Introduction

Ordination is a set of analyses that collectively aim to summarize and present multivariate data, in which multiple dependent or response variables have been recorded, to display differences between samples graphically into fewer dimensions than the original data set. For example, if the abundances of ten species were recorded from a range of different sites then the total variation in site differences could only be graphically represented in a ten-dimensional graph. Obviously, this is not possible to draw on a piece of paper; however, if there were only a few key trends or gradients that the species shared, then it might be possible to derive a smaller set of axes (e.g., two) that could be plotted to summarize most of the variation in the data set. The term 'ordination' reflects the original intent of the approach - to identify single gradients (i.e., ordered responses) of variables (usually species) that might reflect underlying causal ecological processes.

This can be illustrated by a three-species example (Figure 1). If the abundances of only three species were sampled at a range of sites, then the entire variation in the data set could be illustrated in a three-dimensional graph. In this example, each species' abundance has approximately the same variance, so each of the species' axes explains the same amount of variance. The samples occur along an environmental gradient (e.g., wave exposure), indicated by the color of the symbol. There is an obvious trend in the data. As the abundance of species 1 increases, so too do the abundances of species 2 and 3. Ordination aims to reduce the number of dimensions required to display the main patterns, by generating a new set of axes and replotting the sample points on these axes. One way to derive a set of axes might be to find a line that runs through the main trend in the data (Figure 1b), and plot the position of the sample on that line. Note that the order of samples along the gradient has been preserved, and in this case also the approximate distance between different samples. In this example, this single axis explains 82% of the total species variance, in comparison with the 33% that any individual species axis could explain. Of course it is possible to plot multiple axes to enhance interpretation (Figure 1c). In this example, 94% of the total variance can be displayed in two dimensions. The utility of this approach becomes apparent when more than three dependent variables have been recorded in samples.

The need to reduce dimensionality of multivariate data is encountered in a wide range of sciences beside ecology. Multivariate data are encountered in a range of biological, sociological, and physical sciences. Consequently, there is a range of different methods currently in use. The literature detailing approaches and subsequent applications of ordination methods is vast. This article does not aim to provide a comprehensive

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Axis 1 (82% variance)

Axis 1 (82% variance)

Figure 1 Reduced space plots of samples along an exposure gradient. Intensity of fill reflects intensity of exposure. (a) In a three-species data set, all variation in the data can be displayed in three dimensions. The thick lines indicate the main trends in the data. (b) Dimensionality could be reduced to one dimension by projecting samples onto a line that runs through the main trend, while still retaining 82% of the variation in the data. The order and relative position of the samples are preserved. (c) Projecting the data onto two orthogonal axes (or 'trends') enables 94% of the variation to be explained in two dimensions.

review of ordination methods, but will provide an overview of the main groups of methods currently in use in ecology.