  Data dimensionality informs us about data complexity and sets limit on the
structure of successful signal processing pipelines. In this work we revisit
and improve the manifold-adaptive Farahmand-Szepesv\'ari-Audibert (FSA)
dimension estimator, making it one of the best nearest neighbor-based dimension
estimators available. We compute the probability density function of local FSA
estimates, if the local manifold density is uniform. Based on the probability
density function, we propose to use the median of local estimates as a basic
global measure of intrinsic dimensionality, and we demonstrate the advantages
of this asymptotically unbiased estimator over the previously proposed
statistics: the mode and the mean. Additionally, from the probability density
function, we derive the maximum likelihood formula for global intrinsic
dimensionality, if i.i.d. holds. We tackle edge and finite-sample effects with
an exponential correction formula, calibrated on hypercube datasets. We compare
the performance of the corrected-median-FSA estimator with kNN estimators:
maximum likelihood (ML, Levina-Bickel) and two implementations of DANCo (R and
matlab). We show that corrected-median-FSA estimator beats the ML estimator and
it is on equal footing with DANCo for standard synthetic benchmarks according
to mean percentage error and error rate metrics. With the median-FSA algorithm,
we reveal diverse changes in the neural dynamics while resting state and during
epileptic seizures. We identify brain areas with lower-dimensional dynamics
that are possible causal sources and candidates for being seizure onset zones.
