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6.5. Discussion solutions on hard cluster ensembles (Dudoit and Fridlyand, 2003; Fischer and Buhmann, 2003; Greene and Cunningham, 2006). To our knowledge, the only voting-based consensus function for soft cluster ensembles is the Voting-Merging Algorithm (VMA) of (Dimitriadou, Weingessel, and Hornik, 2002), which employs a weighted version of the sum rule for confidence voting. Moreover, all these works share a common point in that they use the Hungarian algorithm for solving the cluster correspondence problem. Additional techniques for cluster disambiguation developed in the consensus clustering literature include the correspondence estimation based on common space cluster representation by clusters clustering or Singular Value Decomposition (Boulis and Ostendorf, 2004), the Soft Correspondence Ensemble Clustering algorithm, which is based on establishing a soft correspondence between clusters (in the sense that a cluster of a given clustering corresponds to every cluster in another clustering with different weights) (Long, Zhang, and Yu, 2005), the cumulative voting approach, that, unlike common one-to-one voting schemes (e.g. Hungarian), computes a probabilistic mapping between clusters (Ayad and Kamel, 2008), or the FullSearch, Greedy and <strong>La</strong>rgeKGreedy cluster alignment algorithms (Jakobsson and Rosenberg, 2007). The first two approaches coincide in that they can be indistinctly applied for aligning the clusters of crisp and fuzzy partitions. Given the key importance of the cluster disambiguation process as a prior step to voting, we plan to evaluate these alternatives to the Hungarian method, so as to investigate their impact on both the quality of the consensus clusterings obtained and the time complexity of the whole consensus process. The comparative performance analysis of the four proposed consensus functions has revealed that they constitute a feasible alternative for conducting consensus clustering processes on soft cluster ensembles, as they are capable of yielding consensus clustering solutions of comparable or superior quality to those obtained by state-of-the-art clustering combiners at a reasonable computational cost. An additional appealing feature of our proposals is that they naturally deliver fuzzy consensus clustering solutions, which makes all sense in a soft clustering scenario —a fact other recent consensus functions for soft cluster ensembles –as the one presented in (Punera and Ghosh, 2007)– do not consider. However, the lack of a fuzzy ground truth has not allowed evaluating the soft consensus clusterings obtained, which constitutes one of the future directions of research of the work conducted in this chapter. As mentioned earlier, this would probably make the differences between the proposed consensus functions more evident, as it would highlight the differences between the distinct voting methods employed. As reported earlier, the sequential application of the cluster disambiguation and the voting processes penalizes the time complexity of our proposals, specially when they are compared to VMA. Thus, in the future, we plan to adopt the iterative cluster alignment plus voting strategy employed by this consensus function, which, in our opinion, will surely reduce the execution time of the proposed voting-based consensus functions without significantly reducing the quality of the consensus functions obtained. Another significant conclusion is that the EAC and HGPA consensus functions yield the lowest quality consensus clusterings, as already noticed in the vast majority of the experiments conducted in the hard clustering scenario. 190
Chapter 6. Voting based consensus functions for soft cluster ensembles 6.6 Related publications Our first approach to voting based soft consensus functions was the derivation of Borda- Consensus (Sevillano, Alías, and Socoró, 2007b). The details of this publication, presented as a poster at the SIGIR 2007 conference held at Amsterdam, are described next. Authors: Xavier Sevillano, Francesc Alías and Joan Claudi Socoró Title: BordaConsensus: a New Consensus Function for Soft Cluster Ensembles In: Proceedings of the 30th ACM SIGIR Conference Pages: 743-744 Year: 2007 Abstract: Consensus clustering is the task of deriving a single labeling by applying a consensus function on a cluster ensemble. This work introduces BordaConsensus, a new consensus function for soft cluster ensembles based on the Borda voting scheme. In contrast to classic, hard consensus functions that operate on labelings, our proposal considers cluster membership information, thus being able to tackle multiclass clustering problems. Initial small scale experiments reveal that, compared to stateof-the-art consensus functions, BordaConsensus constitutes a good performance vs. complexity trade-off. 191
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C.I.F. G: 59069740 Universitat Ramo
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Resum En segmentar de forma no supe
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Abstract When facing the task of pa
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Contents 2.2.6 Consensus functions
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Contents A.5 Consensus functions .
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Contents D.2.5 WDBC data set . . .
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List of Tables 3.13 Relative percen
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List of Tables 5.15 Relative φ (NM
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List of Figures 1.1 Evolution of th
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List of Figures 4.2 Decreasingly or
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List of Figures C.18 Estimated and
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List of Figures C.49 φ (NMI) of th
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List of Figures D.22 φ (NMI) boxpl
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List of Algorithms 6.1 Symbolic des
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List of symbols OΛ: object co-asso
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Chapter 1. Framework of the thesis
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1.1. Knowledge discovery and data m
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1.1. Knowledge discovery and data m
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1.2. Clustering in knowledge discov
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1.3. Multimodal clustering in clust
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1.4. Clustering indeterminacies exe
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1.4. Clustering indeterminacies The
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1.4. Clustering indeterminacies Dat
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1.5. Motivation and contributions o
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Chapter 2. Cluster ensembles and co
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fuzzy consensus clustering solution
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2.1. Related work on cluster ensemb
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2.2. Related work on consensus func
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3.1. Motivation - the computational
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3.1. Motivation An additional and v
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3.2. Random hierarchical consensus
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3.5. Discussion Consensus Consensus
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3.5. Discussion separate clustering
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Chapter 4 Self-refining consensus a
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Chapter 4. Self-refining consensus
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- What do we want to measure? Chapt
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φ (NMI) φ (NMI) φ (NMI) φ (NMI)
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φ (NMI) 0.8 0.78 0.76 0.74 0.72 0
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1. Given a cluster ensemble E conta
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%of experiments relative % φ (NMI)
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Publisher: Springer Series: Lecture
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5.1. Generation of multimodal clust
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5.2. Self-refining multimodal conse
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5.3. Multimodal consensus clusterin
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Page 239 and 240: References Ben-Hur, A., D. Horn, H.
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Page 243 and 244: References Fred, A. and A.K. Jain.
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Page 247 and 248: References Li, S.Z. and G. GuoDong.
Page 249 and 250: References Sebastiani, F. 2002. Mac
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Page 253 and 254: Appendix A Experimental setup A.1 T
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Page 259 and 260: A.2.2 Multimodal data sets Appendix
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B.1. Clustering indeterminacies in
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C.1. Configuration of a random hier
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C.2. Estimation of the computationa
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C.4. Computationally optimal RHCA,
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D.1. Experiments on consensus-based
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D.2. Experiments on selection-based
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E.1. CAL500 data set φ (NMI) 1 0.8
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E.1. CAL500 data set φ (NMI) CSPA
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E.2. InternetAds data set φ (NMI)
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E.3. Corel data set φ (NMI) 1 0.8
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F.1. Iris data set φ (NMI) 1 0.8 0
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F.3. Glass data set CSPA EAC HGPA M
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F.5. WDBC data set φ (NMI) 1 0.8 0
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F.7. MFeat data set φ (NMI) 1 0.8
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F.9. Segmentation data set φ (NMI)
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F.10. BBC data set φ (NMI) 1 0.8 0
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F.11. PenDigits data set φ (NMI) 1
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