  We combine the metrics of distance and isolation to develop the Analytic
Isolation and Distance-based Anomaly (AIDA) detection algorithm. AIDA is the
first distance-based method that does not rely on the concept of
nearest-neighbours, making it a parameter-free model.
  Differently from the prevailing literature, in which the isolation metric is
always computed via simulations, we show that AIDA admits an analytical
expression for the outlier score, providing new insights into the isolation
metric. Additionally, we present an anomaly explanation method based on AIDA,
the Tempered Isolation-based eXplanation (TIX) algorithm, which finds the most
relevant outlier features even in data sets with hundreds of dimensions. We
test both algorithms on synthetic and empirical data: we show that AIDA is
competitive when compared to other state-of-the-art methods, and it is superior
in finding outliers hidden in multidimensional feature subspaces. Finally, we
illustrate how the TIX algorithm is able to find outliers in multidimensional
feature subspaces, and use these explanations to analyze common benchmarks used
in anomaly detection.
