  Statistical data by their very nature are indeterminate in the sense that if
one repeated the process of collecting the data the new data set would be
somewhat different from the original. Therefore, a statistical method, a map
$\Phi$ taking a data set $x$ to a point in some space F, should be stable at
$x$: Small perturbations in $x$ should result in a small change in $\Phi(x)$.
Otherwise, $\Phi$ is useless at $x$ or -- and this is important -- near $x$. So
one doesn't want $\Phi$ to have "singularities," data sets $x$ such that the
the limit of $\Phi(y)$ as $y$ approaches $x$ doesn't exist. (Yes, the same
issue arises elsewhere in applied math.)
  However, broad classes of statistical methods have topological obstructions
of continuity: They must have singularities. We show why and give lower bounds
on the Hausdorff dimension, even Hausdorff measure, of the set of singularities
of such data maps. There seem to be numerous examples.
  We apply mainly topological methods to study the (topological) singularities
of functions defined (on dense subsets of) "data spaces" and taking values in
spaces with nontrivial homology. At least in this book, data spaces are usually
compact manifolds. The purpose is to gain insight into the numerical
conditioning of statistical description, data summarization, and inference and
learning methods. We prove general results that can often be used to bound
below the dimension of the singular set. We apply our topological results to
develop lower bounds on Hausdorff measure of the singular set. We apply these
methods to the study of plane fitting and measuring location of data on
spheres.
  This is not a "final" version, merely another attempt.
