Abstract
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.
Highlights
Symmetry has played and continues playing, a highly significant role in the way of how humans perceive the world [1]
The symmetric behaviour of the 10 metrics is first determined by means of computing the distance between the baseline and each of the 31 possible transformations, in accordance with Equation (20)
Based on the results obtained in our analysis, it can be stated that the majority of the most commonly used classification performance metrics present some type of symmetry
Summary
Symmetry has played and continues playing, a highly significant role in the way of how humans perceive the world [1]. Symmetry plays a key role as it can be discovered in nature [2,3], society [4] and mathematics [5]. Symmetry provides an intuitive way to attain faster and deeper insights into scientific problems. An increasing interest has arisen in detecting and taking advantage of symmetry in various aspects of theoretical and applied computing [6]. Pattern recognition and machine learning procedures are becoming key aspects of modern science [12] and the hottest topics in the scientific literature on computing [13] Several studies involving symmetry have been published in network technology [7], human interfaces [8], image processing [9], data hiding [10]
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