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Unconstrained ordination

Correspondence Analysis (CA) & Detrended Correspondence Analysis (DCA)


Unimodal method of unconstrained ordination. In the multidimensional space of all ordination axes it preserves chi-square distances among samples, which is sometimes considered as one of the least suitable distance measures for ecological data, since it gives high weight to rare species (species with low occurrence, producing many zeros in the dataset). Suffers from artefact called arch effect, which is caused by non-linear correlation between first and higher axes. Popular, even though clumsy way how to remove this artefact is to use detrending in DCA.

In CA, both objects and species are represented by points in the ordination diagram (compare to PCA, where species/descriptors are vectors and sites are points). Similarly to PCA, two types of scaling are available (Borcard et al. 2011):

  • Scaling 1 - the distances among objects (sites) in the reduced ordination space approximate their chi-square distance; any object found near the point representing a species is likely to contain a high contribution of that species.
  • Scaling 2 - the distances among species in the reduced space approximate their chi-square distances; any species that lies close to the point representing an object is more likely to be found in that object or to have higher frequency there.

Detrended version of Correspondence Analysis, removing the arch effect from ordination. The method was (and still is) very popular, because it gives rather nice ordination diagrams, and it has one interesting property: the length of the first axis (in SD units) refers to the heterogeneity or homogeneity of the dataset (a sort of beta diversity measure). As a rule of thumb, Lepš & Šmilauer (2003) and ter Braak & Šmilauer (2002) recommends to use this length for decision between the use of linear (axis shorter than 3 SD) or unimodal (axis longer than 4 SD) ordination methods 1). However, detrending (by segments) resembles using hammer on data - arch is hammered by cutting the first axis into segments and moving the sample points up down along the second axis (you may see rescaling from CA to DCA here). For this reason, the method is sometimes criticized and not recommended for use (see e.g. Legendre & Legendre 1998, Borcard et al. 2011, or Jari Oksanen2)). Despite this criticism, DCA is still one of the most widely used unconstrained ordination methods, at least among vegetation ecologists (seems like zoologists are biased towards NMDS...).

This rule is based on empirical experience and it is provided for orientation. If axis length falls between 3 and 4, both methods should perform OK.
Jari Oksanen, in the manual to decorana function, provides the following arguments: In late 1970s, correspondence analysis became the method of choice for ordination in vegetation science, since it seemed better able to cope with non-linear species responses than principal components analysis. However, even correspondence analysis can produce an arc-shaped configuration of a single gradient. Mark Hill developed detrended correspondence analysis to correct two assumed ‘faults’ in correspondence analysis: curvature of straight gradients and packing of sites at the ends of the gradient. The curvature is removed by replacing the orthogonalization of axes with detrending. In orthogonalization successive axes are made non-correlated, but detrending should remove all systematic dependence between axes. Detrending is performed using a five-segment smoothing window with weights (1,2,3,2,1) on mk segments — which indeed is more robust than the suggested alternative of detrending by polynomials. The packing of sites at the ends of the gradient is undone by rescaling the axes after extraction. After rescaling, the axis is supposed to be scaled by ‘SD’ units, so that the average width of Gaussian species responses is supposed to be one over whole axis. Other innovations were the piecewise linear transformation of species abundances and downweighting of rare species which were regarded to have an unduly high influence on ordination axes. It seems that detrending actually works by twisting the ordination space, so that the results look non-curved in two-dimensional projections (‘lolly paper effect’). As a result, the points usually have an easily recognized triangular or diamond shaped pattern, obviously an artefact of detrending. Rescaling works differently than commonly presented, too. decorana does not use, or even evaluate, the widths of species responses. Instead, it tries to equalize the weighted variance of species scores on axis segments (parameter mk has only a small effect, since decorana finds the segment number from the current estimate of axis length). This equalizes response widths only for the idealized species packing model, where all species initially have unit width responses and equally spaced modes.
en/ca_dca.txt · Last modified: 2017/10/11 07:36 (external edit)