Sequence spaces M ( ϕ ) M(ϕ)M(\phi) and N ( ϕ ) N(ϕ)N(\phi) 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 l p $l_{p}$ distance measures, researchers were motivated to use them in almost every clustering process. Beside l p $l_{p}$ distance measures, there exist several distance measures. Sargent introduced a special type of distance measures m ( ϕ ) $m(\phi)$ and n ( ϕ ) $n(\phi)$ which is closely related to l p $l_{p}$ . In this paper, we generalized the Sargent sequence spaces through introduction of M ( ϕ ) $M(\phi)$ and N ( ϕ ) $N(\phi)$ 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 ) $M(\phi)$ -distance measure (induced by the Sargent sequence space M ( ϕ ) $M(\phi)$ ) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the M ( ϕ ) $M(\phi)$ -distance measure in the k-means algorithm