**Principal Components Analysis PCA**

**53**

**10**

**Principal Components Analysis PCA**

The relationship between samples (columns) in terms of their species cannot

normally be visualised because this would require a plot with as many axes as

there are species (rows). If your study only includes 3 species this is possible, but

is quite impossible given 4 or more species. PCA is a technique that may

summarise the relationship between the samples in a small number of axes that

can be plotted. For such a summarisation to work, there must be some degree of

correlation between the species (descriptive variables) so that the effect of a

number of these variables can be combined into a single axis. For good general

introductions to PCA for non mathematicians see Kent & Coker (1992) and

Legendre & Legendre (1983). See the

References

111

section.

From the ordination drop down menu CAP offers a PCA undertaken on either the

correlation or variance covariance matrix between the descriptors (the variables

in the rows normally species in the working matrix). Once either PCA correlation

or PCA covariance is selected a PCA on the working data set is undertaken.

Output from a PCA is presented under a number of tabbed components that can

each be viewed by clicking on the tab. These are described in turn below:

Variance PCA

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Scores PCA

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Eigenvectors PCA

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Cross products

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PCA plot

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**10.1**

**Variance**

This form presents in the first column the eigenvalues of the dispersion

(correlation or variance covariance) matrix arranged from largest to smallest. In

the second column the cumulative total of the eigenvalues is given. The third

column, labelled % of total variance, gives the cumulative total of the eigenvalues

presented as a percentage of the total sum of the eigenvalues. This gives the

total variance in the dispersal matrix represented by the cumulative total

magnitude of the eigenvalues. If the relationship between the samples (columns)

is to be usefully represented by a small number of axes then the first 3 or 4

eigenvalues should represent a large proportion of the total variance. The amount

of the total % variance represented can be seen in the fourth column.

**10.2**

**Scores**

This table gives the co ordinates of the different samples (columns of the working

data) along each of the axes. These scores are displayed graphically by clicking

on the PCA plot tab.

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