  With the recent advent of a sound mathematical theory for extreme events in
dynamical systems, new ways of analyzing a system's inherent properties have
become available: Studying only the probabilities of extremely close
Poincar\'{e} recurrences, we can infer the underlying attractor's local
dimensionality -- a quantity which is closely linked to the predictability of
individual configurations, as well as the information gained from observing
them. This study examines possible ways of estimating local and global
attractor dimensions, identifies potential pitfalls and discusses conceivable
applications. The Portable University Model of the Atmosphere (PUMA) serves a
test subject of intermediate complexity between simple mathematical toys and
truly realistic atmospheric data-sets. It is demonstrated that the introduction
of a simple, analytical estimator can streamline the procedure and allows for
additional tests of the agreement between theoretical expectation and observed
data. We furthermore show how the newly gained knowledge about local dimensions
can complement classical techniques like principal component analysis and may
assist in separating meaningful patterns from mathematical artifacts.
