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\begin{document}

\title{Which Verification Metrics Are Appropriate for Rare-Event Classification
Problems?}



\author[1]{Azim Ahmadzadeh}%
\author[1]{Dustin Kempton}%
\author[1]{Petrus Martens}%
\author[1]{Rafal Angryk}%
\affil[1]{Georgia State University}%


\vspace{-1em}



  
  \date{June 12, 2024}


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\begin{abstract}
Strong solar flares are indeed rare events, which make the flare
classification task a rare-event problem. Solar energetic particle
events are even rarer space weather events as only a few instances of
them are recorded each year. With the unprecedented growth in employment
of Machine Learning algorithms for rare-event classification/forecast
problems, a proper evaluation of rare-event models becomes a necessary
skill for domain experts to have. This task remains to be an outstanding
challenge as both the learning process and the metrics used for
quantitative verification of models can easily obscure or skew the true
performance of models and yield misleading and biased results. To help
mitigate this effect we introduce a bounded semimetric space that
provides a generic representation for any deterministic performance
verification metric. This space, named Contingency Space, can be easily
visualized and shed light on models' performance as well as on the
metrics' distinct behavior. An arbitrary model's performance can be
mapped to a unique point in this space, which allows comparison of
multiple models at the same time, for a given metric. Using this
geometrical setting we show the difference between a metric's
interpretation of performance and the true performance of the model.
Using this perspective, models which are seemingly different but
practically identical, or only marginally different, can be easily
spotted. By tracking down a learner's performance at each epoch, we can
also compare different learners' learning paths, which provides a deeper
understanding of the utilized algorithms and their challenge in the
learning process. Moreover, in the Contingency Space, a given
verification metric can be represented by a geometrical surface, which
allows a visual comparison between different metrics---a task that
without this concept could be done only by the tedious algebraic
comparison of metrics' formulae. Moreover, using such a surface, for the
first time we can see and quantify the impact of scarcity of data
(intrinsic to rare-even problems) on different metrics. This extra
knowledge provides us with the information we need to choose an
appropriate metric for evaluation of our rare-event models.%
\end{abstract}%



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