EIGENANALYSIS-BASED METHODS

for indirect gradient analysis

Eigenanalysis

Eigenanalysis is central to the mathematical discipline of linear (matrix) algebra, and a thorough understanding of ordination methods requires a training in linear algebra. However, for our present purposes, it suffices to know that:

Eigenanalysis is a mathematical operation on a square, symmetric matrix. A square matrix has the same number of rows as columns. A symmetric matrix is the same if you switch rows and columns. Distance and similarity matrices are nearly always square and symmetric.
It is possible to perform a eigenanalyses analytically (that is, get exact results) only for very small matrices (e.g. three rows and columns). For large matrices, eigenanalysis requires an iterative approach which eventually "closes in" on the answer (in most cases).
The answer to an eigenanalysis consists of a series of eigenvalues and eigenvectors. Each eigenvalue has an eigenvector, and there are as many eigenvectors and eigenvalues as there are rows in the initial matrix. Eigenvalues are usually ranked from the greatest to the least. The first eigenvalue is often called the "dominant" or "leading" eigenvalue. Eigenvalues are also often called "latent values".
The eigenvalue is a measure of the strength of an axis, the amount of variation along an axis, and ideally the importance of an ecological gradient. The precise meaning depends on the ordination method used.
The eigenvectors are the sample scores, if the rows and columns of the initial matrix represent samples.
Some texts (e.g. Digby and Kempton 1987) describe "singular value decomposition" instead of eigenanalysis. The approaches are theoretically identical.
Principal Components Analysis, Correspondence Analysis (Reciprocal Averaging), DCA (Detrended Correspondence Analysis) are all examples of eigenanalysis-based ordination methods.
The direct gradient analysis (constrained ordination) techniques of RDA, CCA, and DCCA (not covered in this page) are also eigenanalysis-based techniques.

Principal Components Analysis (PCA)

See Principal Components Analysis for more information.
PCA is available in most statistical packages.
PCA is often considered a form of "factor analysis".
PCA is a "rigid rotation" of the data matrix: it does not change the positions of points relative to each other; it just changes the coordinate systems.
PCA is mathematically quite elegant.
In PCA, axes are created such that the perpendicular distance from each object to the ordination axes is minimized.
Axes are linear combinations of species/variables. The weights are known as "coefficients" or "loadings".
PCA is good for data that are not in the same units, in which case data must be standardized to zero mean and unit variance (this is known as PCA on the correlation matrix).
PCA can accept negative numbers for variables/species.
PCA can place new points in an old ordination.
PCA can find characteristics of any point in ordination space (though this is not standard in most packages).
Eigenvalues have a meaning: variance explained.
The sum of the eigenvalues will equal the sum of the variance of all variables.
If performed on a correlation matrix, the sum of the eigenvalues will equal the number of variables/species.
If performed on a covariance matrix, the sum of the eigenvalues will equal the sum of the variances of all species.
PCA has a serious problem for vegetation data: the horseshoe effect. This is caused by the curvilinearity of species distributions along gradients. Since species response curves are typically unimodal (i.e. very strongly curvilinear), horseshoe effects are common.

Correspondence Analysis or Reciprocal Averaging (RA)

See Correspondence Analysis for more information.
RA has been discovered independently by different scientists.
Reciprocal Averaging means that sample scores are calculated as a weighted average of species scores, and species scores are calculated as a weighted average of sample scores, and iterations continue until there is no change. However, other algorithms are possible.
RA simultaneously ordinates species and samples. There are as many axes as there are species or samples, whichever is less.
The number of axes worth interpreting is a matter of taste, but eigenvalues can be a guide (e.g. if the third eigenvalue is much less than the second eigenvalue, third and higher axes might not be informative).
RA maximizes the correlation between species scores and sample scores. The eigenvalue is equal to the correlation coefficient.
An eigenvalue of 1.0 implies that one sample (or group of samples) shares no species with all other samples.
One can put new points in a Correspondence Analysis without affecting the rest of the ordination.
As with all the other eigenvector techniques, it is possible to define "passive samples" or "passive species".
RA has a problem: the arch effect. It is also caused by nonlinearity of distributions along gradients.
The arch is not as serious as the horseshoe effect of PCA, because the ends of the gradient are not convoluted.
Another related problem of RA is that the ends of the gradient are compressed.

Detrended Correspondence Analysis (DCA)

See Detrended Correspondence Analysis for more information.
DCA is probably the most widely used ordination technique today. It is an extension of RA.
Both sample scores and species scores are produced.
The first axis has same eigenvalue as RA.
The arch is removed by dividing the first axis into segments, and re-centering samples.
Samples are shifted to equalize beta-diversity: species have, on average, a habitat breadth (as measured by standard deviations) of 1. Thus the axes of DCA are a useful measure of beta diversity.
One cannot easily put in new samples in DCA.
DCA has problems: it is inelegant, sensitive to parameters which determine number of segments, destroys a "true" arch (if it exists), and perhaps creates a "tongue effect" (I suspect that the existence of a tongue effect is more commonly a real attribute of nature, than an artifact of DCA).

Selected References, by type of ordination

PCA

Aitchison, J. 1983. Principal component analysis of compositional data. Biometrika 70:57-65.

Aitchison, J. 1984. Reducing the dimensionality of compositional data sets. J. Math. Geol. 16:617-35.

Allen, T. F. H. 1977. Multiple stable configurations in ordination of phytoplankton community change rates. Ecology 58:1076-84.

Allen, T. F. H., and J. F. Koonce. 1973. Multivariate approaches to algal stratagems and tactics in systems analysis of phytoplankton. Ecology 54:1234-46.

Anderson, D. S., R. B. Davis, and F. Berge. 1986. Relationships between diatom assemblages in lake surface- sediments and limnological characteristics in southern Norway. Pages 97-113 in J. P. Smol, R. W. Battarbee, R. B. Davis and J. Meril"inen, editors. Diatoms and Lake Acidity. W. Junk, Dordrecht.

Andrews, J. T., and H. F. Diaz. 1981. Eigenvector analysis of reconstructed holocene July tmeperature over northern Canada. Quat. Res. 16:373-89.

Bailey, R. C. 1992. Hierarchical analysis of community and habitat structure. Coenoses 7:127-35.

Barla, M. J. 1991. Species composition, richness and diversity of fish assemblages in different habitats of a pampean lake (Argentina). Ann. Limnol. 27:163-73.

Birks, H. J. B. 1985. Recent and possible future mathematical developments in quantitative palaeoecology. Palaeogeog. Palaeoclim. Palaeoecol. 50:107-47.

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.

Butler, J. C. 1976. Principal components analysis using the hypothetical closed array. J. Math. Geol. 8:25-36.

Carleton, T. J., and P. F. Maycock. 1980. Vegetation of the boreal forests south of James Bay: non-centered component analysis of the vascular flora. Ecology 61:1199-212.

Chayes, F., and J. Trochimczyk. 1978. An effect of closure on the structure of principal components. J. Math. Geol. 10:323-33.

Collins, S. L. 1983. Geographic variation in habitat structure of the black-throated green warbler (Dendroica virens). Auk 100:382-9.

Collins, S. L. 1990. Habitat relationships and survivorship of tree seedlings in hemlock-hardwood forest. Can. J. Bot. 68:790-7.

Cornet, C. 1985. Non-marine diatom numerical zonations of a boring from Hinkelsmaar, Eifel (Germany). Ecol. Mediterr. 11:177-8.

Crawford, R. E., C. Hudon, and D. G. Parsons. 1992. An acoustic study of shrimp (Pandalus montagui) distribution near Resolution Island (Eastern Hudson Strait). Can. J. Fish. Aquat. Sci. 49.

Davis, R. B., and D. S. Anderson. 1985. Methods of pH calibration of sedimentary diatom remains for reconstructing history of pH in lakes. Hydrobiol. 120:69- 87.

Diaz, H. F., and J. T. Andrews. 1982. Analysis of the spatial pattern of July temperature departures (1943-1972) over Canada and estimates of the 700 mb midsummer circulation during middle and late holocene. J. Clim. 2:251-65.

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

Dodson, J. R., and R. V. S. Wright. 1989. Humid to arid to subhumid vegetation shift on Pilliga Sandstone, Ulungra Springs, New South Wales. Quat. Res. 32:182-92.

Ehrenfeld, J. G., and J. P. Schneider. 1991. Chamaecyparis thyoides wetlands and suburbanization: effects on hydrology, water quality and plant community composition. J. Appl. Ecol. 28:467-90.

Elek, I. 1988. Some applications of principal component analysis: well-to-well correlation, zonation. Geobyte May 1988:46-55.

Elias, S. A. 1982. Paleoenvironmental interpretation of Holocene insect fossils from northeastern Labrador, Canada. Arct. Alp. Res. 14:311-9.

Espinosa-Garcia, F. J., and J. H. Langenheim. 1990. The endophytic fungal community in leaves of a coastal redwood population - diversity and spatial patterns. New Phytol. 116:89-97.

Fall, P. L., G. Kelso, and V. Markgraf. 1981. Paleoenvironmental reconstruction at Canyon del Muerto, Arizona, based on principal-components analysis. J. Arch. Sci. 8:297-307.

Feoli, E., and L. Orlóci. 1992. The properties and interpretation of observations in vegetation study. Coenoses 6:61-70.

Fernández Alés, R., J. M. Laffarga, and F. Ortega. 1993. Strategies in Mediterranean grassland annuals in relation to stress and disturbance. J. Veg. Sci. 4:313-22.

Foster, D. R., and S. C. Fritz. 1987. Mire development, pool formation and landscape processes on patterned fens in Dalarna, central Sweden. J. Ecol. 75:400-37.

Franceschi, E. A., and J. P. Lewis. 1991. Early stages in the Paran River tall grassland recover after an extraordinary flood. Coenoses 6:47-52.

Gajewski, K. 1987. Climatic impacts on the vegetation of eastern North America during the past 2000 years. Vegetatio 68:179-90.

Garcia, A., G. Carlos, and F. F. Bermudez. 1992. Vegetation, soil and management relations in meadow communities of the Vadeon Valley, Cordillera Cantabrica, Spain. Coenoses 7:55-60.

García-Rubies, A., and M. Zabala. 1990. Effects of total fishing prohibition on the rocky fish assemblages of Medes Islands marine reserve (NW Mediterranean). Sci. Mar. 54:317-28.

Gower, J. C. 1987. Introduction to ordination techniques. Pages 3-64 in P. Legendre and L. Legendre, editors. Developments in Numerical Ecology. Springer-Verlag, Berlin.

Grimm, E. C. 1988. Data analysis and display. Pages 43-76 in B. Huntley and T. Webb III, editors. Vegetation History. Kluwer Academic Publishers, Dordrecht.

Harrison, S., S. J. Ross, and J. H. Lawton. 1992. Beta diversity on geographic gradients in Britain. J. Anim. Ecol. 61:151-8.

Hauser, M., and L. Mucina. 1991. Spatial interpolation methods for interpretation of ordination diagrams. Coenoses 6:151-68.

Held, M. E. 1983. Pattern of beech regeneration in the east-central United States. Bull. Torr. Bot. Club 110:55-62.

Heusser, C. J. 1981. Palynology of the last interglacial- glacial cycle in midlatitiudes of southern Chile. Quat. Res. 16:293-321.

Heusser, L. E. 1983. Contemporary pollen distribution in coastal California and Oregon. Palynol. 7:19-42.

Heusser, L. E., and J. J. Morely. 1985. Correlative 90 kyr northeast Asia-northwest Pacific climate records. Nature 313:470-2.

Hohn, M. E. 1978. Stratigraphic correlation by principal components: effects of missing data. J. Geol. 86:524-32.

Hohn, M. E. 1985. SAS program for quantitative stratigraphic correlation by principals components. Comp. Geosci. 11:471-7.

Hudon, C. 1990. Distribution of shrimp and fish by-catch assemblages in the Canadian Eastern Arctic in relation to water circulation. Can. J. Fish. Aquat. Sci. 47:1710-23.

Jackson, D. A. 1993. Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches. Ecology 74:2204-14.

Kearney, M. S. 1983. Modern pollen deposition in the Athabasca Valley Jasper National Park. Bot. Gaz. 144:450- 9.

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

Klovan, J. E., and A. T. Miesch. 1976. Extended CABFAC and Q-model computer programs for Q-mode factor analysis of compositional data. Comp. Geosci. 1:161-78.

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

Lamb, H. F. 1985. Palynological evidence for postglacial change in the position of tree limit in Labrador. Ecol. Mon. 55:241-58.

Lamb, H. F., and M. E. Edwards. 1988. The Arctic. Pages 519-555 in B. Huntley and T. Webb III, editors. Vegetation History. Kluwer, Dordrecht.

LeBreton, J. D., R. Sabatier, G. Banco, and A. M. Bancou. 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. Pages 85-114 in J. Devillers and W. Karcher, editors. Applied multivariate analysis in SAR and Environmental Studies. Kluwer Academic Publishers, Dordrecht.

Lee, S. 1978. A factor analysis study of the functional significance of angiosperm pollen. Syst. Bot. 3:1-19.

Lobb, M. D., and D. J. Orth. 1991. Habitat use by an assemblage of fish in a large warmwater stream. Trans. Am. Fish. Soc. 120:65-78.

MacDonald, G. M. 1987. Postglacial development of the subalpine-boreal transition forest of western Canada. J. Ecol. 75:303-20.

MacDonald, G. M., and J. C. Ritchie. 1986. Modern pollen spectra from the western interior of Canada and the interpretation of late-Quaternary vegetation development. New Phytol. 103:245-68.

MacDonald, G. M., and N. M. Waters. 1988. The use of most predictable surfaces for the classification and mapping of taxon assemblages. Vegetatio 74:125-35.

Malmgren, B. A., and B. U. Haq. 1982. Assessment of quantitative techniques in paleobiogeography. Marine Micr. 7:213-26.

Markgraf, V., H. L. D. Antoni, and T. A. Ager. 1981. Modern pollen dispersal in Argentina. Palynol. 5:43-63.

Maslov, A. A. 1990. Multi-scaled and multi-species pattern analysis in boreal forest communities. Pages 83-88 in F. Krahulec, A. D. Q. Agnew, S. Agnew and H. J. Willems, editors. Spatial processes in plant communities. Academia, Prague.

Maslow, A. A. 1990. Quantitative Analysis of horizontal structure of forest communities. Nauka, Moscow.

Mathewes, R. W. 1988. Pollen spectra from lake surface sediments in the Swiss Alps. Arct. Alp. Res. 20:310-24.

Mathewes, R. W., and L. E. Heusser. 1980. A 12000 year palynological record of temperature and precipitation trends in south-western British Columbia. Can. J. Bot. 59:707-10.

Matthews, R. A., G. B. Matthews, G. B. Ehinger, and W. J. Ehinger. 1991. Classification and ordination of limnological data: a comparison of analytical tools. Ecol. Mod. 53:167-87.

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

McCarthy, B. C., C. A. Hammer, G. L. Kauffman, and P. D. Cantino. 1987. Vegetation patterns and structure of an old-growth forest in southeastern Ohio. Bull. Torr. Bot. Club 114:33-45.

Mellink, E. 1991a. Bird communities associated with three traditional agroecosystems in the San Luis Potosi Plateau, Mexico. Agric. Ecosyst. Environ. 36:37-50.

Mellink, E. 1991b. Rodent communities associated with three traditional agroecosystems in the San Luis Potosi Plateau, Mexico. Agric. Ecosyst. Environ. 33:363-75.

Meyer, S. E., E. García-Moya, and L. Lagunes-Espinoza. 1992. Topographic and soil surface effects on gypsophile plant community patterns in central Mexico. J. Veg. Sci. 3:429- 38.

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

Morzuch, B. J., and G. A. Ruark. 1991. Principal components regression to mitigate the effects of multicollinearity. For. Sci. 37:191-9.

Nichols, S. 1977. On the interpretation of principal component analysis in ecological contexts. Vegetatio 34:191-7.

Norton, D. A., M. S. McGlone, and T. M. L. Wigley. 1986. Quantitative analyses of modern pollen-climate relationships in New Zealand indigenous forests. N. Z. J. Bot. 24:331-42.

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

Patterson, B. D., P. L. Meserve, and B. K. Lang. 1990. Quantitative habitat associations of small mammals along an elevational transect in temperate rainforests of Chile. J. Mamm. 71:620-33.

Pielou, E. C. 1977. Mathematical Ecology. John Wiley & Sons, New York.

Pimm, S. L. 1991. The Balance of Nature? The University of Chicago Press, Chicago.

Podani, J. 1988. Syn-Tax III. a package of programs for data analysis in community ecology and systematics. Coenoses 3:111-9.

Podani, J. 1991. On the standardization of Procrustes statistics for the comparison of ordinations. Abstr. Bot. 15:43-6.

Prentice, I. C. 1983. Pollen mapping of regional vegetation patterns in south and central Sweden. J. Biogeogr. 10:441- 54.

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.

Rae, J. G. 1985. A multivariate study of resource partitioning in soft bottom lotic Chironomidae. Hydrobiol. 126:275-85.

Real, R., J. M. Vargas, and A. Ant£nez. 1993. Environmental influences on local amphibian diversity: the role of floods on river basins. Biodivers. Conserv. 2:376-99.

Renberg, I., T. Korsman, and H. J. B. Birks. 1993. Prehistoric increases in the pH of acid-sensitive Swedish lakes caused by land-use changes. Nature 362:824-6.

Sabatier, R., J. D. LeBreton, and D. Chessel. 1989. Principal component analysis with instrumental variables as a tool for modelling composition data. Pages 341-352 in R. Coppi and S. Bolasco, editors. Multiway data analysis. Elsevier Science Publishers B.V., North Holland.

Schnitzler, A., R. Carbiener, and J.-M. Sanchez-Perez. 1991. Variation in vernal species composition in alluvial forests of the Rhine valley, eastern France. J. Veg. Sci. 2:485-90.

Soukupov , L. 1990. Coexisting competitors in a sedge- grass marsh, Central Europe. Pages 145-159 in F. Krahulec, A. D. Q. Agnew, S. Agnew and H. J. Willems, editors. Spatial processes in plant communities. Academia, Prague.

Stauffer, D. F., E. O. Garton, and R. K. Steinhorst. 1985. A comparison of principal components from real and random data. Ecology 66:1693-8.

Steinman, A. D., and T. B. Ladewski. 1987. Quantitative shape analysis of Eunotia pectinalis (Bacillariophyceae) and its application to seasonal distribution patterns. Phycologia 26:467-77.

Sun, C. Y., and E. Feoli. 1992. Trajectory analysis of Chinese vegetation types in a multidimensional climatic space. J. Veg. Sci. 3:587-94.

ter Braak, C. J. F. 1983. Principal components biplots and alpha and beta diversity. Ecology 64:454-62.

ter Braak, C. J. F., and I. C. Prentice. 1988. A theory of gradient analysis. Adv. Ecol. Res. 18:271-313.

Ver Hoef, J. M., and D. C. Glenn-Lewin. 1989. Multiscale ordination: a method for detecting pattern at several scales. Vegetatio 82:59-67.

Vetaas, O. R. 1992. Gradients in field-layer vegetation on an arid misty mountain plateau in the Sudan. J. Veg. Sci. 3:527-34.

Wildi, O. 1985. A package for the analysis of vegetation data by microcomputers. Abstr. Bot. 9:153-9.

Wildi, O. 1989. Analysis of the disintegrating group and gradient structure in Swiss riparian forests. Vegetatio 83:179-86.

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.

Wildi, O., and L. Orlóci. 1987. Flexible gradient analysis: a note on ideas and an application. Coenoses 2:61-5.

You Minsheng and Wu Zhongfu. 1985. The ordination of insect communities in paddy fields. J. Fuj. Agric. Coll. 14:95- 102.

Reciprocal Averaging

Anderson, N. J. 1989. A whole-basin diatom accumulation rate for a small eutrophic lake in northern Ireland and its palaeoecological implications. J. Ecol. 77:926-46.

Arnold, G. W., and J. R. Weeldenburg. 1990. Factors determining the number and species of birds in road verges in the wheatbelt of Western Australia. Biol. Cons. 53:295- 315.

Becker, M., M. Bonneau, and F. Le Tacon. 1992. Long-term vegetation changes in an Abies alba forest: natural development compared with response to fertilization. J. Veg. Sci. 3:467-74.

Birks, H. J. B. 1985. Recent and possible future mathematical developments in quantitative palaeoecology. Palaeogeog. Palaeoclim. Palaeoecol. 50:107-47.

Birks, H. J. B. 1987. Multivariate analysis in geology and geochemistry: an introduction. Chem. Intell. Lab. Syst. 2:15-28.

Bolognini, G., and P. L. Nimis. 1993. Phytogeography of Italian deciduous oak woods based on numerical classification of plant distribution ranges. J. Veg. Sci. 4:847-60.

Burel, F. 1992. Effect of landscape structure and dynamics on species diversity in hedgerow networks. Landscape Ecol. 6:161-74.

Burton, A. J., C. W. Ramm, K. S. Pregitzer, and D. D. Reed. 1991. Use of multivariate methods in forest research site selection. Can. J. For. Res. 21:1573-80.

Bush, M. B. 1988. The use of multivariate analysis and modern analogue sites as an aid to the interpretation of data from fossil mollusc assemblages. J. Biogeogr. 15:849- 61.

Charles, D. F. 1985. Relationships between surface sediment diatom assemblages and lakewater characteristics in Adirondack lakes. Ecology 66:994-1011.

Christie, C. E., J. P. Smol, P. Huttunen, and J. Meril"inen. 1988. Chrysophyte scales recorded in lake sediments form eastern Finland. Hydrobiol. 161:237-43.

Clark, J. S. 1986. Dynamism in the barrier-beach vegetation of Great South Beach, New York. Ecol. Mon. 56:97-126.

Collins, S. L. 1983a. Geographic variation in habitat structure for the wood warblers in Maine and Minnesota. Oecol. 59:236-52.

Collins, S. L. 1983b. Geographic variation in habitat structure of the black-throated green warbler (Dendroica virens). Auk 100:382-9.

Delcourt, P. A., and H. R. Delcourt. 1987a. Late-Quaternary dynamics of temperate forests: applications of paleoecology to issues of global environmental change. Quat. Sci. Rev. 6:129-46.

Delcourt, P. A., and H. R. Delcourt. 1987b. Long-term forest dynamics of the temperate zone: a case study of Late-Quaternary forests in eastern North America. Springer-Verlag, New York.

Escoufier, Y. 1987. The duality diagram: a means for better practical applications. Pages 139-156 in P. Legendre and L. Legendre, editors. Developments in Numerical Ecology. Springer Verlag, Berlin.

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

Foster, D. R., and S. C. Fritz. 1987. Mire development, pool formation and landscape processes on patterned fens in Dalarna, central Sweden. J. Ecol. 75:400-37.

Galiano, E. F. 1983. Detection of multi-species patterns in plant populations. Vegetatio 53:129-38.

Gasse, F., and F. Tekaia. 1983. Transfer functions for estimating paleoecological conditions (pH) from East African diatoms. Hydrobiol. 103:85-90.

Gasse, F., B. Stabell, E. Fourtanier, and Y. van Iperen. 1989. Freshwater diatom influx in intertropical Atlantic sediments: relationships with continental records from Africa. Quat. Res. 32:229-43.

Gauch, H. G. J., R. H. Whittaker, and T. R. Wentworth. 1977. A comparative study of reciprocal averaging and other ordination techniques. J. Ecol. 65:157-74.

Gower, J. C. 1987. Introduction to ordination techniques. Pages 3-64 in P. Legendre and L. Legendre, editors. Developments in Numerical Ecology. Springer-Verlag, Berlin.

Grimm, E. C. 1988. Data analysis and display. Pages 43-76 in B. Huntley and T. Webb III, editors. Vegetation History. Kluwer Academic Publishers, Dordrecht.

van Groenewoud, H. 1992. The robustness of Correspondence Analysis, Detrended Correspondence Analysis, and TWINSPAN Analysis. J. Veg. Sci. 3:239-46.

Hatheway, W. H. 1971. Contigency-table analysis of rain forest vegetation. Pages 271-313 in G. P. Patil, E. C. Pielou and W. E. Waters, editors. Statistical Ecology. Volume 3. Pennsylvania State University Press, University Park, Pennsylvania.

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Jackson, D. A., and K. M. Somers. 1991. Putting things in order: the ups and downs of detrended correspondence analysis. Am. Nat. 137:704-12.

Jones, V. J., A. C. Stevenson, and R. W. Battarbee. 1989. Acidification of lakes in Galloway, South West Scotland: a diatom and pollen study of the post-glacial history of the Round Loch of Glenhead. J. Ecol. 77:1-23.

Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren, editors. 1987. Data Analysis in Community and Landscape Ecology. Pudoc, Wageningen, The Netherlands.

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

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

Knox, R. G. 1989. Effects of detrending and rescaling on correspondence analysis: solution stability and accuracy. Vegetatio 83:129-36.

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

Lavorel, S., and J. D. LeBreton. 1992. Evidence for lottery recruitment in Mediterranean old fields. J. Veg. Sci. 3:91-100.

Levassor, C., M. Ortega, and B. Peco. 1990. Seed bank dynamics of Mediterranean pastures subjected to mechanical disturbance. J. Veg. Sci. 1:339-44.

Majer, J. D., and A. E. de Kock. 1992. Ant recolonization of sand mines near Richards Bay, South Africa: an evaluation of progress with rehabilitation. S. Afr. Tyds. Wetens. 88:31-6.

Matthews, R. A., G. B. Matthews, G. B. Ehinger, and W. J. Ehinger. 1991. Classification and ordination of limnological data: a comparison of analytical tools. Ecol. Mod. 53:167-87.

McPherson, G. R., G. A. Rasmussen, D. B. Wester, and R. A. Masters. 1991. Vegetation and soil zonation associated with Juniperus pinchotii Sudw. trees. Great Basin Nat. 51:316-24.

Mehlman, D. W. 1992. Effects of fire on plant community composition of North Florida second growth pineland. Bull. Torr. Bot. Club 119:376-83.

Mellinger, M. 1984. Correspondence analysis in the study of lithogeochemical data: general strategy and the usefulness of various data-coding schemes. J. Chem. Explor. 21:455- 69.

Muller, R. N. 1982. Vegetation patterns in the mixed mesophytic forest of eastern Kentucky. Ecology 63:1901-7.

Oksanen, J. 1988. Impact of habitat, substrate, and microsite classes on the epiphyte vegetation: Interpretation using exploratory and canonical correspondence analysis. Ann. Bot. Fen. 25:59-71.

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