Distance-based Ordination Methods

DISTANCE-BASED ORDINATION METHODS

Distance-based approaches rely on a square, symmetric distance matrix or similarity matrix. See Similarity, Distance and Difference. For polar ordination, it is necessary for data to obey the triangle inequality (i.e. the distance between A and B plus the distance between B and C cannot exceed the distance between A and C). Unlike methods derived from eigenanalysis, distance-based methods do not provide species and sample scores simultaneously.

Polar ordination (PO)

Also known as Bray-Curtis Ordination or Wisconsin Ordination.
PO is perhaps the easiest ordination technique to visualize, and it is possible to perform without a computer.
It is possible for axes to be correlated.
The user must pre-specify samples as end points (alternatively, there are automatic ways endpoints can be specified). These endpoints have a privileged position; all other samples are defined relative to them.
PO is the only technique where one can control the direction of gradients (i.e. left vs. right or up vs. down).
New samples can be added without affecting the ordination.

Principal Coordinates Analysis (PCoA)

This technique has been labelled "Multidimensional Scaling", although this term might be better avoided, since it is now becoming synonymous with "Nonmetric Multidimensional Scaling". PCoA has also been termed "Metric Multidimensional Scaling".
PCoA maximizes the linear correlation between distance measures and distance in the ordination.
One cannot easily put new points in a PCoA.
PCoA (and NMDS) can be useful if one has only a distance (or similarity) matrix (e.g. for DNA hybridization, or for plotless sampling)
The underlying model for PCoA (and NMDS) is that there a fixed number of gradients. In contrast, PCA, RA, and DCA assume that there are potentially many gradients, but of declining importance.
Special case: for Euclidean distance, PCoA = PCA.
PCoA can be expressed as an eigenanalysis.

Nonmetric Multidimensional Scaling (NMDS)

This is sometimes labeled "Multidimensional Scaling (MDS)", although this term has been used for PCoA.
NMDS is very computer intensive; it has only recently become feasible for large data sets on the microcomputer.
NMDS Maximizes rank-order correlation between distance measures and distance in ordination space. Points are moved to minimize "stress". Stress is a measure of the mismatch between the two kinds of distance.
NOTE: Gauch (1982) is WRONG is stating that NMDS assumes species have monotonic relationships to gradients. What it assumes is that dissimilarity is monotonically related to ecological distance.
The user must pre-specify number of dimensions, or examine a plot of "stress" as a function of number of axes, and select the number of dimensions a posteriori.
The configuration will change as the number of axes change.
There is no guarantee that the correct (lowest stress) solution will be found, though it is widely assumed that this is not a big problem.
To increase the likelihood of finding the correct solution, a DCA is often performed first.
A recent variant of NMDS is Local NMDS, which only minimizes stress locally.

Eigenanalysis-based methods

It can be shown that some methods described in eigenanalysis-based ordination methods are also special cases of distance-based methods, where the distance is based on the chi-squared distribution. While this is true, it is worth keeping in mind that the underlying philosophy of the eigenanalysis-based methods is fundamentally different: these methods attempt to faithfully place species along gradients (either inferred or directly related to measured variables), and not to faithfully relate difference to distance.

Selected References

Polar Ordination

Beals, E. W. 1984. Bray-Curtis ordination: an effective strategy for analysis of multivariate ecological data. Adv. Ecol. Res. 14:1-55.

Brooks, A. R., E. S. Nixon, and J. A. Neal. 1993. Woody vegetation of wet creek bottom communities in eastern Texas. Castanea 58:185-96.

Dooley, K. 1983. Description and dynamics of some western Oak forests in Oklahoma. PhD Thesis. University of Oklahoma.

Dunn, C. P., and F. Stearns. 1987. Relationship of vegetation layers to soils in southeastern Wisconsin forested wetlands. Am. Midl. Nat. 118:366-74.

Feoli, E., and L. Orl¢ci. 1992. Thre properties and interpretation of observations in vegetation study. Coenoses 6:61-70.

Holloway, R. G. 1984. Analysis of surface pollen from Alberta, Canada, by polar ordination. Northw. Sci. 58:18- 28.

Kent, M., and P. Coker. 1992. Vegetation description and analysis: a practical approach. Belhaven Press, London.

Mazzoleni, S., D. D. French, and J. Miles. 1991. A comparative study of classification and ordination methods on successional data. Coenoses 6:91-101.

Roberts, D. W. 1987. An anticommutative difference operator for fuzzy sets and relations. Fuzzy Sets Syst. 21:35-42.

Umbanhower, C. E. 1992. Early patterns of revegetation of artificial earthen mounds in a northern mixed prairie. Can. J. Bot. 70:14-150.

Principal Coordinates Analysis

Jackson, D. A., K. M. Somers, and H. H. Harvey. 1989. Similarity coefficients: measures of co-occurrence and association or simply measures of occurrence? Am. Nat. 133:436-53.

Matus, G., and B. T¢thm‚r‚sz. 1990. The effect of grazing on the structure of a sandy grassland. Pages 23-30 in F. Krahulec, A. D. Q. Agnew, S. Agnew and H. J. Willems, editors. Spatial processes in plant communities. Academia, Prague.

Minchin, P. R. 1987. An evaluation of the relative robustness of techniques for ecological ordination. Vegetatio 69:89-107.

Nantel, P., and P. Neumann. 1992. Ecology of ectomycorrhizal-basidiomycete communities on a local vegetation gradient. Ecology 73:99-117.

Wildi, O. 1992. On the use of Mantel's statistic and flexible shortest path adjustment in the analysis of ecological gradients. Coenoses 7:91-101.

Nonmetric Multidimensional Scaling (and its variants)

Belbin, L. 1991. Semi-strong hybrid scaling, a new ordination algorithm. J. Veg. Sci. 2:491-6.

Birks, H. J. B., and J. Deacon. 1973. A numerical analysis of the past and present flora of the British Isles. New Phytol. 72:877-902.

Boulton, A. J., C. G. Peterson, N. B. Grimm, and S. G. Fisher. 1992. Stability of an aquatic macroinvertebrate community in a multiyear hydrologic disturbance regime. Ecology 73:2192-207.

Bowman, D. M. J. S. 1987. Environmental relationships of woody vegetation patterns in the Australian monsoon tropics. Austral. J. Bot. 35:151-69.

Bradfield, G. E., and N. C. Kenkel. 1987. Nonlinear ordination using flexible shortest path adjustment of ecological distances. Ecology 68:750-3.

Brower, J. C., S. A. Millendorf, and T. S. Dyman. 1978. Methods for the quantification of assemblage zones based on multivariate analysis of weighted and unweighted data. Comp. Geosci. 4:221-7.

Digby, P. G. N., and R. A. Kempton. 1987. Population and Community Biology Series: Multivariate Analysis of Ecological Communities. Chapman and Hall, London.

Duncan, R. P., and G. H. Stewart. 1991. The temporal and spatial analysis of tree age distributions. Can. J. For. Res. 21:1703-10.

Espigares, R., and B. Peco. 1993. Mediterranean pasture dynamics: the role of germination. J. Veg. Sci. 4:189-94.

Faith, D. P., P. R. Minchin, and L. Belbin. 1987. Compositional dissimilarity as a robust measure of ecological distance. Vegetatio 69:57-68.

Feoli, E., and L. Orl¢ci. 1992. Thre properties and interpretation of observations in vegetation study. Coenoses 6:61-70.

Friedel, M. H., G. Pickup, and D. J. Nelson. 1993. The interpretation of vegetation change in a spatially and temporally diverse arid Australian landscape. J. Arid Envir. 24:241-60.

Fulton, M. R. 1991. Simulation modelling of the effects of site conditions and disturbance history on a boreal forest landscape. J. Veg. Sci. 2:603-12.

Harrison, S. P., R. S. Clymo, and H. Southall. 1988. Order amid sparse data: patterns of lake level changes in North America during the late Quaternary. J. Math. Geol. 20:167- 88.

Kenkel, N. C. 1988. Introduction to data analysis: a comprehensive program package for personal computers. Coenoses 3:107-9.

Kenkel, N. C., and L. Orl¢ci. 1986. Applying metric and nonmetric scaling to ecological studies: some new results. Ecology 67:919-28.

Kovach, W. L. 1989. Comparisons of multivariate analytical techniques for use in Pre-quaternary plant paleoecology. Rev. Palaeo. Palyn. 60:255-82.

Kruskal, J. B. 1964a. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29:1-27.

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

MacNally, R. C. 1990. Modelling distributional patterns of woodland birds along a continental gradient. Ecology 71:360-74.

Minchin, P. R. 1987. An evaluation of the relative robustness of techniques for ecological ordination. Vegetatio 69:89-107.

Mosk t, C. 1991. Multivariate plexus concept in the study of complex ecological data: an application of the analysis of bird-habitat relationships. Coenoses 6:79-89.

Oksanen, J., and P. Huttunen. 1989. Finding a common ordination for several data sets by individual differences scaling. Vegetatio 83:137-45.

Prentice, I. C. 1986. Multivariate methods for data analysis. Pages 775-797 in B. E. Berglund, editor. Handbook of Holocene Palaeoecology and Palaeohydrology. John Wiley & Sons, Chichester.

Rydgren, K. 1993. Herb-rich spruce forests in W Nordland, N Norway: an ecological and methodological study. Nord. J. Bot. 13:667-90.

Sibson, R. 1972. Order invariant methods for data analysis. J. Roy. Stat. Soc. 34:311-38.

Spitaleri, R. M., I. Napoleone, and L. Contoli. 1991. A numerical approach for the evaluation of biotopes for conservational purposes. Coenoses 6:169-71.

West, N. E. 1990. Relative influence of observer error and plot randomization on detection of vegetation change. Coenoses 5:45-9.

Woods, K. D. 1993. Effects of invasion by Lonicera tatarica L. on herbs and tree seedlings in four New England forests. Am. Midl. Nat. 130:62-74.

This page was created and is maintained by Michael Palmer.
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