Melvin F. Janowitz

Ordinal and percentile clustering

Abstract The central purpose of this paper is to put together the basic ideas of two separate theories - the theory of ordinal clustering, as developed by Janowitz et al., and the theory of probabilistic metric spaces, as developed by Schweizer et al. The principal result is a new theory of clustering, called percentile clustering, in which the clustering is based, not on some average or other typical value of the data, but directly on the distributed data itself. A secondary outgrowth is a generalized theory of ordinal clustering. The paper begins with a brief but essentially self-contained exposition of the main ideas of the two theories mentioned above. It then combines them to lay the foundations of the theory of percentile clustering. It goes on to develop a number of specific algorithms with a small, artificial data set. The paper concludes by applying the new cluster methods to two concrete examples. The first of these is a data set concerning combat deaths in the Vietnam War; the second is a data set supplied by N. Creel and dealing with the classification of species of gibbons. In both instances results obtained with various standard clustering techniques are also presented.

Semiflat L-cluster methods

Abstract In previous work, an order theoretic model for cluster analysis has been developed, and flat cluster methods characterized in terms of their compatibility with respect to the set of residuated mappings on the non-negative reals. This suggests the possibility of classifying cluster methods according to their compatibility properties with respect to residuated mappings. Such a program is herewith initiated. Semiflat cluster methods are defined and characterized by their compatibility with respect to those residuated mappings θ for which θ + (0) = 0. It is also shown how these methods fit into an earlier graph theoretic model for cluster analysis that was developed by Jardine and Sibson.

On the *-order for Rickart *-rings

It is shown that a Rickart *-ring forms a pseudo upper semilattice under the *-order and that a Baer *-ring in addition forms a complete lower semilattice.

Boolean Products of Lattices

A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with properties of its individual stalks.

On the semilattice of weak orders of a set

Abstract Order theoretic and combinatorial properties of the semilattice of weak orders on a set are developed. In the case of a finite set, an order theoretic characterization of this semilattice is obtained.

Neutral consensus functions

Abstract Neutral consensus methods have been studied in detail by a number of authors in specific contexts. This paper introduces a general mathematical model that allows for a unified approach to their study via the notion of stability families. Special attention is paid to conditions that produce abstract versions of Arrows Theorem.

An Ordered Set Approach to Neutral Consensus Functions

Several authors have investigated consensus functions from the viewpoint of stability families. It will be shown that the stability family approach extends and enriches the ordered set approach to consensus theory. Our investigation will center on the notions of sup and inf-irreducible elements of a finite ordered set. We will see that irreducible elements give rise to natural internal stability families for a given ordered set X. We will investigate the relationship between these internal stability families and abstract stability families. Special attention will be paid to the study of different types of neutrality, thus extending earlier work of B. Monjardet. We will be particularly interested in the change in the available neutral consensus functions on an ordered set when a given internal stability family is changed.

Induced social welfare functions

Abstract The transitive closure of a complete quasi-transitive relation is a weak order. This fact is used to define various social welfare functions that are generalizations of the Pareto extension rule as well as certain median consensus rules. The resulting functions are axiomatically characterized, and some of their properties are developed. A relation is shown between these functions and a generalized form of compatibility.

Compatibility in a graph-theoretic setting

Abstract In ordinal models for cluster analysis, it has proved useful to study the effect of the action of dissimilarity coefficients by certain isotone mappings. The current work places these results in a graph-theoretic setting, and shows how the resulting theory extends the earlier results obtained in the ordinal models.

Preservation of weak order equivalence

Abstract A study is made of the exact nature of a class of cluster methods that are suitable for use with data having only ordinal significance.