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7.4. Voting based soft consensus functions 7.4 Voting based soft consensus functions The outcome of a fuzzy clustering process is much more informative than its crisp counterpart, as it indicates the strength of association of each object to each cluster. Despite this fact, soft clustering combination strategies are a minority in the consensus clustering literature. Allowing for this, we have made several proposals in this area, aiming to extend all our previous proposals to the more generic framework of soft clustering. There exists a pretty evident parallelism between the strength of association of each object to each cluster in a fuzzy clustering solution and the degree of preference of a voter for a candidate in an election. This fact directly allows the application of certain voting methods for consolidating soft clusterings, considering the clusters as the candidates, the cluster ensemble components as voters, and the clusterization of each object as an election. However, given the ambiguous identification of clusters inherent to clustering, a cluster alignment between the cluster ensemble components is required prior to voting. In this work, we have proposed four consensus functions for soft cluster ensembles, which are the result of applying as many voting strategies for combining the clusterings in the ensemble. In particular, we have employed two confidence voting methods –the sum and product rules, which give rise to the SumConsensus (SC) and ProductConsensus (PC) consensus functions–, and two positional voting techniques —the Borda and Condorcet voting strategies that constitute the basis of the BordaConsensus (BC) and CondorcetConsensus (CC) clustering combiners. The main difference between these two families of voting methods lies in the fact that the former operate directly on the object-to-cluster association values that make up the cluster ensemble components, whereas the latter operate on the candidates ranking according to the voters’ preferences. For disambiguating the clusters, we have employed the classic Hungarian algorithm (Kuhn, 1955). The experiments conducted have evaluated our four consensus functions (SC, PC, BC and CC), comparing them with several state-of-the-art soft consensus functions in terms of their computational complexity and the quality of the consensus clusterings they yield. In terms of execution time, confidence voting consensus functions are faster than their positional voting counterparts, as the candidate ranking process penalizes the latter from a computational standpoint. In this sense, CC is the slowest proposal, due to the exhaustive pairwise candidate confrontation implicit in the Condorcet voting method. Contrarily, the more computationally efficient PC and SC consensus functions are as fast or faster than CSPA, EAC, HGPA and MCLA in a 81% of the experiments conducted —however, they are slower than VMA in a 92% of the cases. If the quality of the hardened version of the fuzzy consensus clusterings (measured in terms of φ (NMI) with respect to the ground truth) is used as the comparison factor, we observe that the four proposed consensus functions yield (statistically significantly) better results than any of the state-of-the-art consensus functions in an average 72% of the experiments conducted, which is a clear indicator of the goodness of our proposals. It is important to highlight that it has been impossible to evaluate directly the fuzzy consensus clusterings output by the four proposed consensus functions, due to the unavailability of soft labels in the data sets employed. As a future direction of research, we plan conducting this fuzzy evaluation, and we conjecture that greater differences between SC, PC, BC and CC will be observed, as the differences between the results of the voting strategies they are based upon are somewhat masked when the fuzzy consensus clusterings they yield are 200
Chapter 7. Conclusions hardened. Ours is not the first approach to soft consensus clustering based on voting. In fact, the VMA consensus function employs a weighted version of the sum voting rule (Dimitriadou, Weingessel, and Hornik, 2002). However, to our knowledge, BordaConsensus (firstly introduced in (Sevillano, Alías, and Socoró, 2007b)) and CondorcetConsensus are pioneer positional voting based consensus functions. As aforementioned, our proposals deal with clusterings with a constant number of clusters k, and it would be of paramount interest to adapt them to combine clusterings with different number of clusters. A possible way to do so would consist in completing those clusterings with fewer clusters with dummy clusters (Ayad and Kamel, 2008), as suggested in the VMA consensus function (Dimitriadou, Weingessel, and Hornik, 2002). Besides this, possibly the clearest direction for future research in this area consists of adapting the simultaneous cluster disambiguation and voting mechanism of VMA, which would probably i) reduce the time complexity of the proposed consensus functions, and ii) require introducing some adjustments to the voting methods employed. Moreover, we are also interested in exploring other existing techniques for solving the cluster disambiguation problem, analyzing their impact both in terms of the quality of the consensus clustering solutions obtained and the overall computational complexity of the consensus function. 201
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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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3.3. Deterministic hierarchical con
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3.4. Flat vs. hierarchical consensu
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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.
Page 245 and 246: References Ingaramo, D., D. Pinto,
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 257 and 258: A.2.1 Unimodal data sets Appendix A
Page 259 and 260: A.2.2 Multimodal data sets Appendix
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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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