A comment on "Alternative c-means clustering algorithms"

In their paper [1], Wu and Yang proposed two alternative (hard) c-means and fuzzy c-means clustering algorithms, by replacing the original Euclidean distance in the c-means and fuzzy c-means algorithm with a new Gaussian function based distance. Although the new distance function is more robust than the Euclidean distance, we will show in this comment that the distance Wu and Yang used is not a metric. Suppose x, y∈X, and X is a compact subset of R. Let d(x, y) denote the distance function between two vectors x and y, then the Euclidean distance is represented by d(x, y) =||x-y|| , where ||•|| denote the two norm. Wu and Yang [1] defined a new distance as ) 2 || || exp( 1 ) , ( y x y x d − − − = β . (1) To prove a distance function d(x, y) is a metric, the necessary and sufficient condition is that d(x, y) satisfies the following three conditions [2] (i) 0 ) , ( , , 0 ) , ( = ≠ ∀ > x x d y x y x d , (ii) ) , ( ) , ( x y d y x d = , (iii) z y z d z x d y x d ∀ + ≤ ), , ( ) , ( ) , ( . Wu and Yang [1] claimed that the distance function in equation (1) is a metric, i.e, the above three conditions were satisfied. However, we will show in Theorem 1 that it is not true.