  An extensible statistical framework for detecting anomalous time series
including those with heavy-tailed distributions and non-stationarity in
higher-order moments is introduced based on penalized likelihood distributional
regression. Specifically, generalized additive models for location, scale, and
shape are used to infer sample path representations defined by a parametric
distribution with parameters comprised of basis functions. Akaike weights are
then applied to each model and time series, yielding a probability measure that
can be effectively used to classify and rank anomalous time series. A
mathematical exposition is also given to justify the proposed Akaike weight
scoring under a suitable model embedding as a way to asymptotically identify
anomalous time series. Studies evaluating the methodology on both multiple
simulations and real-world datasets also confirm that high accuracy can be
obtained detecting many different and complex types of shape anomalies. Both
code implementing GAWS for running on a local machine and the datasets
referenced in this paper are available online.
