Sequence spaces [Formula: see text] and [Formula: see text] with application in clustering

Area of Science:

• Mathematics
• Computer Science
• Data Mining

Background:

• Distance measures are fundamental to clustering algorithms.
• Euclidean distance is widely used but has limitations.
• Sargent's sequence spaces offer alternative distance measures.

Purpose of the Study:

• To generalize Sargent sequence spaces by introducing new [Formula: see text] and [Formula: see text] spaces.
• To investigate the properties of these new sequence spaces, including their BK-space status and dual relationships.
• To evaluate the performance of an induced [Formula: see text]-distance measure in the k-means clustering algorithm.

Main Methods:

• Generalization of Sargent sequence spaces to define new [Formula: see text] and [Formula: see text] spaces.
• Mathematical proofs to establish the BK-space properties and dual relationships of the new spaces.
• Implementation of an induced [Formula: text]-distance measure within the k-means clustering algorithm.
• Clustering analysis of the two-moon dataset using both Euclidean and the new [Formula: text]-distance measures.

Main Results:

• Successfully generalized Sargent sequence spaces, introducing novel [Formula: see text] and [Formula: text] spaces.
• Demonstrated that the newly introduced spaces are BK-spaces, with one being the dual of the other.
• The k-means algorithm using the induced [Formula: text]-distance measure achieved effective clustering of the two-moon dataset.
• The [Formula: text]-distance measure proved more efficacious than the Euclidean distance for this specific clustering task.

Conclusions:

• The generalized Sargent sequence spaces provide a foundation for new distance measures.
• The induced [Formula: text]-distance measure shows significant potential for enhancing k-means clustering performance.
• This study highlights the efficacy of tailored distance measures in improving data clustering outcomes.