Analysis on comparison of distances derived by one-norm and two-norm with weight functions

Julian, Hung and Lin recently published a paper in Pattern Recognition Letters 33 (2012) 1219–1223 discussing the similarity measures of Mitchell (2003). However, after careful analysis we have found that the theorem in their paper is incomplete. In this paper, we will first point out the incomplete theorem. Second, we will provide a patch work to prove that the distance with the weighted function derived by one-norm is less than that derived by two-norm. Third, we will show that their original problem is only a corollary of Jensen’s inequality (1906). Our findings will help researchers develop more reasonable similarity measures and its application to pattern recognition under fuzzy intuitionistic set environment.