Bicriterion methods for partitioning dissimilarity matrices

Combining diversity and dispersion criteria for anticlustering: A bicriterion approach

Most partitioning methods used in psychological research seek to produce homogeneous groups (i.e., groups with low intra-group dissimilarity). However, there are also applications where the goal is to provide heterogeneous groups (i.e., groups with high intra-group dissimilarity). Examples of these anticlustering contexts include construction of stimulus sets, formation of student groups, assignment of employees to project work teams, and assembly of test forms from a bank of items. Unfortunately, most commercial software packages are not equipped to accommodate the objective criteria and constraints that commonly arise for anticlustering problems. Two important objective criteria for anticlustering based on information in a dissimilarity matrix are: a diversity measure based on within-cluster sums of dissimilarities; and a dispersion measure based on the within-cluster minimum dissimilarities. In many instances, it is possible to find a partition that provides a large improvement in one of these two criteria with little (or no) sacrifice in the other criterion. For this reason, it is of significant value to explore the trade-offs that arise between these two criteria. Accordingly, the key contribution of this paper is the formulation of a bicriterion optimization problem for anticlustering based on the diversity and dispersion criteria, along with heuristics to approximate the Pareto efficient set of partitions. A motivating example and computational study are provided within the framework of test assembly.

Model selection for minimum-diameter partitioning

The minimum-diameter partitioning problem (MDPP) seeks to produce compact clusters, as measured by an overall goodness-of-fit measure known as the partition diameter, which represents the maximum dissimilarity between any two objects placed in the same cluster. Complete-linkage hierarchical clustering is perhaps the best-known heuristic method for the MDPP and has an extensive history of applications in psychological research. Unfortunately, this method has several inherent shortcomings that impede the model selection process, such as: (1) sensitivity to the input order of the objects, (2) failure to obtain a globally optimal minimum-diameter partition when cutting the tree at K clusters, and (3) the propensity for a large number of alternative minimum-diameter partitions for a given K. We propose that each of these problems can be addressed by applying an algorithm that finds all of the minimum-diameter partitions for different values of K. Model selection is then facilitated by considering, for each value of K, the reduction in the partition diameter, the number of alternative optima, and the partition agreement among the alternative optima. Using five examples from the empirical literature, we show the practical value of the proposed process for facilitating model selection for the MDPP.

Cross validation issues in multiobjective clustering

The implementation of multiobjective programming methods in combinatorial data analysis is an emergent area of study with a variety of pragmatic applications in the behavioural sciences. Most notably, multiobjective programming provides a tool for analysts to model trade offs among competing criteria in clustering, seriation, and unidimensional scaling tasks. Although multiobjective programming has considerable promise, the technique can produce numerically appealing results that lack empirical validity. With this issue in mind, the purpose of this paper is to briefly review viable areas of application for multiobjective programming and, more importantly, to outline the importance of cross-validation when using this method in cluster analysis.

Stability analysis in K-means clustering

This paper develops a new procedure, called stability analysis, for K-means clustering. Instead of ignoring local optima and only considering the best solution found, this procedure takes advantage of additional information from a K-means cluster analysis. The information from the locally optimal solutions is collected in an object by object co-occurrence matrix. The co-occurrence matrix is clustered and subsequently reordered by a steepest ascent quadratic assignment procedure to aid visual interpretation of the multidimensional cluster structure. Subsequently, measures are developed to determine the overall structure of a data set, the number of clusters and the multidimensional relationships between the clusters.

Clustering, seriation, and subset extraction of confusion data

The study of confusion data is a well established practice in psychology. Although many types of analytical approaches for confusion data are available, among the most common methods are the extraction of 1 or more subsets of stimuli, the partitioning of the complete stimulus set into distinct groups, and the ordering of the stimulus set. Although standard commercial software packages can sometimes facilitate these types of analyses, they are not guaranteed to produce optimal solutions. The authors present a MATLAB *.m file for preprocessing confusion matrices, which includes fitting of the similarity-choice model. Two additional MATLAB programs are available for optimally clustering stimuli on the basis of confusion data. The authors also developed programs for optimally ordering stimuli and extracting subsets of stimuli using information from confusion matrices. Together, these programs provide several pragmatic alternatives for the applied researcher when analyzing confusion data. Although the programs are described within the context of confusion data, they are also amenable to other types of proximity data.