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Co-clustering means the simultaneous clustering of the rows and columns of a two-dimensional data table (biclustering, two-way clustering), in contrast to separately clustering the rows and the columns. Practical applications may be met, e.g., in economics, social sciences, bioinformatics, etc. Various co-clustering models, criteria, and algorithms have been proposed that differ with respect to the considered data types (real-valued, integers, binary data, contingency tables), and also the meaning of rows and columns (samples, variables, factors, time,...). This paper concentrates on the case where rows correspond to (independent) samples or objects, and columns to (typically dependent) variables. We emphasize that here, in general, different similarity or homogeneity concepts must be used for rows and columns. We propose two probabilistic co-clustering approaches: a situation where clusters of objects and of variables refer to two different distribution parameters, and a situation where clusters of ‘highly correlated’ variables (by regression to a latent class-specific factor) are crossed with object clusters that are distinguished by additive effects only. We emphasize here the classical ‘classification approach’, where maximum likelihood criteria are optimized by generalized alternating k-means type algorithms.

Background/Aims From quantitative trait locus mapping in inbred mice, we identified the Nr1h4 gene encoding the nuclear bile salt receptor FXR as a candidate gene for the cholesterol gallstone susceptibility locus Lith7 . Here, we investigated further an association of the gene encoding FXR and gallstone susceptibility in mice and humans. Methods The Nr1h4 gene was sequenced in inbred mouse strains with susceptible and resistant Lith7 alleles. Quantitative RT-PCR was employed to determine mRNA expression levels. Gallstone carriers and control subjects of three different populations comprising 1004 individuals were genotyped for polymorphisms of the orthologous human gene detected by sequencing. Results Expression and sequence analyses in inbred mice were consistent with Nr1h4 underlying Lith7 . In the human populations, we identified three frequent haplotypes that accounted for >95% of all haplotypes observed. In a Mexican population, the most common haplotype NR1H4 _ 1 was associated with gallstone prevalence. In contrast, NR1H4 _ 1 displayed no association with gallstone prevalence in a German population, whereas in a Chilean population we observed a trend towards a protective effect of NR1H4 _ 1 . Conclusions Our study in an inbred mouse model and in three ethnically distinct populations indicates complex interactions of NR1H4 alleles and other risk factors for the development of cholelithiasis.

This paper deals with the construction of optimum partitions ${\cal B} = (B_1,...,B_m)$ of $I\hspace{-4.0pt}R^p$ for a clustering criterion which is based on a convex function of the class centroids $E[X\vert X\in B_i]$ as a generalization of the classical SSQ clustering criterion for n data points. We formulate a dual optimality problem involving two sets of variables and derive a maximum-support-plane (MSP) algorithm for constructing a (sub-)optimum partition as a generalized k-means algorithm. We present various modifications of the basic criterion and describe the corresponding MSP algorithm. It is shown that the method can also be used for solving optimality problems in classical statistics (maximizing Csiszar’s $\phi$ -divergence) and for simultaneous classification of the rows and columns of a contingency table.

The committee election problem is to choose from a finite set S of candidates a nonempty subset of committee members as the consequence of an election in which each voter expresses a preference for a candidate in S. We use ideas of vote concentration to formulate families of committee election rules, which may exhibit several natural, intuitively appealing properties. One concept of vote concentration, a typification of committee strength, ensures that the associated committee election rule is strategy-proof and so is not subject to voter manipulation.

Two-mode clustering methods provide a simultaneous clustering of the rows and columns of a rectangular data matrix. By the simultaneity of the methods they may reveal the value, interaction or dependence information as implied by the data. Keywords: rectangular data matrix; simultaneous clustering; coupled clustering; two-way; two-mode

A committee election rule is proposed that is based on a measure of vote concentration. Relationships are established between properties of the committee selection function, and properties of the function used to measure vote concentration.

The committee election problem is to choose from a finite set S of candidates a nonempty subset T of committee members as the consequence of an election in which each voter expresses a preference for a candidate in S. Solutions of this problem can be modelled by functions which map each partition of 1 (i.e., normalized vote tallies of candidates who have been ordered canonically by tally) into a nonempty subset of positive integers (i.e., sizes of committees). To solve this problem, we recently described a parameterized voting scheme, the ratio-of-sums or ras"p consensus rule, in which p controls the degree to which votes must be concentrated in elected committees. It is desirable to identify the attainable results of such rules so as to understand their properties and to facilitate their comparison. For all p, we characterize the attainable ras"p results in the general case where the partition's parts are real, and in the special case where p as well as its parts are rational.

The committee election problem is to choose from a finite set S of candidates a nonempty subset T of committee members as the consequence of an election in which each voter expresses a preference for a candidate in S . Consensus rules for electing committees can be modeled as optimization problems based on criteria involving distances, angles or ratios of quantities derived from the election tallies. We describe three such consensus rules and discuss their features and interrelationships.

This paper considers the problem of clustering n observed time series \(\mathbf{x}_{k} =\{\ x_{k}(t)\ \vert \ t \in \mathcal{T}\}\), k = 1, …, n, with time points t in a suitable time range \(\mathcal{T}\), into a suitable number m of clusters \(C_{1},\ldots,C_{m} \subset \{ 1,\ldots,n\}\) each one comprising time series with a ‘similar’ structure. Classical approaches might typically proceed by first computing a dissimilarity matrix and then applying a traditional, possibly hierarchical clustering method. In contrast, here we will present a brief survey about various approaches that start by defining probabilistic clustering models for the time series, i.e., with class-specific distribution models, and then determine a suitable (hopefully optimum) clustering by statistical tools like maximum likelihood and optimization algorithms. In particular, we will consider models with class-specific Gaussian processes and Markov chains.

This paper discusses cluster analysis in a probabilistic and inferential framework as opposed to more exploratory, heuristic or algorithmic approaches. It presents a broad survey on probabilistic models for partition-type, hierarchical and tree-like clustering structures and points to the relevant literature. It is shown how suitable clustering criteria or grouping methods may be derived from these models in the case of vector-valued data, dissimilarity matrices and similarity relations. In particular, we discuss hypothesis testing for homogeneity or for a grouping structure, the asymptotic distribution of test statistics, the use of random graph theory and combinatorial methods for simulating random dendrograms. Our presentation of hierarchies includes, e.g., Markovian branching processes and phylogenetic inference based on molecular sequence data.

Classical data analysis considers data vectors with real-valued or categorical components. In contrast, Symbolic Data Analysis (SDA) deals with data vectors whose components are intervals, sets of categories, or even frequency distributions. SDA generalizes common methods of multivariate statistics to the case of symbolic data tables. This paper presents a brief survey on basic problems and methods of this fast-developing branch of data analysis. As an alternative to the current more or less heuristic approaches, we propose a new probabilistic approach in this context. Our presentation concentrates on visualization, dissimilarities, and partition-type clustering for symbolic data.

This paper surveys some historical issues related to the well-known k-means algorithm in cluster analysis. It shows to which authors the different versions of this algorithm can be traced back, and which were the underlying applications. We sketch various generalizations (with references also to Diday’s work) and thereby underline the usefulness of the k-means approach in data analysis.

Two-mode clustering methods provide a simultaneous clustering of the rows and columns of a rectangular data matrix. By the simultaneity of the methods they may reveal the value, interaction or dependence information as implied by the data. Keywords: rectangular data matrix; simultaneous clustering; coupled clustering; two-way; two-mode

In this paper we present a structured overview of methods for two-mode clustering, that is, methods that provide a simultaneous clustering of the rows and columns of a rectangular data matrix. Key structuring principles include the nature of row, column and data clusters and the type of model structure or associated loss function. We illustrate with analyses of symptom data on archetypal psychiatric patients. Onderzoeksgroep Kwantitatieve- en persoonlijkheidspsychologie. Afdeling Ontwikkelings- en differentiële psychologie. Onderzoeksgroep Hogere cognitie en individuele verschillen. Afdeling Psychodiagnostiek en psychologische begeleiding. ispartof: Statistical Methods in Medical Research vol:13 issue:5 pages:363-394 ispartof: location:England status: published

We consider the simultaneous clustering of the rows and columns of a contingency table such that the dependence between row clusters and column clusters is maximized in the sense of maximizing a general dependence measure. We use Csiszar’s o-divergence between the given two-way distribution and the independence case with the same margins. This includes the classical χ2 measure, Kullback-Leibler’s discriminating information, and variation distance. By using the general theory of ‘convexity-based clustering criteria’ (Bock 1992, 2002a, 2002b) we derive a k-means-like clustering algorithm that uses ‘maximum support-plane partitions’ (in terms of likelihood ratio vectors) in the same way as classical SSQ clustering uses ‘minimum-distance partitions’.

In the present chapter we propose an extension of the standard principal component analysis method which takes as input a symbolic data matrix \(\underline X = ({\xi _{ij}})\) of interval type (Chouakria 1994, 1995, Cazes 1997; see section 3.2). Each ‘value’ \({\xi _{ij}}\) is an interval containing all the possible values of the feature Y j for an object i ∈ E (or i ∈ Ω). Instead of representing each object i and its description x i by a single point on a factorial plane in \({\mathbb{R}^2}\) (or \({\mathbb{R}^s}\)) as in classical principal component analysis (PCA), the proposed method visualizes each object i by a rectangle in \({\mathbb{R}^2}\) . Whereas the classical PCA is briefly sketched in section 9.1, we describe our generalized method in section 9.2. Thereby, we present a typical example concerning oils and fats in order to illustrate the effectiveness of the proposed symbolic PCA method.

One of the most common tasks in (classical as well as symbolic) data analysis is the detection and construction of ‘homogeneous’ groups C1, C2,… of objects in a population Ω or E such that objects from the same group show a high similarity whereas objects from different groups are typically more dissimilar. Such groups are usually called ‘clusters’ and must be constructed on the basis of the (classical or symbolic) data which were recorded for the objects.

The classical methods of statistical data analysis were designed for a relatively simple situation. First, the data were obtained for single individuals (persons, objects, products,…) which were the basic entities of the data gathering process by using interviews, experiments, archives, etc. Second, the recorded data concerned a list of one or several well defined variables and, third, these variables were single-valued insofar as the observation of each variable for a given individual resulted in just one single ‘value’ or ‘category’ such as in the statements: ‘the height of a person is 170 cm’, ‘the colour of a car is red’ etc. Depending on the situation, these variables were classified into quantitative (continuous or discrete) and qualitative (ordinal or nominal) ones.