The Minkowski central partition as a pointer to a suitable distance exponent and consensus partitioning

We generate optimal Minkowski partitions at various values of the exponent p.We define the Minkowski profile based on the average similarity between partitions.Minkowski profile is highly correlated with ARI vectors related to the ground truth.We define the central Minkowski partition which can serve as a consensus partition.The Silhouette width should be used for selecting the optimal Minkowski exponent. The Minkowski weighted K-means (MWK-means) is a recently developed clustering algorithm capable of computing feature weights. The cluster-specific weights in MWK-means follow the intuitive idea that a feature with low variance should have a greater weight than a feature with high variance. The final clustering found by this algorithm depends on the selection of the Minkowski distance exponent. This paper explores the possibility of using the central Minkowski partition in the ensemble of all Minkowski partitions for selecting an optimal value of the Minkowski exponent. The central Minkowski partition appears to be also a good consensus partition. Furthermore, we discovered some striking correlation results between the Minkowski profile, defined as a mapping of the Minkowski exponent values into the average similarity values of the optimal Minkowski partitions, and the Adjusted Rand Index vectors resulting from the comparison of the obtained partitions to the ground truth. Our findings were confirmed by a series of computational experiments involving synthetic Gaussian clusters and real-world data.