  This work proposes and investigates a novel method for anomaly detection and
shows it to be competitive in a variety of Euclidean and non-Euclidean
situations. It is based on an extension of the depth quantile functions (DQF)
approach. The DQF approach encodes geometric information about a point cloud
via functions of a single variable, whereas each observation in a data set is
associated with a single such function. Plotting these functions provides a
very beneficial visualization aspect. This technique can be applied to any data
lying in a Hilbert space.
  The proposed anomaly detection approach is motivated by the geometric insight
of the presence of anomalies in data being tied to the existence of antimodes
in the data generating distribution. Coupling this insight with novel
theoretical understanding into the shape of the DQFs gives rise to the proposed
adaptive DQF (aDQF) methodology. Applications to various data sets illustrate
the DQF and aDQF's strong anomaly detection performance, and the benefits of
its visualization aspects.
