The index of cluster validity is very useful in cluster algorithm. The number of clusters to choose in any index of cluster validity is the most important parameter in the sense that other parameters of the cluster algorithm really have second effect. In this paper, an important theorem on the variance ratio of index of cluster validity is proven, which shows that the variance ratio is closely related to a characteristic scale in clustering process and linearly linear relation as the characteristic scale is varied. Likewise two basic principles for designing the indexes of clustering validity are presented. Based on these results and the divisibility of clusters in dataset, four new equations about the lower and upper bounds of real number of clusters are derived. Furthermore, facing the densityand grid-based clustering algorithms two new approaches for computing the indexes of cluster validity are proposed, and these approaches show some new and instructive design ways. Also, two experiments are used to verify the effectiveness of these approaches in this paper.
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