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BenchMetrics: a systematic benchmarking method for binary classification performance metrics

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Abstract

This paper proposes a systematic benchmarking method called BenchMetrics to analyze and compare the robustness of binary classification performance metrics based on the confusion matrix for a crisp classifier. BenchMetrics, introducing new concepts such as meta-metrics (metrics about metrics) and metric space, has been tested on fifteen well-known metrics including balanced accuracy, normalized mutual information, Cohen’s Kappa, and Matthews correlation coefficient (MCC), along with two recently proposed metrics, optimized precision and index of balanced accuracy in the literature. The method formally presents a pseudo-universal metric space where all the permutations of confusion matrix elements yielding the same sample size are calculated. It evaluates the metrics and metric spaces in a two-staged benchmark based on our proposed eighteen new criteria and finally ranks the metrics by aggregating the criteria results. The mathematical evaluation stage analyzes metrics’ equations, specific confusion matrix variations, and corresponding metric spaces. The second stage, including seven novel meta-metrics, evaluates the robustness aspects of metric spaces. We interpreted each benchmarking result and comparatively assessed the effectiveness of BenchMetrics with the limited comparison studies in the literature. The results of BenchMetrics have demonstrated that widely used metrics have significant robustness issues, and MCC is the most robust and recommended metric for binary classification performance evaluation.

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Availability of data and material

The authors confirm that all data and materials support their published claims and comply with field standards (see Appendix A).

Code availability

The authors confirm that all software application or custom code support their published claims and comply with field standards (see Appendix A).

Notes

  1. Note that ‘performance metrics’ that are in [0, 1] or [− 1, 1] directly represents the success of a classifier (e.g., Accuracy or True Positive Rate). Those metrics are the instruments published in the literature to report, evaluate, and compare classifiers. Whereas, ‘performance measures’ that are usually not published represent other aspects such as dataset or classifier’s output characteristics (e.g., PREV is the ratio of positive examples in a dataset and BIAS is the ratio of positive outcomes of a classifier). Some instruments indicating the performance in an unbounded interval [0, ∞) or (− ∞, ∞) are also ‘measures’ that are not applicable to publish and compare classification performances in the literature (e.g., Odds Ratio or Discriminant Power) because of limitations in interpretability.

  2. Sample sizes (permutations/metric space sizes): Sn = 25 (3,276); Sn = 50 (23,426); Sn = 75 (76,076); Sn = 100 (176,851); Sn = 125 (341,376); Sn = 150 (585,276); Sn = 175 (924,176); Sn = 200 (1,373,701); Sn = 250 (2,667,126).

  3. For ten negative samples (e.g., i = 1, …, 10): ci = 0 and example pi = 0.49 then | ci – pi |= 0.49. For remaining ten positive samples (e.g., i = 11, …, 20): ci = 1 and example pi = 0.51 then | ci – pi |= 0.49. Hence, MAE = 0.49.

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The authors did not receive support from any organization for the submitted work.

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Authors and Affiliations

  1. ASELSAN, Ankara, Turkey

    Gürol Canbek

  2. Informatics Institute, Middle East Technical University, Ankara, Turkey

    Gürol Canbek & Tugba Taskaya Temizel

  3. Computer Engineering Department, Gazi University, Ankara, Turkey

    Seref Sagiroglu

Authors
  1. Gürol Canbek
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  2. Tugba Taskaya Temizel
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  3. Seref Sagiroglu
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Contributions

GC: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing—original draft, Writing—review and editing, Visualization. TTT: Validation, Writing—review and editing, Supervision. SS: Conceptualization, Validation, Writing—review & editing, Supervision.

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Correspondence to Gürol Canbek.

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Appendices

Appendix A Developed online research tools and data

Appendix B Binary classification performance instrument list

Table 16 lists the instruments and their abbreviations and equations. Note that more information can be found in [40]. See Table 12 for the recently proposed metrics’ equations.

Table 16 Performance measures and metrics (names, alternative names, abbreviations, and equations)
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Canbek, G., Taskaya Temizel, T. & Sagiroglu, S. BenchMetrics: a systematic benchmarking method for binary classification performance metrics. Neural Comput & Applic 33, 14623–14650 (2021). https://doi.org/10.1007/s00521-021-06103-6

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