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Classifiers with rejection are essential in real-world applications wheremisclassifications and their effects are critical. However, if no problemspecific cost function is defined, there are no established measures to assessthe performance of such classifiers. We introduce a set of desired propertiesfor performance measures for classifiers with rejection, based on which wepropose a set of three performance measures for the evaluation of theperformance of classifiers with rejection that satisfy the desired properties.The nonrejected accuracy measures the ability of the classifier to accuratelyclassify nonrejected samples; the classification quality measures the correctdecision making of the classifier with rejector; and the rejection qualitymeasures the ability to concentrate all misclassified samples onto the set ofrejected samples. From the measures, we derive the concept of relativeoptimality that allows us to connect the measures to a family of cost functionsthat take into account the trade-off between rejection and misclassification.We illustrate the use of the proposed performance measures on classifiers withrejection applied to synthetic and real-world data.

Performance measures for classification systems with rejection