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Gini objective functions for three-way classifications

机译：基尼目标函数的三向分类

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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. (C) 2016 Elsevier Inc. All rights reserved.
机译：三向分类的目的是将对象的宇宙划分为三个不相交的区域，即接受，拒绝和非承诺区域。我们可以从这些区域中得出不同类型的分类规则。存在评估区域质量的不同措施。基于某些指标（例如基尼系数）的三个区域划分是三向分类中的挑战之一。当使用基尼系数来评估三向区域的杂质时，在将各个区域向首选测量水平变化时可能存在矛盾。当区域改变时，一个区域的杂质减少了，而另一区域的杂质增加了。减少一个区域的杂质而不增加其他区域的杂质是不可能的。在本文中，我们制定了基尼目标函数来平衡三向区域杂质之间的矛盾。详细讨论了三个基尼目标函数，即最小化三个区域的总杂质，同时最小化立即和非承诺决策区域的杂质，以及同时最小化接受，拒绝和非承诺区域的杂质。这些基尼目标函数表达了三向区域的不同首选情况。可以通过找到这些基尼目标函数的解来获得表示杂质之间权衡的平衡三通区域。一个示例显示了如何以及通过调整这些区域的杂质以满足某些Gini目标函数来获得三通区域。建议使用建议的基尼目标函数可以诱导出更有效和适用的三向区域。 （C）2016 Elsevier Inc.保留所有权利。

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