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A novel similarity, neighborhood counting measure, was recently proposed which counts the number of neighborhoods of a pair of data points. This similarity can handle numerical and categorical attributes in a conceptually uniform way, can be calculated efficiently through a simple formula, and gives good performance when tested in the framework of k-nearest neighbor classifier. In particular it consistently outperforms a combination of the classical Euclidean distance and Hamming distance. This measure was also shown to be related to a probability formalism, G probability, which is induced from a target probability function P. It was however unclear how G is related to P, especially for classification. Therefore it was not possible to explain some characteristic features of the neighborhood counting measure. In this paper we show that G is a linear function of P, and G-based Bayes classification is equivalent to P-based Bayes classification. We also show that the k-nearest neighbor classifier, when weighted by the neighborhood counting measure, is in fact an approximation of the G-based Bayes classifier, and furthermore, the P-based Bayes classifier. Additionally we show that the neighborhood counting measure remains unchanged when binary attributes are treated as categorical or numerical data. This is a feature that is not shared by other distance measures, to the best of our knowledge. This study provides a theoretical insight into the neighborhood counting measure.
Decision support, Clustering, classification, and association rules
Fionn Murtagh, Hui Wang, "A Study of the Neighborhood Counting Similarity", IEEE Transactions on Knowledge & Data Engineering, vol. 20, no. , pp. 449-461, April 2008, doi:10.1109/TKDE.2007.190721
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