Minimum spanning ordination — a graphic-analytical technique for three-dimensional ordination display

Problems of display and interpretation often associated with ordination techniques are briefly discussed. Minimum spanning ordination (MSO) is put forward as offering a meaningful compromise between the limitations of two-dimensional representation and three-dimensional complexity. The method incorporates the use of a minimum spanning tree together with a graphic spherical perspective of points in the third dimension. It is suggested that where this approach is used with reduction analysis, i.e. with centroids of representative clusters, interpretation is much improved and inherent distortions likely to result from techniques such as principal component analysis are more readily exposed.     

Comparisons of three ordination techniques

Summary Simulated coenoclines were used to test performance of several techniques for ordinating samples by species composition: Wisconsin polar or Bray-Curtis ordination with Euclidean distance (ED) and the complements of percentage similarity (PD) and coefficient of community (CD) as distance measures, Principal components analysis, and polar and non-polar or indirect use of Discriminant function analysis. In general the Bray-Curtis technique gave the best ordinations, and PD was the best distance measure. Euclidean distance gave greater distortion than PD in all tests; CD may be better than PD only for some sample sets of high alpha and beta diversity and high levels of noise or sample error. Principal components ordinations are increasingly distored as beta diversity increases, and are highly vulnerable to effects of both noise and sample clustering. Discriminant function analysis was found generally unsuitable for ordination of samples by species composition, but likely to be useful for sample classification.     

An algorithm for predictive ordination

Summary An algorithm is described for predictive ordination. The functional form of species response is required, but it need not be Gaussian. Any integrable function is acceptable.The project has been supported by N.S.E.R.C. of Canada funds.     

Non-linear ordination in several dimensions

A method is described for fitting a model in which site characteristics are represented by a set of orthogonal axes, while the probability that a particular species will be present, and its quantity if it is present, are each related to the axes of the system by symmetrical hypersurfaces with an optimum, decreasing asymptotically to zero as conditions depart from the optimum. The method needs initial estimates of the positions of the sample sites within the axis system. Given these starting points, they can be progressively improved by an iterative procedure. Results are reported of an extensive series of tests using artificial data, and of an analysis of field data from brigalow woodland in Queensland.     

On information flow in ordination

Summary The path of information flow is described and major sources of information loss are identified in ordinations. These sources are shown to be related to the resemblance function and to the transformations which produce the ordiantion co-ordinates. Preventive steps to minimize information loss are suggested. The results which I present in this paper are outcomes of a research project supported by a National Research Council of Canada grant.     

Canonical Analysis

Several fundamental properties of canonical variates are developed. In addition, the equivalence of the redundancy coefficient and the composite coefficient of determination is demonstrated; the latter is used in evaluating the prediction of input variables by canonical variates.     

The development of numerical classification and ordination

Summary The paper reviews the constraints and influences which have affected the development of numerical classification and ordination of vegetation.Initial development of ordination techniques and their reception by ecologists was hindered by the mistaken idea that ordination involved acceptance of variation in vegetation as a continuum, as well as by a general suspicion of mathematical approaches.Three distinct approaches to ordination, largely unrecognised at the time, are apparent in earlier work: direct gradient analysis, reduction in dimensionality and path-seeking (catenation) (Dale 1975).Modifications of simple initial techniques made them more efficient at the cost of increased computation. Acceptance of heavier computation as computers increased in capacity and speed turned attention to prineipal component analysis and the superficially similar factor analysis. These have been widely misunderstood largely because they were initially applied in the same way as in the analysis of psychological data, in which different constraints and objectives apply. The initial failure to recognise that principal component analysis involves a preliminary data transformation, the form of which depends on answers to biological, not mathematical, questions, was particularly unfortunate.Principal component analysis has limitations as a technique of ordination resulting from its assumptions of linearity and additivity of plant responses. Attempts to devise more effective techniques raise questions about the practical importance of non-linearity if the objective is data-exploration rather than elucidating the nature of species-response curves and about the adequacy of using simulated data as test data when we do not know how to simulate realistic data.Data-exploration has been more prominent in practical uses of ordination but many methodological developments have concentrated rather on species-response curves.Numerical classification also met obstacles to its acceptance additional to a general aversion to numerical techniques. The first numerical techniques were presented in the context of the relationships of a particular set of data, rather than of a generally valid system, which was the more familiar concept in non-numerical classification.Both numerical and non-numerical classification aim to produce as homogeneous groups as possible. The distinctive contribution of numerical methods is to allow the data to indicate the most efficient criteria of classification; this was an unfamiliar idea.The strategy of classification may be either divisive or agglomerative and either monothetic or polythetic. Choice of strategy in earlier work was not only constrained by computational limitation but may also have been influenced by an author's previous experience of non-numerical classification. As with ordination, the distinction between preliminary data transformation and subsequent analysis was at first not appreciated.Numerical classification has been influenced by parallel numerical developments in formal taxonomy. Because objectives and assumptions are not always the same, this influence has not been altogether helpful.The limitations of real data suggest that developments of technique are at risk of becoming too concerned with refinements of methodology. Increasingly complex methods and increasing availability of programmes for such methods carry the risk that they may be used without adequate understanding of what they do.     

On the use of ordination models in phytosociology

Summary Q-type ordination methods are emphasized in general phytosociological studies. A new modification of the simple ordination method is introduced as principal axes ordination. Its relation to the theoretically preferred principal components analysis is discussed. A new simple similarity coefficient based on both similarity and disimilarity components is proposed.Four examples of application are given: ordination of structural characters, of stands, of successional relationships and of vegetation types.The concept of isocene is introduced; examples of isocene patterns in ordination models are given.The prospects of synecological and synsystematical interpretation of phytosociological ordination models for the Zürich-Montpellier school are discussed.

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Zusammenfassung Ordinationsmethoden vom Q-Typus werden empfohlen in allgemeinen phytosoziologischen Untersuchungen. Eine Modifikation der einfachen Ordinationsmethode wird eingeführt als Prinzipale Achsen Ordination. Ihre Beziehungen mit der theoretisch vorzuziehenden Prinzipalen Komponenten Analyse werden diskutiert. Ein neuer Similaritätskoeffizient, basiert auf Similarität und Dissimilarität, wird vorgeschlagen.Vier Beispiele von Anwendungen werden gegeben: Ordination von Strukturmerkmalen, von Aufnahmen, von Sukzessionsbeziehungen und von Vegetationstypen.Der Isozön-Begriff wird introduziert. Beispiele von Isozön-Karten werden gegeben.Die Perspektiven der synökologischen und synsystematischen Interpretation von phytosoziologischen Ordinationsmodellen für die Zürich-Montpellier Schule werden diskutiert.

     

A paired-quadrat method for use in multiscale ordination

Multiscale ordination is a technique for examining spatial patterns of several species at several scales. We present a paired-quadrat method (paired quadrat covariance; PQC) to be used in multiscale ordination and test it with artificial data. Multiscale ordination with PQC successfully extracted the salient features of the data set. The method appears to be more sensitive than blocked-quadrat techniques for extracting small-scale patterns. We suggest that PQC will be useful as a complement to existing procedures or as a tool for analysing data from scattered quadrat arrangements.Abbreviations PQC = Paired Quadrat Covariance - PQV = Paired Quadrat Variance - TTLC = Two-Term Local Covariance - TTLQV = Two-Term Local Quadrat Variance     

Detrended correspondence analysis: An improved ordination technique

Summary Detrended correspondence analysis (DCA) is an improvement upon the reciprocal averaging (RA) ordination technique. RA has two main faults: the second axis is often an arch or horseshoe distortion of the first axis, and distances in the ordination space do not have a consistent meaning in terms of compositional change (in particular, distances at the ends of the first RA axis are compressed relative to the middle). DCA corrects these two faults. Tests with simulated and field data show DCA superior to RA and to nonmetric multidimensional sealing in giving clear, interpretable results. DCA has several advantages. (a) Its performance is the best of the ordination techniques tested, and both species and sample ordinations are produced simultaneously. (b) The axes are scaled in standard deviation units with a definite meaning, (c) As implemented in a FORTRAN program called DECORANA, computing time rises only linearly with the amount of data analyzed, and only positive entries in the data matrix are stored in memory, so very large data sets present no difficulty. However, DCA has limitations, making it best to remove extreme outliers and discontinuities prior to analysis. DCA consistently gives the most interpretable ordination results, but as always the interpretation of results remains a matter of ecological insight and is improved by field experience and by integration of supplementary environmental data for the vegetation sample sites.This research was supported by the Institute of Terrestrial Ecology, Bangor, Wales, and by a grant from the National Science Foundation to R.H. Whittaker. We thank R.H. Whittaker for encouragement and comments, S.B. Singer for assistance with the Cornell computer, and H.J.B. Birks, S.R. Sabo, T.C.E. Wells, and R.H. Whittaker for data sets used for ordination tests.     

Semi-strong Hybrid Scaling,a new ordination algorithm

There is a small group of association measures that appear optimal for comparing sites on the basis of their species composition. These measures can accurately estimate affinity between sites when they are ecologically similar. Once sites share few or no species, these measures always under-estimate the ‘ecological distance’ between them. A new ordination algorithm called Semi-strong Hybrid Scaling, SHS, uses these features in an attempt to provide a better configuration of the sites. The new method is evaluated by a direct comparison of the structure in simulated data with the Hybrid method of Faith, Minchin & Belbin (1987). To evaluate SHS and compare it with Hybrid Scaling, 3240 datasets were generated using the COMPAS simulator (Minchin 1987). The data were designed to simulate as closely as possible, what is known of the distribution of species on environmental gradients. The factors included the dimensionality of the data, the number of sites and species, the shape of the species response surfaces, positioning of the sites in the simulation space, carrying capacity and level of noise. Recovery of the simulated site positions by SHS and Hybrid Scaling was evaluated using Procrustes rotation. SHS produced a better recovery in 88% of the datasets.     

Effects of sample distribution along gradients on eigenvector ordination

In general, disproportionately heavy sampling of the ends of a gradient increases the interpretability of eigenvector ordinations. More specifically, correspondence analysis (CA) and detrended correspondence analysis (DCA) best reproduce the original positions of samples in simulated coenoclines when samples are clustered toward the ends of the axis. Principal components analysis (PCA) reproduces the original sample positions less well than either CA or DCA and shows no improvement as samples are increasingly clustered toward the ends of the axis. PCA and CA show less curvature of one dimensional data into the second axis when sampling favors the ends of the axis.     

Data manipulation and gradient length estimation in DCA ordination

Abstract. The effects of different kinds of data manipulation on gradient length estimation by non-linear rescal-ing (as in DCA ordination) are evaluated by considering the first axis inDCA ordinations of 11 field data sets from four investigations. Gradient length estimates are dependent on the range of the abundance scale; the more the scale favours the quantitative aspect (abundance) of the data over the qualitative aspect (presence), the longer the DCA axes. The gradient length estimate decreases when infrequent species are deleted. A new formula is proposed to replace the option for downweighting of rare species in DCA, as the option presently available has some undesirable properties. Some implications for interpretation of gradient length estimates by non-linear rescaling in general (and in DCA in particular) and for comparison of gradient length estimates between studies, are discussed. The potential of non-linear rescaling of gradients for estimation of β diversity is emphasized.