Sequence spaces M(ϕ) and N(ϕ) with application in clustering

Abstract

Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of lp distance measures, researchers were motivated to use them in almost every clustering process. Beside lp distance measures, there exist several distance measures. Sargent introduced a special type of distance measures m(ϕ) and n(ϕ) which is closely related to lp. In this paper, we generalized the Sargent sequence spaces through introduction of M(ϕ) and N(ϕ) sequence spaces. Moreover, it is shown that both spaces are BK-spaces, and one is a dual of another. Further, we have clustered the two-moon dataset by using an induced M(ϕ) -distance measure (induced by the Sargent sequence space M(ϕ)) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the M(ϕ) -distance measure in the k-means algorithm.