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FAO/IAEA INTERREGIONAL TRAINING COURSE ON MUTANT
GERMPLASM CHARACTERISATION
MULTIVARIATE ANALYSES
15.9. Principal Component Analysis (PCA)
If a multivariate dataset is represented as a set of coordinates in an n-dimensional data space
(1 axis per variable), PCA can reduce the dimensionality of the transformed data and supply a
lower-dimensional projection when viewed from its most informative viewpoint, using only
the first few principal components. For a seemingly random distribution of data points in the
n-dimensional results space, PCA starts with finding the analytical plane by slicing the results
space into lower dimensional representations of uncorrelated parameters (eigenvectors).
In mathematical terms, PCA is a procedure to transform a set of potentially correlated
observations into a set of uncorrelated data points: principal components (in number less than
or equal to the original variables). This orthogonal transformation is defined in such a way
that the first principal component accounts for as much of the variability in the data as
possible (maximum variance), and each succeeding component in turn has the highest
variance possible under the constraint that it is uncorrelated with (orthogonal to) the preceding
components. Principal components are guaranteed to be independent only if the data set is
jointly normally distributed.
PCA is the simplest of the true eigenvector-based multivariate analyses. It might be visualized
as uncovering the internal structure of the data in a way which best explains their variance.
Sensitive to the relative scaling of the original variables, it can be done by eigenvalue
decomposition of a data covariance matrix or singular value decomposition of a data matrix,
usually after mean centring the data for each attribute. The results of a PCA are usually
discussed in terms of component scores (the transformed variable values corresponding to a
particular data point) and loadings (the weight by which each standardized original variable is
to be multiplied to get the component score). PCA is closely related to factor analysis; and
some statistical packages deliberately merge the two techniques. True factor analysis makes
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