Weighted MCC: A Robust Measure of Multiclass Classifier Performance for Observations with Individual Weights

TL;DR

This paper introduces a weighted MCC that accounts for individual observation weights in classification tasks, improving the evaluation of classifiers by emphasizing performance on more important data points.

Contribution

The paper proposes a novel weighted MCC and related measures that are sensitive to observation weights and robust to small weight changes, enhancing classifier evaluation.

Findings

Weighted MCC emphasizes high-weight observations in classifier assessment.

Weighted measures are robust to small changes in observation weights.

Unweighted measures do not reflect the importance of weighted observations.

Abstract

Several performance measures are used to evaluate binary and multiclass classification tasks. But individual observations may often have distinct weights, and none of these measures are sensitive to such varying weights. We propose a new weighted Pearson-Matthews Correlation Coefficient (MCC) for binary classification as well as weighted versions of related multiclass measures. The weighted MCC varies between $-1$ and $1$. But crucially, the weighted MCC values are higher for classifiers that perform better on highly weighted observations, and hence is able to distinguish them from classifiers that have a similar overall performance and ones that perform better on the lowly weighted observations. Furthermore, we prove that the weighted measures are robust with respect to the choice of weights in a precise manner: if the weights are changed by at most $\epsilon$, the value of the weighted measure changes at most by a factor of $\epsilon$ in the binary case and by a factor of $\epsilon^2$ in the multiclass case. Our computations demonstrate that the weighted measures clearly identify classifiers that perform better on higher weighted observations, while the unweighted measures remain completely indifferent to the choices of weights.