Voting-Based Consensus of Data Partitions

Over the past few years, there has been a renewed interest in the consensus problem for ensembles of partitions. Recent work is primarily motivated by the developments in the area of combining multiple supervised learners. Unlike the consensus of supervised classifications, the consensus of data partitions is a challenging problem due to the lack of globally defined cluster labels and to the inherent difficulty of data clustering as an unsupervised learning problem. Moreover, the true number of clusters may be unknown. A fundamental goal of consensus methods for partitions is to obtain an optimal summary of an ensemble and to discover a cluster structure with accuracy and robustness exceeding those of the individual ensemble partitions. The quality of the consensus partitions highly depends on the ensemble generation mechanism and on the suitability of the consensus method for combining the generated ensemble. Typically, consensus methods derive an ensemble representation that is used as the basis for extracting the consensus partition. Most ensemble representations circumvent the labeling problem. On the other hand, voting-based methods establish direct parallels with consensus methods for supervised classifications, by seeking an optimal relabeling of the ensemble partitions and deriving an ensemble representation consisting of a central aggregated partition. An important element of the voting-based aggregation problem is the pairwise relabeling of an ensemble partition with respect to a representative partition of the ensemble, which is refered to here as the voting problem. The voting problem is commonly formulated as a weighted bipartite matching problem. In this dissertation, a general theoretical framework for the voting problem as a multi-response regression problem is proposed. The problem is formulated as seeking to estimate the uncertainties associated with the assignments of the objects to the representative clusters, given their assignments to the clusters of an ensemble partition. A new voting scheme, referred to as cumulative voting, is derived as a special instance of the proposed regression formulation corresponding to fitting a linear model by least squares estimation. The proposed formulation reveals the close relationships between the underlying loss functions of the cumulative voting and bipartite matching schemes. A useful feature of the proposed framework is that it can be applied to model substantial variability between partitions, such as a variable number of clusters. A general aggregation algorithm with variants corresponding to cumulative voting and bipartite matching is applied and a simulation-based analysis is presented to compare the suitability of each scheme to different ensemble generation mechanisms. The bipartite matching is found to be more suitable than cumulative voting for a particular generation model, whereby each ensemble partition is generated as a noisy permutation of an underlying labeling, according to a probability of error. For ensembles with a variable number of clusters, it is proposed that the aggregated partition be viewed as an estimated distributional representation of the ensemble, on the basis of which, a criterion may be defined to seek an optimally compressed consensus partition. The properties and features of the proposed cumulative voting scheme are studied. In particular, the relationship between cumulative voting and the well-known co-association matrix is highlighted. Furthermore, an adaptive aggregation algorithm that is suited for the cumulative voting scheme is proposed. The algorithm aims at selecting the initial reference partition and the aggregation sequence of the ensemble partitions the loss of mutual information associated with the aggregated partition is minimized. In order to subsequently extract the final consensus partition, an efficient agglomerative algorithm is developed. The algorithm merges the aggregated clusters such that the maximum amount of information is preserved. Furthermore, it allows the optimal number of consensus clusters to be estimated. An empirical study using several artificial and real-world datasets demonstrates that the proposed cumulative voting scheme leads to discovering substantially more accurate consensus partitions compared to bipartite matching, in the case of ensembles with a relatively large or a variable number of clusters. Compared to other recent consensus methods, the proposed method is found to be comparable with or better than the best performing methods. Moreover, accurate estimates of the true number of clusters are often achieved using cumulative voting, whereas consistently poor estimates are achieved based on bipartite matching. The empirical evidence demonstrates that the bipartite matching scheme is not suitable for these types of ensembles

Clustering based on weighted ensemble

The clustering is an ill-posed problem and it has been proven that there is no algorithm that would satisfy all the assumptions about good clustering. This is why numerous clustering algorithms exist, based on various theories and approaches, one of them being the well-known Kohonen’s self-organizing map (SOM). Unfortunately, after training the SOM there is no explicitly obtained information about clusters in the underlying data, so another technique for grouping SOM units has to be applied afterwards. In the thesis, a contribution towards a two-level clustering of the SOM is presented, employing principles of Gravitational Law. The proposed algorithm for gravitational clustering of the SOM (gSOM) is capable of discovering complex cluster shapes, not only limited to the spherical ones, and is able to automatically determine the number of clusters. Experimental comparison with other clustering techniques is conducted on synthetic and real-world data. We show that gSOM achieves promising results especially on gene-expression data. As there is no clustering algorithm that can solve all the problems, it turns out as very beneficial to analyse the data using multiple partitions of them – an ensemble of partitions. Cluster-ensemble methods have emerged recently as an effective approach to stabilize and boost the performance of the single-clustering algorithms. Basically, data clustering with an ensemble involves two steps: generation of the ensemble with single-clustering methods and the combination of the obtained solutions to produce a final consensus partition of the data. To alleviate the consensus step the weighted cluster ensemble was proposed that tries to assess the relevance of ensemble members. One way to achieve this is to employ internal cluster validity indices to perform partition relevance analysis (PRA). Our contribution here is two-fold: first, we propose a novel cluster validity index DNs that extends the Dunn’s index and is based on the shortest paths between the data points considering the Gabriel graph on the data; second, we propose an enhancement to the weighted cluster ensemble approach by introducing the reduction step after the assessment of the ensemble partitions is done. The developed partition relevance analysis with the reduction step (PRAr) yields promising results when plugged in the three consensus functions, based on the evidence accumulation principle. In the thesis we address all the major stages of data clustering: data generation, data analysis using single-clustering algorithms, cluster validity using internal end external indices, and finally the cluster ensemble approach with the focus on the weighted variants. All the contributions are compared to the state-of-art methods using datasets from various problem domains. Results are positive and encourage the inclusion of the proposed algorithms in the machine-learning practitioner’s toolbox

Clustering based on weighted ensemble

The clustering is an ill-posed problem and it has been proven that there is no algorithm that would satisfy all the assumptions about good clustering. This is why numerous clustering algorithms exist, based on various theories and approaches, one of them being the well-known Kohonen’s self-organizing map (SOM). Unfortunately, after training the SOM there is no explicitly obtained information about clusters in the underlying data, so another technique for grouping SOM units has to be applied afterwards. In the thesis, a contribution towards a two-level clustering of the SOM is presented, employing principles of Gravitational Law. The proposed algorithm for gravitational clustering of the SOM (gSOM) is capable of discovering complex cluster shapes, not only limited to the spherical ones, and is able to automatically determine the number of clusters. Experimental comparison with other clustering techniques is conducted on synthetic and real-world data. We show that gSOM achieves promising results especially on gene-expression data. As there is no clustering algorithm that can solve all the problems, it turns out as very beneficial to analyse the data using multiple partitions of them – an ensemble of partitions. Cluster-ensemble methods have emerged recently as an effective approach to stabilize and boost the performance of the single-clustering algorithms. Basically, data clustering with an ensemble involves two steps: generation of the ensemble with single-clustering methods and the combination of the obtained solutions to produce a final consensus partition of the data. To alleviate the consensus step the weighted cluster ensemble was proposed that tries to assess the relevance of ensemble members. One way to achieve this is to employ internal cluster validity indices to perform partition relevance analysis (PRA). Our contribution here is two-fold: first, we propose a novel cluster validity index DNs that extends the Dunn’s index and is based on the shortest paths between the data points considering the Gabriel graph on the data; second, we propose an enhancement to the weighted cluster ensemble approach by introducing the reduction step after the assessment of the ensemble partitions is done. The developed partition relevance analysis with the reduction step (PRAr) yields promising results when plugged in the three consensus functions, based on the evidence accumulation principle. In the thesis we address all the major stages of data clustering: data generation, data analysis using single-clustering algorithms, cluster validity using internal end external indices, and finally the cluster ensemble approach with the focus on the weighted variants. All the contributions are compared to the state-of-art methods using datasets from various problem domains. Results are positive and encourage the inclusion of the proposed algorithms in the machine-learning practitioner’s toolbox

A Method to Improve the Analysis of Cluster Ensembles

Clustering is fundamental to understand the structure of data. In the past decade the cluster ensembleproblem has been introduced, which combines a set of partitions (an ensemble) of the data to obtain a singleconsensus solution that outperforms all the ensemble members. However, there is disagreement about which arethe best ensemble characteristics to obtain a good performance: some authors have suggested that highly differentpartitions within the ensemble are beneï¬ cial for the ï¬ nal performance, whereas others have stated that mediumdiversity among them is better. While there are several measures to quantify the diversity, a better method toanalyze the best ensemble characteristics is necessary. This paper introduces a new ensemble generation strategyand a method to make slight changes in its structure. Experimental results on six datasets suggest that this isan important step towards a more systematic approach to analyze the impact of the ensemble characteristics onthe overall consensus performance.Fil: Pividori, Milton Damián. Universidad Tecnologica Nacional. Facultad Regional Santa Fe. Centro de Investigacion y Desarrollo de Ingenieria en Sistemas de Informacion; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional. Universidad Nacional del Litoral. Facultad de Ingeniería y Ciencias Hídricas. Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional; ArgentinaFil: Stegmayer, Georgina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional. Universidad Nacional del Litoral. Facultad de Ingeniería y Ciencias Hídricas. Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional; Argentina. Universidad Tecnologica Nacional. Facultad Regional Santa Fe. Centro de Investigacion y Desarrollo de Ingenieria en Sistemas de Informacion; ArgentinaFil: Milone, Diego Humberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional. Universidad Nacional del Litoral. Facultad de Ingeniería y Ciencias Hídricas. Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional; Argentin

Combining Multiple Clusterings via Crowd Agreement Estimation and Multi-Granularity Link Analysis

The clustering ensemble technique aims to combine multiple clusterings into a probably better and more robust clustering and has been receiving an increasing attention in recent years. There are mainly two aspects of limitations in the existing clustering ensemble approaches. Firstly, many approaches lack the ability to weight the base clusterings without access to the original data and can be affected significantly by the low-quality, or even ill clusterings. Secondly, they generally focus on the instance level or cluster level in the ensemble system and fail to integrate multi-granularity cues into a unified model. To address these two limitations, this paper proposes to solve the clustering ensemble problem via crowd agreement estimation and multi-granularity link analysis. We present the normalized crowd agreement index (NCAI) to evaluate the quality of base clusterings in an unsupervised manner and thus weight the base clusterings in accordance with their clustering validity. To explore the relationship between clusters, the source aware connected triple (SACT) similarity is introduced with regard to their common neighbors and the source reliability. Based on NCAI and multi-granularity information collected among base clusterings, clusters, and data instances, we further propose two novel consensus functions, termed weighted evidence accumulation clustering (WEAC) and graph partitioning with multi-granularity link analysis (GP-MGLA) respectively. The experiments are conducted on eight real-world datasets. The experimental results demonstrate the effectiveness and robustness of the proposed methods.Comment: The MATLAB source code of this work is available at: https://www.researchgate.net/publication/28197031

Paradigm of tunable clustering using binarization of consensus partition matrices (Bi-CoPaM) for gene discovery

Copyright @ 2013 Abu-Jamous et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.Clustering analysis has a growing role in the study of co-expressed genes for gene discovery. Conventional binary and fuzzy clustering do not embrace the biological reality that some genes may be irrelevant for a problem and not be assigned to a cluster, while other genes may participate in several biological functions and should simultaneously belong to multiple clusters. Also, these algorithms cannot generate tight clusters that focus on their cores or wide clusters that overlap and contain all possibly relevant genes. In this paper, a new clustering paradigm is proposed. In this paradigm, all three eventualities of a gene being exclusively assigned to a single cluster, being assigned to multiple clusters, and being not assigned to any cluster are possible. These possibilities are realised through the primary novelty of the introduction of tunable binarization techniques. Results from multiple clustering experiments are aggregated to generate one fuzzy consensus partition matrix (CoPaM), which is then binarized to obtain the final binary partitions. This is referred to as Binarization of Consensus Partition Matrices (Bi-CoPaM). The method has been tested with a set of synthetic datasets and a set of five real yeast cell-cycle datasets. The results demonstrate its validity in generating relevant tight, wide, and complementary clusters that can meet requirements of different gene discovery studies.National Institute for Health Researc

Yeast gene CMR1/YDL156W is consistently co-expressed with genes participating in DNA-metabolic processes in a variety of stringent clustering experiments

© 2013 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0/, which permits unrestricted use, provided the original author and source are credited.The binarization of consensus partition matrices (Bi-CoPaM) method has, among its unique features, the ability to perform ensemble clustering over the same set of genes from multiple microarray datasets by using various clustering methods in order to generate tunable tight clusters. Therefore, we have used the Bi-CoPaM method to the most synchronized 500 cell-cycle-regulated yeast genes from different microarray datasets to produce four tight, specific and exclusive clusters of co-expressed genes. We found 19 genes formed the tightest of the four clusters and this included the gene CMR1/YDL156W, which was an uncharacterized gene at the time of our investigations. Two very recent proteomic and biochemical studies have independently revealed many facets of CMR1 protein, although the precise functions of the protein remain to be elucidated. Our computational results complement these biological results and add more evidence to their recent findings of CMR1 as potentially participating in many of the DNA-metabolism processes such as replication, repair and transcription. Interestingly, our results demonstrate the close co-expressions of CMR1 and the replication protein A (RPA), the cohesion complex and the DNA polymerases α, δ and ɛ, as well as suggest functional relationships between CMR1 and the respective proteins. In addition, the analysis provides further substantial evidence that the expression of the CMR1 gene could be regulated by the MBF complex. In summary, the application of a novel analytic technique in large biological datasets has provided supporting evidence for a gene of previously unknown function, further hypotheses to test, and a more general demonstration of the value of sophisticated methods to explore new large datasets now so readily generated in biological experiments.National Institute for Health Researc