5.1 Why ordination?

Ordination is a graphical representation of the similarity of sampling units and/or attributes in resemblance space. An example of an ordination in two-dimensional space has been shown in Figure 4.1 (b), where the axes represent two plant species and the data points releves. This graph displays the similarity of the two releves involved, a rather trivial case as it presents the full configuration given in the raw numbers without improving insight into the system. Hence, ordination is a tool for analysing and visualizing complex data sets including a high number of sampling units with many attributes involved.

Recognizing patterns in large multivariate data sets inevitably means operating in a resemblance space of high dimension. When considering four species, for example, the configuration of resemblance is three-dimensional.

Figure 5.1 Three-dimensional representation of similarity relationships (Mueller-Dombois & Ellenberg 1974).

While four dimensions are already difficult to display graphically, in vegetation ecology large data sets often include hundreds of species, requiring hundreds of dimensions to be analysed. The aim of ordination is to reduce this number, to derive a graph that can be plotted or inspected dynamically as a three- or four-dimensional rotating point cloud. This is why ordination has always played a key role in vegetation ecology, and Figure 5.1 is a historical example of this effort, taken from Mueller-Dombois & Ellenberg (1974), illustrating the effort to reveal all the important relationships found in a multi-dimensional configuration. Although the methods have since evolved, the mode of interpretation has remained unchanged.

Most methods for gaining the desired insight into the similarity patterns of multivariate data sets roughly proceed through the steps illustrated in Figure 5.2:

• Centre the attributes in a data matrix to shift the origin of the new coordinate system into the centre of the point cloud.

Figure 5.2 Main functions of PCA. (a) The data table (artificial data) with original scores (species 1 and 2, white background), centred species vectors (species 1 and 2, light grey background) and centred as well as rotated scores (axes 1 and 2, dark grey). (b) Representation of the point configuration in x- and y-space. (c) Species scores as a function of releve order (response functions).

Figure 5.2 Main functions of PCA. (a) The data table (artificial data) with original scores (species 1 and 2, white background), centred species vectors (species 1 and 2, light grey background) and centred as well as rotated scores (axes 1 and 2, dark grey). (b) Representation of the point configuration in x- and y-space. (c) Species scores as a function of releve order (response functions).

• Rotate the point cloud such that the maximum possible variance is found along the first axis.

• Continue rotating the point cloud, while keeping the first axis fixed, and maximize the remaining variance on the second axis.

• Continue this process until all the axes are processed.

• Represent the result graphically by omitting higher dimensions.

In this procedure, conforming to principal component analysis (PCA), the point pattern hidden in raw data is maintained. As will be shown later, there are methods that are changing the pattern. Hence, choosing the proper method and understanding what it does to data is crucial, and offers flexibility in defining the goal of the analysis.

5.2 Principal component analysis (PCA)

Principal component analysis is a basic procedure that operates as described in Section 5.1. First of all, it strictly relies on linear correlation of attributes.

Figure 5.3 Projecting data into ordination space in PCA. The scalar product of the species by releve matrix X' and the species by axes matrix a (the Eigenvectors) yields the axes by releve matrix Y' (the ordination scores).

It operates in the orthogonal Euclidean space and searches for useful projections of point clouds. Because it is based on the concept of variance partitioning and the variance is maximized along the axes, the result it finds is reproducible - even when using different computers and computer programs (e.g. independent of any initial order of data). Orthogonality (i.e. absence of correlation) also means that the variance carried by the axes is additive. Whatever projection of a point is chosen, the variance explained by the graph is equal to the sum of the explanatory power of the axes involved.

Whereas centring the data is a trivial task, finding the best method of rotation is more demanding. As seen in Figure 4.1, metric data can be used directly as coordinates, where data points are sampling units and species are axes. In mathematical terms, generating a new projection is just a transformation of a coordinate system and is achieved by multiplying two matrices. In the case of PCA, this is shown in Figure 5.3, where matrix X contains the original data and X' the centred. This is multiplied by a new square matrix a, according to:

Matrix a holds the Eigenvectors. It is a squared matrix with the number of species by the number of axes as dimensions. X' and a have one dimension in common, the species. The new matrix Y' still has the releves as rows, but the attributes are now the new axes. The matrix of Eigenvectors a is obtained from the original data by Eigenanalysis (Batschelet 1975), yielding the desired properties of the final result - orthogonality of axes - and maximizing variance on first axes. Eigenanalysis is performed on the variance or correlation matrix of the species, and as a result there are as many Eigenvalues as there are species (although some may be zero).

Figure 5.4 Numerical example of PCA. The centred data matrix (left) is multiplied by the matrix of Eigenvectors (centre) to yield the ordination coordinates (right). The variances of the ordination axes are the corresponding Eigenvalues.

The numerical example illustrated in Figure 5.4 is carried over from Figure 5.2. The elements of the Eigenvector matrix are correlation coefficients (by definition) between the original attributes (the species) and the new ordination axes. Element 0.35 signifies that the first species has a positive correlation of r = 0.35 with the first ordination axis. The correlation with the second axis is r = -0.937. The second species correlates with the first axis by r = 0.937 and with the second axis by r = 0.35. Hence, the Eigenvectors are a useful tool for interpreting the final ordination.

In Figure 5.4 the variances of the attributes are also shown. In the original data, species 2 has the highest variance with 10.8. According to the definitions in PCA, the highest variance in the ordination is attributed to the first axis. This is 11.95 and is the first Eigenvalue in the Eigenanalysis. Because variance on any axis is a linear combination of the variance of many species, it generally exceeds the variance of any individual species. However, the total variance remains unchanged as the point pattern as a whole is not affected by PCA, and only its projection is adjusted.

The result of PCA deserves careful interpretation, as illustrated in Figure 5.5, a data set consisting of 63 sampling units (releves) and 119 attributes (species) ('Schlaenggli', see Appendix B). The environmental factors are not analysed in this example (but will be in later sections). The absolute magnitudes of the Eigenvalues depend on the size of the sample and are therefore not useful in the interpretation. The relative proportions are most crucial, as they inform us about the explanatory power of axes. Here, the x-axis explains 20.6% of the variance and the y-axis 8.0%. The ordination shown in Figure 5.5 uses the scores of the first two axes of PCA as coordinates, hence explaining 28.6% of the total variance, as the

Data set Schlaenggti'

(WOdi 1977) Releves Species: Site factors: Eigenvalues 1-3:

Selected Eigenvectors:

63 119 21

Selected Eigenvectors:

Oxycoccus quadripetalus


-0 117

Carex ecliinata


0 167

Arnica montana



Festuca rubra


0 169

Carex pulicsns



Sphagnum recurvum



Viola paluBtris

0 014

0 193

Galium uliginosum



Stachys officinalis



PCA Ordination

Viola palustris

PCA Ordination

V* V_\

♦ . *♦ ♦

♦ .

Carex pul i carls >

Stachys officinalis

Festuca rubra

Carex pul i carls >

Stachys officinalis

Viola palustris

Festuca rubra

Carex echlnata

Sphagnum reçurvum m-

Arnica montana

Oxycoccus quadri petal us


Galium uliginosum

Carex echlnata

Sphagnum reçurvum m-

Arnica montana

Oxycoccus quadri petal us

Figure 5.5 Main results of a PCA using real data. The Eigenvectors are used to help in the interpretation of the ordination by pointing in the direction of the centres of species occurrences.

variances are strictly additive. A three-dimensional plot would explain another 6% - a total of 34.6%.

Is 28.6% explained variance good or poor for a two-dimensional ordination? Peres-Neto et al. (2005) have written a review on papers discussing the issue of 'nontrivial axes'. The authors suggest a randomization test to identify the number of relevant axes. It may be safe, however, to screen for patterns beyond this number. The proportion of explained variance depends on the type of data analysed and of course on the number of axes considered for viewing. The total dimensionality of the data set is 63 (although there are 119 species involved, 63 data points can be presented in a maximum of 63 dimension without loss of information. A detailed inspection of the Eigenvalues would show that all beyond 63 are zero!). For this size of sample, experience suggests that 28.6% usually reveals the dominating pattern, which in this case is a classical horseshoe, indicating that a (nonlinear) gradient exists. However, it is good practice to inspect the third and the fourth dimension as well. From many more examples it can be infered that data sets of several hundreds of releves usually result in a first Eigenvalue explaining around 10% of the total variance or even less (see Chapter 11 for examples). As a rule the explanatory power of the first axis will decrease as the sample size is increasing.

The interpretation of the point cloud is simple as it displays the similarity space, with the only complication arising from the high dimensionality. If any two data points are in close neighbourhood then they are similar. However, they may still be distant in the third dimension, which is not visible. Even in simple cases it is suggested that a computer program which displays three-dimensional point clouds be used, either as a stereogram or as a spinning graph.

For proper interpretation of PCA results the Eigenvectors (also known as component coefficients) have to be considered as well (Figure 5.5). As explained above, they are the correlations of the species with the ordination axes. Due to the very high dimensionality of the resemblance space, most correlations are rather low, with none even reaching r = 0.2. Geometrically, correlation coefficients are cosines of vectors (see Figure 4.3). Therefore, they can be used for drawing species vectors (Figure 5.5, lower-right graph). Their scaling differs entirely from the ordination diagram, but the graph of vectors can be enlarged or reduced to fit into ordination by superimposing the origins of the two diagrams (lower-left graph). The species arrows now point in the directions of their centres of occurrence. As done here, selecting just a few species for display will avoid a proliferation of information in the graph.

The detailed interpretation of Figure 5.5 proceeds as follows: the releves (i.e data points) in the lower-right quadrant of the ordination are characterized by high values of Oxycoccus quadripetalus. In the releves in the lower-left quadrant Stachys officinalis and Galium uliginosum occur frequently. The horseshoe-shaped point cloud reveals a gradient from the lower left (with high soil pH values, not shown here) towards the lower right (with low pH values). Assessing the relationship to pH, however, requires other methods such as constrained ordination (Section 7.5).

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