  Dimensionality is a major concern in analyzing large data sets. Some well
known dimension reduction methods are for example principal component analysis
(PCA), invariant coordinate selection (ICS), sliced inverse regression (SIR),
sliced average variance estimate (SAVE), principal hessian directions (PHD) and
inverse regression estimator (IRE). However, these methods are usually adequate
of finding only certain types of structures or dependencies within the data.
This calls the need to combine information coming from several different
dimension reduction methods. We propose a generalization of the Crone and
Crosby distance, a weighted distance that allows to combine subspaces of
different dimensions. Some natural choices of weights are considered in detail.
Based on the weighted distance metric we discuss the concept of averages of
subspaces as well to combine various dimension reduction methods. The
performance of the weighted distances and the combining approach is illustrated
via simulations.
