This paper further extends the ‘kernel’-based approach to clustering proposed by E. Diday in early 70s. According to this approach, a cluster’s centroid can be represented by parameters of any analytical model, such as linear regression equation, built over the cluster. We address the problem of producing regression-wise clusters to be separated in the input variable space by building a hybrid clustering criterion that combines the regression-wise clustering criterion with the conventional centroid-based one.

Directions for representing and comparing hierarchies are discussed.

Clustering methods that are invariant under monotone dissimilarity transformations are analyzed.

Most recent theories and methods concerning such concepts as ultrametric, tree metric, Robinson matrix, pyramid, and weak hierarchy are presented.

A linear theory for binary hierarchy is proposed to allow decomposing the data entries, as well as covariances, by the clusters.

A review of clustering concepts and algorithms is provided emphasizing: (a) output cluster structure, (b) input data kind, and (c) criterion.

A dozen cluster structures is considered including those used in either supervised or unsupervised learning or both.

The techniques discussed cover such algorithms as nearest neighbor, K-Means (moving centers), agglomerative clustering, conceptual clustering, EM-algorithm, high-density clustering, and back-propagation.

Interpretation is considered as achieving clustering goals (partly, via presentation of the same data with both extensional and intensional forms of cluster structures).

The appeal of metric evaluation of research impact has attracted considerable interest in recent times. Although the public at large and administrative bodies are much interested in the idea, scientists and other researchers are much more cautious, insisting that metrics are but an auxiliary instrument to the qualitative peer-based judgement. The goal of this article is to propose availing of such a well positioned construct as domain taxonomy as a tool for directly assessing the scope and quality of research. We first show how taxonomies can be used to analyze the scope and perspectives of a set of research projects or papers. Then we proceed to define a research team or researcher’s rank by those nodes in the hierarchy that have been created or significantly transformed by the results of the researcher. An experimental test of the approach in the data analysis domain is described. Although the concept of taxonomy seems rather simplistic to describe all the richness of a research domain, its changes and use can be made transparent and subject to open discussions.

Bilinear clustering for mixed — quantitative, nominal and binary — variables is proved to be a theory-motivated extension of K-Means method.

Decomposition of the data scatter into “explained” and “residual” parts is provided (for each of the two norms: sum of squares and moduli).

Contribution weights are derived to attack machine learning problems (conceptual description, selecting and transforming the variables, and knowledge discovery).

The explained data scatter parts related to nominal variables appear to coincide with the chi-squared Pearson coefficient and some other popular indices, as well.

Approximation (bi)-partitioning for contingency tables substantiates and extends some popular clustering techniques.

The issue of determining “the right number of clusters” in K-Means has attracted considerable interest, especially in the recent years. Cluster intermix appears to be a factor most affecting the clustering results. This paper proposes an experimental setting for comparison of different approaches at data generated from Gaussian clusters with the controlled parameters of between- and within-cluster spread to model cluster intermix. The setting allows for evaluating the centroid recovery on par with conventional evaluation of the cluster recovery. The subjects of our interest are two versions of the “intelligent” K-Means method, ik-Means, that find the “right” number of clusters by extracting “anomalous patterns” from the data one-by-one. We compare them with seven other methods, including Hartigan’s rule, averaged Silhouette width and Gap statistic, under different between- and within-cluster spread-shape conditions. There are several consistent patterns in the results of our experiments, such as that the right K is reproduced best by Hartigan’s rule – but not clusters or their centroids. This leads us to propose an adjusted version of iK-Means, which performs well in the current experiment setting.

Entity-to-variable data table can be represented geometrically in three different settings of which one (row-points) pertains to conventional clustering, another (column-vectors), to conceptual clustering, and the third one (matrix space), to approximation clustering.

Two principles for standardizing the conditional data tables are suggested as related to the data scatter.

Standardizing the aggregable data is suggested based on the flow index concept introduced.

Graph-theoretic concepts related to clustering are considered.

Low-rank approximation of data, including the popular Principal component and Correspondence analysis techniques, are discussed and extended into a general Sequential fitting procedure, SEFIT, which will be employed for approximation clustering.