Principal coordinates analysis PCOA

This method is similar to PCA, but since it accepts almost any kind of similarity or distance measure, it is of broad practical use. The method, first published by Gower (1966), is not only known as principal coordinates analysis but also as principal axis analysis and metric multidimensional scaling. In PCOA, when the releves are ordinated, the Eigenvalues and Eigenvectors are derived from the similarity matrix of the releves. This differs from PCA, where the species-similarity matrix is used to derive the coordinates of the releves. As a minor disadvantage, there are no Eigenvectors available to help in the interpretation of the ordination.

The position of the data points is initially defined by a distance or similarity matrix, which may be either metric or nonmetric. If distances are given then the elements dij are transformed according to:

The elements of S are interpreted as direction cosine and have to be adjusted for range. The new matrix A has the elements:

where s, and are the row and column means of S and s is the same of the grand total. Then the Eigenvalues, ,..., xn, and the corresponding Eigenvectors, ..., fin, of A are found. The Eigenvectors are adjusted to the Eigenvalues to satisfy the condition:

These are now the ordination coordinates. The question arises whether and how they deviate from PCA coordinates of the same data set. This is shown in an example using the data set previously presented in Figure 5.5. The cover-abundance scores are first changed into a rank scale and then scalar is transformed according to x' = x0 5. For PCA, the correlation matrix of the species is computed; for PCOA it is the matrix of the releves. The resulting Eigenvalues are as follows:

PCA %

PCOA %

PCA, variance

PCOA, variance

A.1

20.62

27.96

24.5

12.2

8.07

9.43

9.61

4.11

6.07

5.99

7.23

2.61

Clearly, the Eigenvalues differ in size and proportion. The resulting ordinations are shown in Figure 5.6. The two point clouds are superimposed after (heuristic) linear adjustment of the scale (scores of PCOA are multiplied by a factor of 5.57). It can be seen that the overall shape of the point cloud, a horseshoe, really is the same. The individual points, however, are slightly displaced. Since PCA reproduces the geometrical configuration of points,

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Figure 5.6 Comparison of PCA and PCOA using the 'SchLaenggLi' data set of 63 relevés. Data points from PCA (crosses) and from PCOA (triangles) superimposed after adjustment of scale.

Figure 5.6 Comparison of PCA and PCOA using the 'SchLaenggLi' data set of 63 relevés. Data points from PCA (crosses) and from PCOA (triangles) superimposed after adjustment of scale.

there has to be a minor (but for ecological interpretations, unimportant) distortion in the ordination of PCOA.

Why this distortion? Depending on the initial resemblance measure used, matrix A (Formula (5.3)) is usually not strictly metric. PCOA will then extract the metric portion from A and the corresponding positive Eigenvalues express the explanatory power of the axes. The remaining nonmetric part appears in the form of negative Eigenvalues. This cannot be displayed in an ordination diagram.

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