  Principal component analysis (PCA) is an important tool in exploring data.
The conventional approach to PCA leads to a solution which favours the
structures with large variances. This is sensitive to outliers and could
obfuscate interesting underlying structures. One of the equivalent definitions
of PCA is that it seeks the subspaces that maximize the sum of squared pairwise
distances between data projections. This definition opens up more flexibility
in the analysis of principal components which is useful in enhancing PCA. In
this paper we introduce scales into PCA by maximizing only the sum of pairwise
distances between projections for pairs of datapoints with distances within a
chosen interval of values [l,u]. The resulting principal component
decompositions in Multiscale PCA depend on point (l,u) on the plane and for
each point we define projectors onto principal components. Cluster analysis of
these projectors reveals the structures in the data at various scales. Each
structure is described by the eigenvectors at the medoid point of the cluster
which represent the structure. We also use the distortion of projections as a
criterion for choosing an appropriate scale especially for data with outliers.
This method was tested on both artificial distribution of data and real data.
For data with multiscale structures, the method was able to reveal the
different structures of the data and also to reduce the effect of outliers in
the principal component analysis.
