  This work investigates into cost behaviors of binary classification measures
in a background of class-imbalanced problems. Twelve performance measures are
studied, such as F measure, G-means in terms of accuracy rates, and of recall
and precision, balance error rate (BER), Matthews correlation coefficient
(MCC), Kappa coefficient, etc. A new perspective is presented for those
measures by revealing their cost functions with respect to the class imbalance
ratio. Basically, they are described by four types of cost functions. The
functions provides a theoretical understanding why some measures are suitable
for dealing with class-imbalanced problems. Based on their cost functions, we
are able to conclude that G-means of accuracy rates and BER are suitable
measures because they show "proper" cost behaviors in terms of "a
misclassification from a small class will cause a greater cost than that from a
large class". On the contrary, F1 measure, G-means of recall and precision, MCC
and Kappa coefficient measures do not produce such behaviors so that they are
unsuitable to serve our goal in dealing with the problems properly.
