Gini objective functions for three-way classifications

The three-way classifications aim to divide the universe of objects into three disjoint regions, i.e., acceptance, rejection, and non-commitment regions. We can induce different types of classification rules from these regions. There exist different measures to evaluate the quality of regions. The partition of the three regions based on certain measures such as Gini coefficient is one of the challenges in three-way classifications. When using Gini coefficients to evaluate the impurities of three-way regions, there may exist contradiction on changing of various regions towards the preferred measure levels. The impurity of one region decreases at the expense of the increase of other regions' impurities when regions change. It is impossible to decrease one region's impurity without increasing the other regions' impurities. In this paper, we formulate Gini objective functions to balance the contradictions among the impurities of three-way regions. Three Gini objective functions, i.e., minimizing the overall impurity of three regions, minimizing impurities of immediate and non-commitment decision regions simultaneously, and minimizing impurities of acceptance, rejection and non-commitment regions simultaneously, are discussed in detail. These Gini objective functions express different preferred situations of three-way regions. The balanced three-way regions representing the trade-off among impurities can be obtained by finding the solutions to these Gini objective functions. An example shows how and what three-way regions are obtained by tuning impurities of these regions to satisfy certain Gini objective functions. It is suggested that with the proposed Gini objective functions more efficient and applicable three-way regions may be induced.