Comparisons presented in this chapter concern 'adaptive' methods aimed at improving on already existing ordinations, mainly correcting the horseshoe. This imposes at least two restrictions on comparison. First, resulting ordinations have to be opposed to the initial configurations: for example, detrended correspondence analysis (DCA) versus ordinary correspondence analysis (CA), nonmetric multidimensional scaling (NMDS) versus principal components analysis (PCA) and flexible shortest-path adjustment (FSPA) versus its underlying principal coordinates analysis (PCOA) ordination. Second, the three methods are flexible by nature, offering options for conducting the process by various means. I am presenting just one among an almost infinite number of solutions, and readers should be aware that running computer programs differently may alter the results.

Another issue is measuring quality; that is, performance of ordinations. Generally performance of an ordination is high when the proportion of variance explained by environmental factors is high. That is what constrained ordination really measures (Section 7.5), but the results primarily express a property of data used; therefore the comparison eventually becomes an evaluation of data rather than of method. Ordinations can always be compared by stress functions, as shown in Section 7.3.1. However, these consider regular patterns and statistical noise simultaneously, whereas in practice priority is usually given to revealing striking patterns. That is what I do below, taking a small data set ('Schlaenggli') exhibiting an obvious pattern (a gradient) and revealing another - a group pattern - with the aim of displaying both for visual inspection. Using minimum-variance clustering analysis the number of releve groups is set to three to facilitate distinction of symbols.

The first example compares CA and DCA (Figure 5.10). In DCA 26 segments are used and 4 iteration cycles applied. Only releve data points are displayed. Both ordinations nicely resolve the group pattern and DCA succeeds in stretching the horseshoe. Hence, while maintaining the order along the x-axis, the y-axis is compressed by DCA.

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H' Group no. 1 • Group no. 2 ,••, Group no. 3

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Figure 5.10 Comparison of CA (Left) and DCA (right). Data point groups are from minimum-variance cluster analysis.

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Figure 5.11 Comparison of PCA (left) and NMDS (right). Data point groups are from minimum-variance cluster analysis.

The second example is about NMDS, with the initial configuration being PCA derived from a correlation matrix. The NMDS ordination is optimized for two dimensions (Figure 5.11). The strong horseshoe from PCA ordination is still visible, as is the group structure. While NMDS does not alter the ordination too much, it must be noted that replacing the correlation coefficient by the frequently used Bray-Curtis index (see for example Gauch 1982) would change the result considerably.

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Figure 5.12 Comparison of PCOA (left) and flexible shortest-path adjustment (right). Data point groups are from minimum-variance cluster analysis.

The third example compares PCOA ordination with its analogue processed by FSPA. Resemblance is the correlation coefficient (Section 4.4), of which FSPA takes the one-complement, a distance measure. I choose 47.8% of the total of 1953 distances to be recalculated in order to change the pattern remarkably. As can be seen in Figure 5.12, FSPA stretches the horseshoe while retaining the group structure. Because this predominantly affects the first axis, the first Eigenvalue accounts for an astonishing À1 = 53.14% of total variance, and the two dimensions for 62.17%.

It does not come as a surprise that all methods produce usable results. A main issue in all examples is the flexibility with which initial configurations, alternative pathways and the number of iterations can be chosen. This adds uncertainty to the methods not existent in PCA, PCOA and CA. While the need for stretching of a horseshoe is debatable, it becomes clear that the quality of the data used is far more important than the selection of the best ordination method.

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