 1989. Effects of increasing water temperatures and flow rates on epilithic fauna in a cooling-water discharge basin. J. Appl. Ecol. 26:935–956.Google Scholar
  84. Snoeijs, P. J. M. and I. C. Prentice, 1989. Effects of cooling water discharge on the structure and dynamics of epilithic algal communities in the northern Baltic. Hydrobiologia 184:99–123.Google Scholar
  85. Soetaert, K., M. Vincx, J. Wittoeck, M. Tulkens and D. Vangansbeke, 1994. Spatial patterns of Westerschelde meiobenthos. Estuarine Coastal and Shelf Science 39:367–388.Google Scholar
  86. Stevenson, A. C., H. J. B. Birks, R. J. Flower and R. W. Battarbee, 1989. Diatom-based pH reconstruction of lake acidification using canonical correspondence analysis. Ambio 18:228–233.Google Scholar
  87. Stewart-Oaten, A., W. M. Murdoch and K. P. Parker, 1986. Environmental impact assessment: “Pseudoreplication” in time? Ecology 67:929–940.Google Scholar
  88. Sundbäck, K. and P. Snoeijs, 1991. Effects of nutrient enrichment on microalgal community composition in a coastal shallow-water sediment system: an experimental study. Bot. Mar. 34:341–358.Google Scholar
  89. Takane, Y., H. Yanai and S. Mayekawa, 1991. Relationships among several methods of linearly constrained correspondence analysis. Psychometrika 56:667–684.Google Scholar
  90. ter Braak, C. J. F., 1985. Correspondence analysis of incidence and abundance data: properties in terms of a unimodal reponse model. Biometrics 41:859–873.Google Scholar
  91. ter Braak, C. J. F., 1986. Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology 67:1167–1179.Google Scholar
  92. ter Braak, C. J. F., 1987a. The analysis of vegetation-environment relationships by canonical correspondence analysis. Vegetatio 69:69–77.Google Scholar
  93. ter Braak, C. J. F., 1987b. Ordination. In: Jongman R. H. G., C. J. F. ter Braak and O. F. R. van Tongeren (eds.), Data analysis in communityy and landscape ecology, Pudoc, Wageningen (reprinted by Cambridge University Press, Cambridge, 1995), pp. 91–173.Google Scholar
  94. ter Braak, C. J. F., 1988a. Partial canonical correspondence analysis. In: Bock H. H. (ed.), Classification and related methods of data analysis, North-Holland, Amsterdam, pp. 551–558.Google Scholar
  95. ter Braak, C. J. F., 1988b. CANOCO — a FORTRAN program for canonical community ordination by [partial] [detrended] [canonical] correspondence analysis, principal components analysis and redundancy analysis (version 2.1). Report LWA-88-02, Agricultural Mathematics Group, Wageningen, 95 pp.Google Scholar
  96. ter Braak, C. J. F., 1990a. Interpreting canonical correlation analysis through biplots of structural correlations and weights. Psychometrika 55:519–531.Google Scholar
  97. ter Braak, C. J. F., 1990b. Update notes: CANOCO version 3.1, Agricultural Mathematics Group, Wageningen, 35 pp.Google Scholar
  98. ter Braak, C. J. F., 1992. Permutation versus bootstrap significance tests in multiple regression and ANOVA. In: Jöckel K.-H., G. Rothe and W. Sendler (eds.), Bootstrapping and related techniques, Springer Verlag, Berlin, pp. 79–85.Google Scholar
  99. ter Braak, C. J. F., 1994. Canonical community ordination. Part I: Basic theory and linear methods. Ecoscience 1:127–140.Google Scholar
  100. ter Braak, C. J. F., 1995a. Non-linear methods for multivariate statistical calibration and their use in palaeoecology: a comparison of inverse (k-Nearest Neighbours, PLS and WA-PLS) and classical approaches. Chemometrics Intell. Lab. Syst. 28:165–180.Google Scholar
  101. ter Braak, C. J. F., 1995 b. Canonical community ordination. Part II: The correspondence analysis family. in prep.Google Scholar
  102. ter Braak, C. J. F. and S. Juggins, 1993. Weighted averaging partial least squares regression (WA-PLS): an improved method for reconstructing environmental variables from species assemblages. Hydrobiologia 269:485–502.Google Scholar
  103. ter Braak, C. J. F., S. Juggins, H. J. B. Birks and H. Van der Voet, 1993. Weighted averaging partial least squares regression (WA-PLS): definition and comparison with other methods for speciesenvironment calibration. In: Patil G. P. and C. R. Rao (eds.), Multivariate Environmental Statistics, North-Holland, Amsterdam, pp. 525–560.Google Scholar
  104. ter Braak, C. J. F. and C. W. N. Looman, 1994. Biplots in reduced-rank regression. Biom. J. 36:983–1003.Google Scholar
  105. ter Braak, C. J. F. and I. C. Prentice, 1988. A theory of gradient analysis. Adv. ecol. res. 18:271–317.Google Scholar
  106. ter Braak, C. J. F. and H. van Dam, 1989. Inferring pH from diatoms: a comparison of old and new calibration methods. Hydrobiologia 178:209–223.Google Scholar
  107. Underwood, A. J., 1992. Beyond BACI: the detection of environmental impacts on populations in the real, but variable, world. J. Exp. Mar. Biol. Ecol. 161:145–178.Google Scholar
  108. van Nes, E. H. and H. Smit, 1993. Multivariate analysis of macrozoobenthos in Lake Volkerak-Zoommeer (the Netherlands): changes in an estuary before and after closure. Achiv. Hydrobiol. 127:185–203.Google Scholar
  109. Verdonschot, P. F. M., 1989. The role of oligochaetes in the management of waters. Hydrobiologia 180:213–227.Google Scholar
  110. Verdonschot, P. F. M. and C. J. F. ter Braak, 1994. An experimental manipulation of oligochaete communities in mesocosms treated with chlorpyrifos or nutrient additions: multivariate analysis with Monte Carlo permutation tests. Hydrobiologia 278:251–266.Google Scholar
  111. von Tümpling, W., 1966. Über die statistische Sicherheit soziologischer Methoden in der biologischen Gewässeranalyse. Limnologica (Berlin) 4:235–244.Google Scholar
  112. Walker, I. R., J. P. Smol, D. R. Engstrom and H. J. B. Birks, 1991. An assessment of Chironomidae as quantitative indicators of past climatic change. Can. J. Fish. Aquat. Sci. 48:975–987.Google Scholar
  113. Washington, H. G., 1984. Diversity, biotic and similarity indices: a review with special relevance to aquatic ecosystems. Water Res. 18:653–694.Google Scholar
  114. Whittaker, R. H., S. A. Levin and R. B. Root, 1973. Niche, habitat and ecotope. Amer. Nat. 107:321–338.Google Scholar
  115. van Wijngaarden, R. P. A., P. J. van den Brink, J. H. Oude Voshaar and P. Leeuwangh, 1995. Ordination techniques for analysing response of biological communities to toxic stress in experimental ecosystems. Ecotoxicol. 4:61–77.Google Scholar
  116. Wold, H., 1982. Soft modeling: the basic design and some extensions. In: Joreskog K. G. and H. Wold (eds.), Systems under indirect observations II, North-Holland, Amsterdam, pp. 1–54.Google Scholar
  117. Zelinka, M. and P. Marvan, 1961. Zur Präzisierung der biologischen Klassifikation der Reinheit fliessender Gewässer. Arch. Hydrobiol. 57:389–407.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Cajo J. F. ter Braak
    • 1
    • 2
  • Piet F. M. Verdonschot
    • 2
  1. 1.DLO Agricultural Mathematics GroupsWageningenthe Netherlands
  2. 2.DLO Institute for Forestry and Nature ResearchWageningenthe Netherlands

Personalised recommendations