论模糊C均值算法的模糊指标

模糊C均值算法(FCM)是经常使用的聚类算法之一．模糊指标m的选取对FCM的性能有重要影响．但使用模糊C均值算法时，理论上如何选取模糊指标m一直是一个问题．该文指出当一个数据被聚集成c个子类时，每个子类一般情形下应有不同的类中心．据此作者通过研究FCM算法的收敛点集的性质，得到了FCM算法的平凡解的稳定性判据，由此证明了如何选取模糊指标m理论上依赖于数据本身，并给出了理论上选取模糊指标m的规则．实验结果说明了该文给出的规则是有效的．

On the Fuzziness Index of the FCM Algorithms

The fuzzy c-means algorithm (FCM) is a widely used clustering algorithm. It is well known that the fuzziness index m has a significant impact on the performance of the FCM. However, it is an open problem how to select an appropriate fuzziness index m in theory when implementing the FCM. In this paper, we point out that each subset is often expected to have a different prototype (or cluster center) than others when the data set is clustered into c (c>1) subsets in general cases. But the FCM has a trivial solution--the mass center of the data set. According to the above assumption, the mass center of the data set is not expected to be stable. We get a simpler criterion to judge whether the trivial solution of the FCM is stable or not. As such criterion is related to the fuzziness index m,we also prove that the optimal choice of the fuzziness index m depends on the data set itself. Therefore, a theoretical approach to choose the appropriate fuzziness index m is obtained. Finally, we carry out numerical experiments in order to verify if our method is effective or not.The experimental results show that these rules are effective.