Analysis and Comparison of Classification Metrics

A variety of different performance metrics are commonly used in the machine learning literature for the evaluation of classification systems. Some of the most common ones for measuring quality of hard decisions are standard and balanced accuracy, standard and balanced error rate, F-beta score, and Matthews correlation coefficient (MCC). In this document, we review the definition of these and other metrics and compare them with the expected cost (EC), a metric introduced in every statistical learning course but rarely used in the machine learning literature. We show that both the standard and balanced error rates are special cases of the EC. Further, we show its relation with F-score and MCC and argue that EC is superior to these traditional metrics, being more elegant, general, and intuitive, as well as being based on basic principles from statistics. The metrics above measure the quality of hard decisions. Yet, most modern classification systems output continuous scores for the classes which we may want to evaluate directly. Metrics for measuring the quality of system scores include the area under the ROC curve, equal error rate, cross-entropy, Brier score, and Bayes EC or Bayes risk, among others. The last three metrics are special cases of a family of metrics given by the expected value of proper scoring rules (PSRs). We review the theory behind these metrics and argue that they are the most principled way to measure the quality of the posterior probabilities produced by a system. Finally, we show how to use these metrics to compute the system's calibration loss and compare this metric with the standard expected calibration error (ECE), arguing that calibration loss based on PSRs is superior to the ECE for a variety of reasons.