## Abstract

Pseudo-centroid clustering replaces the traditional concept of a centroid expressed as a center of gravity with the notion of a *pseudo-centroid* (or a *coordinate free centroid*) which has the advantage of applying to clustering problems where points do not have numerical coordinates (or categorical coordinates that are translated into numerical form). Such problems, for which classical centroids do not exist, are particularly important in social sciences, marketing, psychology and economics, where distances are not computed from vector coordinates but rather are expressed in terms of characteristics such as affinity relationships, psychological preferences, advertising responses, polling data and market interactions, where distances, broadly conceived, measure the similarity (or dissimilarity) of characteristics, functions or structures. We formulate a K-PC algorithm analogous to a K-Means algorithm and focus on two key types of pseudo-centroids, *MinMax-centroids* and (weighted) *MinSum-centroids*, and describe how they, respectively, give rise to a K-MinMax algorithm and a K-MinSum algorithm which are analogous to a K-Means algorithm. The K-PC algorithms are able to take advantage of problem structure to identify special *diversity-based* and *intensity-based* starting methods to generate initial pseudo-centroids and associated clusters, accompanied by theorems for the intensity-based methods that establish their ability to obtain best clusters of a selected size from the points available at each stage of construction. We also introduce a regret-threshold PC algorithm that modifies the K-PC algorithm together with an associated diversification method and a new criterion for evaluating the quality of a collection of clusters.

## Keywords

Clustering Centroids K-Means K-Medoids Advanced starting methods Metaheuristics## Notes

### Acknowledgments

This study was not funded.

### Compliance with ethical standards

#### Conflict of interest

The author declares that he has no conflict of interest.

#### Ethical approval

This article does not contain any studies with human or animal participants.

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