Kernelized Cluster Validity Measures and Application to Evaluation of Different Clustering Algorithms

Many cluster validity measures have been proposed up to now, and it is realized that no universally best measure exists. In this paper we propose kernelized validity measures where a kernel means the kernel function used in support vector machines. Two measures are considered: one is the sum of the traces of the fuzzy covariances within clusters. Why we consider the trace instead of the determinant is that the calculation of the determinant will be ill-posed when kernelized, while the trace is sound and easily computed. The second is a kernelized Xie-Beni´s measure. These two measures are applied to the determination of the number of clusters having nonlinear boundaries generated by kernelized clustering algorithms. Another application of the measures is the evaluation of robustness of different algorithms with respect to variations of initial values and changes of a parameter.