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Symmetry 2019, 11(1), 47;

Exploring Symmetry of Binary Classification Performance Metrics

Ingeniería del Diseño; Escuela Politécnica Superior. Universidad de Sevilla, 41011 Sevilla, Spain
Tecnología Electrónica; Escuela Ingeniería Informática. Universidad de Sevilla, 41012 Sevilla, Spain
Author to whom correspondence should be addressed.
Received: 14 November 2018 / Revised: 13 December 2018 / Accepted: 24 December 2018 / Published: 3 January 2019
(This article belongs to the Special Issue Symmetry in Computing Theory and Application)


Selecting the proper performance metric constitutes a key issue for most classification problems in the field of machine learning. Although the specialized literature has addressed several topics regarding these metrics, their symmetries have yet to be systematically studied. This research focuses on ten metrics based on a binary confusion matrix and their symmetric behaviour is formally defined under all types of transformations. Through simulated experiments, which cover the full range of datasets and classification results, the symmetric behaviour of these metrics is explored by exposing them to hundreds of simple or combined symmetric transformations. Cross-symmetries among the metrics and statistical symmetries are also explored. The results obtained show that, in all cases, three and only three types of symmetries arise: labelling inversion (between positive and negative classes); scoring inversion (concerning good and bad classifiers); and the combination of these two inversions. Additionally, certain metrics have been shown to be independent of the imbalance in the dataset and two cross-symmetries have been identified. The results regarding their symmetries reveal a deeper insight into the behaviour of various performance metrics and offer an indicator to properly interpret their values and a guide for their selection for certain specific applications. View Full-Text
Keywords: performance metrics; classification; computational symmetry; machine learning performance metrics; classification; computational symmetry; machine learning

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Luque, A.; Carrasco, A.; Martín, A.; Lama, J.R. Exploring Symmetry of Binary Classification Performance Metrics. Symmetry 2019, 11, 47.

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