Decision regions and decision boundaries of generalized K mean algorithm based on various norm criteria

Abstract

This paper derives the decision regions and the decision boundaries of the generalized K mean algorithms based on the L1 norm criterion, the L2 norm criterion and the L∞ norm criterion. The decision boundaries of these three generalized K mean algorithms are all linear hyperplanes. However, the total numbers of the decision boundaries of the generalized K mean algorithms based on both the L1 norm criterion and the L∞ norm criterion are more than that based on the L2 norm criterion. On the other hand, the decision regions of the generalized K mean algorithm based on the L2 norm criterion are convex while that based on both the L1 norm criterion and the L∞ norm criterion are in general nonconvex. The computer numerical simulations on a toy example demonstrate the above phenomena. Besides, two examples are illustrated. The first example on the patent image retrieval system shows that the recognition accuracies of using the generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ norm criterion are 58.25%, 61% and 58.75%, respectively. The second example on the electromyogram based Parkinson’s disease detection system shows that the recognition accuracies of using the generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ norm criterion are 60%, 60% and 67%, respectively, if the signals are classified directly in the time domain. On the other hand, the recognition accuracies of the generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ norm criterion are 60%, 87% and 60%, respectively, if the signals are classified directly in the discrete cosine transform domain. The improvements are due to the nonconvexity of the decision regions of the generalized K mean algorithm based on the L1 norm criterion and the L∞ norm criterion.