Comments on “aggregation of equivalence relations” by P. C. Fishburn and A. Rubinstein

The theorem of the paper Aggregation of Equivalence Relations, by Fishburn and Rubinstein, states a result already known. This theorem improves a result from Mirkin (1975) and appears as a corollary occurring in Leclerc (1984).

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Resume L'unique théorème de l'article Aggregation of Equivalence Relations de Fishburn et Rubinstein est déja connu. Il améliore, en fait, un résultat de Mirkin (1975) et apparait en tant que corollaire dans Leclerc (1984).

     

On some significance tests in cluster analysis

We investigate the properties of several significance tests for distinguishing between the hypothesisH of a homogeneous population and an alternativeA involving clustering or heterogeneity, with emphasis on the case of multidimensional observationsx ₁, ...,x _n ^p . Four types of test statistics are considered: the (s-th) largest gap between observations, their mean distance (or similarity), the minimum within-cluster sum of squares resulting from a k-means algorithm, and the resulting maximum F statistic. The asymptotic distributions underH are given forn and the asymptotic power of the tests is derived for neighboring alternatives.     

The weighted sum of split and diameter clustering

In this paper, we propose a bicriterion objective function for clustering a given set ofN entities, which minimizes [d–(1–)s], where 01, andd ands are the diameter and the split of the clustering, respectively. When =1, the problem reduces to minimum diameter clustering, and when =0, maximum split clustering. We show that this objective provides an effective way to compromise between the two often conflicting criteria. While the problem is NP-hard in general, a polynomial algorithm with the worst-case time complexityO(N ²) is devised to solve the bipartition version. This algorithm actually gives all the Pareto optimal bipartitions with respect to diameter and split, and it can be extended to yield an efficient divisive hierarchical scheme. An extension of the approach to the objective [(d ₁+d ₂)–2(1–)s] is also proposed, whered ₁ andd ₂ are diameters of the two clusters of a bipartition.This research was supported in part by the National Science and Engineering Research Council of Canada (Grant OGP 0104900). The authors wish to thank two anonymous referees, whose detailed comments on earlier drafts improved the paper.     

The median procedure for n-trees

Let (X,d) be a metric space The functionM:X ^k 2 ^x defined by is the minimum } is called themedian procedure and has been found useful in various applications involving the notion of consensus Here we present axioms that characterizeM whenX is a certain class of trees (hierarchical classifications), andd is the symmetric difference metricWe would like to thank the referees and Editor for helpful comments     

Algebraic Self-Dualityc as the ”Ultimate Explanation”

Shahn Majids philosophy of physics is critically presented. In his view the postulate that the universe should be self-explaining implies that no fundamental theory of physics is complete unless it is self-dual. Majid shows that bicrossproduct Hopf algebras have this property. His philosophy is compared with other approaches to the ultimate explanation and briefly analyzed.     

Ultrametric tree representations of incomplete dissimilarity data

The least squares algorithm for fitting ultrametric trees to proximity data originally proposed by Carroll and Pruzansky and further elaborated by De Soete is extended to handle missing data. A Monte Carlo evaluation reveals that the algorithm is capable of recovering an ultrametric tree underlying an incomplete set of error-perturbed dissimilarities quite well.Geert De Soete is Aangesteld Navorser of the Belgian National Fonds voor Wetenschappelijk Onderzoek.     

Optimal variable weighting for hierarchical clustering: An alternating least-squares algorithm

This paper presents the development of a new methodology which simultaneously estimates in a least-squares fashion both an ultrametric tree and respective variable weightings for profile data that have been converted into (weighted) Euclidean distances. We first review the relevant classification literature on this topic. The new methodology is presented including the alternating least-squares algorithm used to estimate the parameters. The method is applied to a synthetic data set with known structure as a test of its operation. An application of this new methodology to ethnic group rating data is also discussed. Finally, extensions of the procedure to model additive, multiple, and three-way trees are mentioned.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek.     

Additive clustering and qualitative factor analysis methods for similarity matrices

We review methods of qualitative factor analysis (QFA) developed by the author and his collaborators over the last decade and discuss the use of QFA methods for the additive clustering problem. The QFA method includes, first, finding a square Boolean matrix in a fixed set of Boolean matrices with simple structures to approximate a given similarity matrix, and, second, repeating this process again and again using residual similarity matrices. We present convergence properties for three versions of the method, provide cluster interpretations for results obtained from the algorithms, and give formulas for the evaluation of factor shares of the initial similarities variance.I am indebted to Professor P. Arabie and the referees for valuable comments and editing of the text.     

Metric unidimensional scaling and global optimization

For the problem of metric unidimensional scaling, the number of local minima is estimated. For locating the globally optimal solution we develop an approach, called the smoothing technique. Although not guaranteed inevitably to locate the global optimum, the smoothing technique did so in all computational experiments where the global optimum was known.     

OVWTRE: A program for optimal variable weighting for ultrametric and additive tree fitting

The author is supported as Bevoegdverklaard Navorser of the Belgian National Fonds voor Wetenschappelijk Onderzoek.     

Clustering and isolation in the consensus problem for partitions

We examine the problem of aggregating several partitions of a finite set into a single consensus partition We note that the dual concepts of clustering and isolation are especially significant in this connection. The hypothesis that a consensus partition should respect unanimity with respect to either concept leads us to stress a consensus interval rather than a single partition. The extremes of this interval are characterized axiomatically. If a sufficient totality of traits has been measured, and if measurement errors are independent, then a true classifying partition can be expected to lie in the consensus interval. The structure of the partitions in the interval lends itself to partial solutions of the consensus problem Conditional entropy may be used to quantify the uncertainty inherent in the interval as a whole     

Least squares algorithms for constructing constrained ultrametric and additive tree representations of symmetric proximity data

A mathematical programming algorithm is developed for fitting ultrametric or additive trees to proximity data where external constraints are imposed on the topology of the tree. The two procedures minimize a least squares loss function. The method is illustrated on both synthetic and real data. A constrained ultrametric tree analysis was performed on similarities between 32 subjects based on preferences for ten odors, while a constrained additive tree analysis was carried out on some proximity data between kinship terms. Finally, some extensions of the methodology to other tree fitting procedures are mentioned.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek.     

Fitting an extended INDSCAL model to three-way proximity data

The INDSCAL individual differences scaling model is extended by assuming dimensions specific to each stimulus or other object, as well as dimensions common to all stimuli or objects. An alternating maximum likelihood procedure is used to seek maximum likelihood estimates of all parameters of this EXSCAL (Extended INDSCAL) model, including parameters of monotone splines assumed in a quasi-nonmetric approach. The rationale for and numerical details of this approach are described and discussed, and the resulting EXSCAL method is illustrated on some data on perception of musical timbres.     

Efficient algorithms for divisive hierarchical clustering with the diameter criterion

Divisive hierarchical clustering algorithms with the diameter criterion proceed by recursively selecting the cluster with largest diameter and partitioning it into two clusters whose largest diameter is smallest possible. We provide two such algorithms with complexitiesO( N ²) andO(N ²logN) respectively, where denotes the maximum number of clusters in a partition andN the number of entities to be clustered. The former algorithm, an efficient implementation of an algorithm of Hubert, allows to find all partitions into at most clusters and is inO(N ²) for fixed . Moreover, if in each partitioning the size of the largest cluster is bounded byp times the number of entities in the set to be partitioned, with 1/2<=p<1, it provides a complete hierarchy of partitionsO(N ² logN) time. The latter algorithm, a refinement of an algorithm of Rao allows to build a complete hierarchy of partitions inO(N ² logN) time without any restriction. Comparative computational experiments with both algorithms and with an agglomerative hierarchical algorithm of Benzécri are reported.

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Résumé Les algorithmes de classification hiérarchique descendante utilisant le critère du diamètre, sélectionnent récursivement la classe de plus grand diamètre et la partitionnent en deux classes, dont le plus grand diamètre est le plus, petit possible. Nous proposons deux tels algorithmes, avec des complexités enO ( N²) etO(N ² logN) respectivement, où désigne le nombre maximum de classes d'une partition etN le nombre d'objets à classifier. Le premier algorithme, une implantation d'un algorithme de Hubert, permet de construire des partitions avec au plus classes et est enO(N ²) pour fixé. De plus, si dans chaque bipartition le nombre d'objets de la plus grande classe, est borné parp fois le nombre d'objets de l'ensemble à partitionner, où 1/2≤p<1, cet algorithme permet de construire une hiérarchie complète de partitions en tempsO(N ² logN). Le second algorithme, un raffinement d'un algorithme de Rao, permet de construire une hiérarchie complète de partitions en tempsO(N ² logN) sans aucune restriction On présente également des résultats de calcul comparatifs pour les deux algorithmes et pour l'algorithme de classification hiérarchique ascendante de Benzécri.

     

Maximum sum-of-splits clustering

ConsiderN entities to be classified, and a matrix of dissimilarities between pairs of them. The split of a cluster is the smallest dissimilarity between an entity of this cluster and an entity outside it. The single-linkage algorithm provides partitions intoM clusters for which the smallest split is maximum. We study here the average split of the clusters or, equivalently, the sum of splits. A (N ²) algorithm is provided to determine maximum sum-of-splits partitions intoM clusters for allM betweenN – 1 and 2, using the dual graph of the single-linkage dendrogram.

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Résumé SoientN objets à classifier et une matrice de dissimilarit és entre paires de ces objets. L'écart d'une classe est la plus petite dissimilarité entre un objet de cette classe et un objet en dehors d'elle. L'algorithme du lien simple fournit des partitions enM classes dont le plus petit écart est maximum. On étudie l'écart moyen des classes, ou, ce qui est équivalent, la somme des écarts. On propose un algorithme en (N ²) pour déterminer des partitions enM classes dont la somme des écarts est maximum pourM allant deN – 1 à 2, basé sur le graphe dual du dendrogramme de la méthode du lien simple.

     

An evaluation of five algorithms for generating an initial configuration for SINDSCAL

Five different methods for obtaining a rational initial estimate of the stimulus space in the INDSCAL model were compared using the SINDSCAL program for fitting INDSCAL. The effect of the number of stimuli, the number of subjects, the dimensionality, and the amount of error on the quality and efficiency of the final SINDSCAL solution were investigated in a Monte Carlo study. We found that the quality of the final solution was not affected by the choice of the initialization method, suggesting that SINDSCAL finds a global optimum regardless of the initialization method used. The most efficient procedures were the methods proposed by by de Leeuw and Pruzansky (1978) and by Flury and Gautschi (1986) for the simultaneous diagonalization of several positive definite symmetric matrices, and a method based on linearly constraining the stimulus space using the CANDELINC approach developed by Carroll, Pruzansky, and Kruskal (1980).Geert De Soete is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. The authors gratefully acknowledge the helpful comments and suggestions of the reviewers.     

Clustering the rows and columns of a contingency table

A number of ways of investigating heterogeneity in a two-way contingency table are reviewed. In particular, we consider chi-square decompositions of the Pearson chi-square statistic with respect to the nodes of a hierarchical clustering of the rows and/or the columns of the table. A cut-off point which indicates significant clustering may be defined on the binary trees associated with the respective row and column cluster analyses. This approach provides a simple graphical procedure which is useful in interpreting a significant chi-square statistic of a contingency table.The author gratefully acknowledges the constructive comments of the referees and the editor.     

N-trees as nestings: Complexity,similarity, and consensus

Interpreting a taxonomic tree as a set of objects leads to natural measures of complexity and similarity, and sets natural lower bounds on a consensus tree Interpretations differing as to the kind of objects constituting a tree lead to different measures and consensus Subset nesting is preferred over the clusters (strict consensus) and even the triads interpretations because of its superior expression of shared structure Algorithms for computing the complexity and similarity of trees, as well as a consensus index onto [0,1], are presented for this interpretation The full consensus is defined as the only tree which includes all the nestings shared in a profile of rival trees and whose clusters reflect only nestings shared in the profile The full consensus is proved to exist uniquely for each profile, and to equal the Adams consensusThe author is grateful for the many helpful comments on presentation from Frances McA Adams, William H E Day, and Christopher A Meacham