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
53
Scores PCA
53
Eigenvectors PCA
54
Cross products
54
PCA plot
55
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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