Fuzzy clustering with Minkowski distance

Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance.Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L_1-distance and Bobrowski and Bezdek (1991) also used the L_infty-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski distance, Groenen and Jajuga (2001) introduced a majorization algorithm to minimize the error. One of the advantages of iterative majorization is that it is a guaranteed descent algorithm, so that every iteration reduces the error until convergence is reached.However, their algorithm was limited to the case of Minkowski parameter between 1 and 2, that is, between the L_1-distance and the Euclidean distance. Here, we extend their majorization algorithm to any Minkowski distance with Minkowski parameter greater than (or equal to) 1. This extension also includes the case of the L_infty-distance. We also investigate how well this algorithm performs and present an empirical application.

Fuzzy clustering with Minkowski distance

Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance. Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L_1-distance and Bobrowski and Bezdek (1991) also used the L_infty-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski distance, Groenen and Jajuga (2001) introduced a majorization algorithm to minimize the error. One of the advantages of iterative majorization is that it is a guaranteed descent algorithm, so that every iteration reduces the error until convergence is reached. However, their algorithm was limited to the case of Minkowski parameter between 1 and 2, that is, between the L_1-distance and the Euclidean distance. Here, we extend their majorization algorithm to any Minkowski distance with Minkowski parameter greater than (or equal to) 1. This extension also includes the case of the L_infty-distance. We also investigate how well this algorithm performs and present an empirical application

Self-organization and clustering algorithms

Kohonen's feature maps approach to clustering is often likened to the k or c-means clustering algorithms. Here, the author identifies some similarities and differences between the hard and fuzzy c-Means (HCM/FCM) or ISODATA algorithms and Kohonen's self-organizing approach. The author concludes that some differences are significant, but at the same time there may be some important unknown relationships between the two methodologies. Several avenues of research are proposed

Unsupervised Learning with Normalised Data and Non-Euclidean Norms

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Applying subclustering and Lp distance in Weighted K-Means with distributed centroids

We consider the Weighted K-Means algorithm with distributed centroids aimed at clustering data sets with numerical, categorical and mixed types of data. Our approach allows given features (i.e., variables) to have different weights at different clusters. Thus, it supports the intuitive idea that features may have different degrees of relevance at different clusters. We use the Minkowski metric in a way that feature weights become feature re-scaling factors for any considered exponent. Moreover, the traditional Silhouette clustering validity index was adapted to deal with both numerical and categorical types of features. Finally, we show that our new method usually outperforms traditional K-Means as well as the recently proposed WK-DC clustering algorithm.Peer reviewe