Section 9.1 introduces the basic ideas and technical elements
behind principal
components. No particular assumption will be made on except that
the mean vector and the covariance matrix exist. When reference is made to
a data matrix
in Section 9.2,
the empirical mean and covariance matrix
will be used. Section 9.3 shows how to interpret
the principal components
by studying their correlations with the original components of
. Often
analyses are performed in practice by looking at two-dimensional
scatterplots.
Section 9.4 develops inference techniques on principal components.
This is particularly
helpful in establishing the appropriate dimension reduction and
thus in determining the quality of the resulting
lower dimensional representations.
Since principal component analysis is performed on covariance matrices,
it is not scale
invariant. Often, the measurement units of the components of
are quite
different, so it is reasonable to standardize the measurement units.
The normalized version of principal components is defined
in Section 9.5.
In Section 9.6 it is discovered that the empirical principal
components are the factors of appropriate transformations of the data matrix.
The classical way of defining principal components through linear
combinations with respect to the largest variance is described
here in geometric terms, i.e., in
terms of the optimal fit within subspaces generated by the columns and/or
the rows of
as was discussed in Chapter 8.
Section 9.9 concludes with additional examples.