Abstract
Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of l_{p} distance measures, researchers were motivated to use them in almost every clustering process. Beside l_{p} distance measures, there exist several distance measures. Sargent introduced a special type of distance measures m(phi) and n(phi) which is closely related to l_{p}. In this paper, we generalized the Sargent sequence spaces through introduction of M(phi) and 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(phi)-distance measure (induced by the Sargent sequence space M(phi)) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the M(phi)-distance measure in the k-means algorithm.
Highlights
1 Introduction Clustering is a well-known procedure to deal with an unsupervised learning problem appearing in pattern recognition
Clustering is a process of organizing data into groups called clusters so that objects in the same cluster are similar to one another, but are dissimilar to objects in other clusters [ ]
The main contribution in the field of clustering analysis was the pioneering work of MacQueen [ ] and Bezdek [ ]
Summary
Clustering is a well-known procedure to deal with an unsupervised learning problem appearing in pattern recognition. We define Sargent’s spaces for double sequences x = {xmn} For this we first suppose U to be the set whose elements are finite sets of distinct elements of N × N obtained by σ × ς , where σ ∈ Cs and ς ∈ Ct for each s, t ≥. For φ ∈ , we define the following sequence spaces: M(φ) = x = {xmn} ∈. (iv) By using (iii) we have φ∈Lp Mφ ⊆ Lp. for obtaining the complementary relation Lp ⊆ φ∈Lp Mφ , let us suppose that x ∈ Lp. limm,n→∞ xmn = , and there is an element u of S(x) such that {|umn|} is a non-increasing sequence. 4.3 Two-moon dataset clustering by using M(φ)-distance measure in k-means algorithm Two-moon dataset is a well-known nonconvex data set. M(φ)distance measure substantially improves the clustering accuracy
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