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.

Recognition of anomalous events is a challenging but critical task in many scientific and industrial fields, especially when the properties of anomalies are unknown. In this paper, we present a new anomaly concept called "unicorn" or unique event and present a new, model-independent, unsupervised detection algorithm to detect unicorns. The Temporal Outlier Factor (TOF) is introduced to measure the uniqueness of events in continuous data sets from dynamic systems. The concept of unique events differs significantly from traditional outliers in many aspects: while repetitive outliers are no longer unique events, a unique event is not necessarily outlier in either pointwise or collective sense; it does not necessarily fall out from the distribution of normal activity. The performance of our algorithm was examined in recognizing unique events on different types of simulated data sets with anomalies and it was compared with the standard Local Outlier Factor (LOF). TOF had superior performance compared to LOF even in recognizing traditional outliers and it also recognized unique events that LOF did not. Benefits of the unicorn concept and the new detection method were illustrated by example data sets from very different scientific fields. Our algorithm successfully recognized unique events in those cases where they were already known such as the gravitational waves of a black hole merger on LIGO detector data and the signs of respiratory failure on ECG data series. Furthermore, unique events were found on the LIBOR data set of the last 30 years.