A comparative study of reciprocal averaging and other ordination techniques
TL;DR: Comparison of ordination performance of reciprocal averaging with non-standardized and standardized principal components analysis (PCA) and polar or Bray-Curtis ordination (PO) found that RA is much superior to PCA at high beta diversities and on the whole preferable toPCA at low Beta diversities.
Abstract: SUMMARY Reciprocal averaging is a technique of indirect ordination, related both to weighted averages and to principal components analysis and other eigenvector techniques. A series of tests with simulated community gradients (coenoclines), simulated community patterns (coenoplanes), and sets of vegetation samples was used to compare ordination performance of reciprocal averaging (RA) with non-standardized and standardized principal components analysis (PCA) and polar or Bray-Curtis ordination (PO). Of these, non-standardized PCA is most vulnerable to effects of beta diversity, giving distorted ordinations of sample sets with three or more half-changes. PO and RA give good ordinations to five or more half-changes, and standardized PCA is intermediate. Sample errors affect all these techniques more at low than at high beta diversity, but PCA is most vulnerable to effects of sample errors. All three techniques could ordinate well a small (1-5 x 1-5 half-changes) simulated community pattern; and PO and RA could ordinate larger patterns (4 5 x 4-5 half-changes) well. PCA distorts larger community patterns into complex surfaces. Given a rectangular pattern (1-5 x 4-5 halfchanges), RA distorts the major axis of sample variation into an arch in the second axis of ordination. Clusters of samples tend to distort PCA ordinations in rather unpredictable ways, but they have smaller effects on RA, and none on PO. Outlier samples do not affect PO (unless used as endpoints), but can cause marked deterioration in RA and PCA ordinations. RA and PO are little subject to the involution of axis extremes that affects nonstandardized PCA. Despite the arch effect, RA is much superior to PCA at high beta diversities and on the whole preferable to PCA at low beta diversities. Second and higher axes of PCA and RA may express ecologically meaningless, curvilinear functions of lower axes. When curvilinear displacements are combined with sample error, axis interpretation is difficult. None of the techniques solves all the problems for ordination that result from the curvilinear relationships characteristic of community data. For applied ordination research consideration of sample set properties, careful use of supporting information to evaluate axes, and comparison of results of RA or PCA with PO and direct ordination are suggested.
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TL;DR: 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.
Abstract: Studies by ourselves and others (Swan 1970, Austin & Noy-Meir 1972, Beals 1973, Hill 1973, 1974, Austin 1976a, b, Fasham 1977, Gauch Whittaker & Wentwarth 1977, Noy-Meir & Whittaker 1977, Orloci 1978, Gauch, Whittaker & Singer 1979) have found faults with all ordination techniques currently in use, at least when applied to ecological data specifying the occurrences of species in community samples. These faults certainly do not make existing techniques useless; but they mean that results must be interpreted with caution. Even with the best techniques, the underlying structure of the data is often poorly expressed.
3,628 citations
TL;DR: In this article, the authors evaluated the robustness of quantitative measures of compositional dissimilarity between sites using extensive computer simulations of species' abundance patterns over one and two dimensional configurations of sample sites in ecological space.
Abstract: The robustness of quantitative measures of compositional dissimilarity between sites was evaluated using extensive computer simulations of species’ abundance patterns over one and two dimensional configurations of sample sites in ecological space. Robustness was equated with the strength, over a range of models, of the linear and monotonic (rank-order) relationship between the compositional dissimilarities and the corresponding Euclidean distances between sites measured in the ecological space. The range of models reflected different assumptions about species’ response curve shape, sampling pattern of sites, noise level of the data, species’ interactions, trends in total site abundance, and beta diversity of gradients.
1,530 citations
TL;DR: In this paper, the performance of the BMWP system across 268 sites on 41 rivers providing a wide range of physical and chemical features has been appraised and changes in score and ASPT with respect to season and sampling effort have been examined.
1,059 citations
TL;DR: In this article, the authors demonstrate the use of piecewise regression as a statistical technique to model ecological thresholds and demonstrate the need for a careful study of the likelihood surface when fitting and interpreting the results from piecewise-regression models.
Abstract: We demonstrate the use of piecewise regression as a statistical technique to model ecological thresholds. Recommended procedures for analysis are illustrated with a case study examining the width of edge effects in two understory plant communities. Piece-wise regression models are “broken-stick” models, where two or more lines are joined at unknown points, called “breakpoints.” Breakpoints can be used as estimates of thresholds and are used here to determine the width of edge effects. We compare a sharp-transition model with three models incorporating smooth transitions: the hyperbolic-tangent, bent-hyperbola, and bent-cable models. We also calculate three types of confidence intervals for the breakpoint estimate: an interval based on the computed standard error of the estimate from the fitting procedure, an empirical bootstrap confidence interval, and a confidence interval derived from an inverted F test. We recommend use of the inverted F test confidence interval when sample sizes are large, and cautious use of bootstrapped confidence intervals when sample sizes are smaller. Our analysis demonstrates the need for a careful study of the likelihood surface when fitting and interpreting the results from piecewise-regression models.
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TL;DR: It is shown that nature of unit variation is a naajor problenl in systematies, and that whether this variation is diserete, continuous, or in some other form, there is a need for appliGation of (uantitative and statistical methods.
Abstract: INTRODUCTION A renewed interest in objeetive and quantitative approaehes to the elassifieation of plant communities has led, within the past decade, to an extensive exalllination of systematic theory and technique. This examination, ineluding the work of Sorenson (1948), Motyka et al. (1950), Curtis & McIntosh (1951), Brown & Curtis (1952), Ramensky (1952), Whittaker (1954, 1956), Goodall (1953a, 1954b)? deVries (1953), Guinoehet (1954, 1955), Webb (1954), Eughes (1954) and Poore (1956) has acconlpanied theoretie studies in taxonomy [Fisher (1936), Womble (1951), Clifford & Binet (1954), Gregg (1954)] and in statisties (Isaaeson 1954). It is a Gonclusion of many of these studies that nature of unit variation is a naajor problenl in systematies, and that whether this variation is diserete, continuous, or in some other form, there is a need for appliGation of (uantitative and statistical methods. In eeologic elassifieation, an inereased use of ordinate systellls, sr hiGh has been stimulated by the developnlent of more effieient sampling teehniques and the collection of stand data on a large seale, has prompted the proposal of the term \"ordination\" ( Goodall 1953b ) . Goodall (1954a) has defined ordination as \"an arrangenlent of units in a unior multi-dinlensional order\" as synonylllous with \"Ordnung,\" (Ramensky
9,549 citations
TL;DR: The fundamental hypothesis is that dissimilarities and distances are monotonically related, and a quantitative, intuitively satisfying measure of goodness of fit is defined to this hypothesis.
Abstract: Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.
6,875 citations
TL;DR: The numerical methods required in the approach to multi-dimensional scaling are described and the rationale of this approach has appeared previously.
Abstract: We describe the numerical methods required in our approach to multi-dimensional scaling. The rationale of this approach has appeared previously.
4,561 citations
4,445 citations
TL;DR: In this paper, the authors derived necessary and sufficient conditions for a solution to exist in real Euclidean space for a multivariate multivariate sample of size n as points P1, P2,..., PI in a Euclidian space and discussed the interpretation of the distance A(Pi, Pj) between the ith and jth members of the sample.
Abstract: SUMMARY This paper is concerned with the representation of a multivariate sample of size n as points P1, P2, ..., PI in a Euclidean space. The interpretation of the distance A(Pi, Pj) between the ith andjth members of the sample is discussed for some commonly used types of analysis, including both Q and R techniques. When all the distances between n points are known a method is derived which finds their co-ordinates referred to principal axes. A set of necessary and sufficient conditions for a solution to exist in real Euclidean space is found. Q and R techniques are defined as being dual to one another when they both lead to a set of n points with the same inter-point distances. Pairs of dual techniques are derived. In factor analysis the distances between points whose co-ordinates are the estimated factor scores can be interpreted as D2 with a singular dispersion matrix.
3,746 citations