  Principal component analysis is a useful dimension reduction and data
visualization method. However, in high dimension, low sample size asymptotic
contexts, where the sample size is fixed and the dimension goes to infinity,a
paradox has arisen. In particular, despite the useful real data insights
commonly obtained from principal component score visualization, these scores
are not consistent even when the sample eigen-vectors are consistent. This
paradox is resolved by asymptotic study of the ratio between the sample and
population principal component scores. In particular, it is seen that this
proportion converges to a non-degenerate random variable. The realization is
the same for each data point, i.e. there is a common random rescaling, which
appears for each eigen-direction. This then gives inconsistent axis labels for
the standard scores plot, yet the relative positions of the points (typically
the main visual content) are consistent. This paradox disappears when the
sample size goes to infinity.
