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4.3. Selection-based self-refining φ (NMI) 1 0.5 0 E λref λ c 2 λ c 5 λ c 10 φ (NMI) HGPA 1 0.5 0 (c) HGPA φ (NMI) 1 0.5 0 E E λref λ c 2 λ c 5 λ c 10 λ c 15 λ c 15 λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 CSPA (a) CSPA λref λ c 2 λ c 5 λ c 10 φ (NMI) KMSAD 1 0.5 0 E (f) KMSAD λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 λref λ c 2 λ c 5 λ c 10 λ c 15 λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 φ (NMI) MCLA 1 0.5 0 (d) MCLA φ (NMI) 1 0.5 0 E E λref λ c 2 λ c 5 λ c 10 λ c 15 λ c 15 λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 λref λ c 2 λ c 5 λ c 10 EAC (b) EAC φ (NMI) SLSAD 1 0.5 0 E (g) SLSAD λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 λref λ c 2 λ c 5 λ c 10 λ c 15 λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 ALSAD (e) ALSAD λ c 15 λ c 20 λ c 30 λ c 40 λ c 50 λ c 60 λ c 75 λ c 90 Figure 4.3: φ (NMI) boxplots of the cluster ensemble E, the selected cluster ensemble component λref and the self-refined consensus clustering solutions λc p i on the Zoo data collection across all the consensus functions employed. The green dashed vertical line identifies the clustering solution selected by the supraconsensus function in each experiment. ing sections present a separate study of the results yielded by the self-refining procedure itself and the supraconsensus function that, a posteriori, must select the best self-refining consensus clustering solution. 4.3.1 Evaluation of the selection-based self-refining process As far as the evaluation of the selection-based consensus self-refining procedure is concerned, four analysis have been conducted. For starters, we have measured the percentage of experiments in which the procedure of self-refining yields a better quality clustering than the cluster ensemble component selected as a reference (i.e. the one maximizing its φ (ANMI) with respect to the cluster ensemble E, referred to as λref). The results, averaged across all the unimodal data collections employed in this work, are presented in table 4.12 as a function of the consensus function employed. In average, the self-refining procedure, when conducted on select cluster ensembles created upon the selection of λref, yields better clustering solutions in a 56% of the conducted experiments. This figure is notably lower than what was obtained when the select cluster ensemble 124
Chapter 4. Self-refining consensus architectures Consensus function CSPA EAC HGPA MCLA ALSAD KMSAD SLSAD 54.5 28.2 10.3 69.6 81.8 82 65.4 Table 4.12: Percentage of self-refining experiments in which one of the self-refined consensus clustering solutions is better than the selected cluster ensemble component reference λref, averaged across the twelve data collections. Consensus function CSPA EAC HGPA MCLA ALSAD KMSAD SLSAD 26.9 9.1 20.8 15.2 15.5 11.4 7.8 Table 4.13: Relative φ (NMI) gain percentage between the top quality self-refined consensus clustering solutions with respect to the maximum φ (ANMI) cluster ensemble component, averaged across the twelve data collections. creation uses a previously derived consensus clustering solution (it was 83%). This is due to the fact that the cluster ensemble component selection usually results in using a reference clustering λref of higher quality than the consensus clustering solution λc. Secondly, in those experiments where self-refined consensus solutions are better than λref, we have measured the relative degree of improvement achieved (quantified in terms of relative percentage φ (NMI) increase). The results, presented in table 4.13, show notable quality improvements, averaging a 15.2% relative φ (NMI) gain across all data sets and consensus functions. These quality gains obtained are much smaller than those obtained on the self-refining experiments based on a previously derived consensus clustering solution (see section 4.2), again due to the superior quality of the reference clustering the self-refining procedure is based upon. Next, the maximum and median φ (NMI) components of the cluster ensemble E –referred to as BEC (best ensemble component) and MEC (median ensemble component), respectively– are compared to either the top quality self-refined consensus clustering solution or λref, (depending on which has the largest φ (NMI) with respect to the ground truth). As in the previous section, we have evaluated i) the percentage of experiments where the maximum φ (NMI) consensus clustering solution attains a higher quality to that of the BEC and MEC, and ii) the relative percentage φ (NMI) variation between them and the top quality consensus clustering solution. Once more, all the results presented correspond to an average across all the experiments conducted on the twelve unimodal data collections. On one hand, table 4.14 presents the aforementioned magnitudes referred to the best cluster ensemble component. In average, the highest quality clustering (either λref or one of the self-refined consensus solutions) is better that the BEC in a 14.1% of the conducted experiments, achieving an average relative percentage φ (NMI) gain of 1.6%. It is important to notice that these results are pretty similar to those obtained when self-refining is based on a previously derived consensus clustering solution (see section 4.2), as these percentages were equal to 10.6% and 1.8%, respectively. On the other hand, table 4.15 presents the results of the same experiment, but referred to the median ensemble component (or MEC). In this case, the selection and self-refining procedure yields clusterings better than the MEC in 98% of the occasions, attaining an aver- 125
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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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3.3. Deterministic hierarchical con
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6.3. Voting based consensus functio
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6.4. Experiments λc = 1 1 1 3 3 3
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6.4. Experiments Data set Soft clus
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6.4. Experiments CSPA EAC HGPA MCLA
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6.5. Discussion solutions on hard c
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Chapter 7 Conclusions The contribut
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Chapter 7. Conclusions of-the-art c
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Chapter 7. Conclusions Though put f
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Chapter 7. Conclusions clustering r
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Chapter 7. Conclusions hardened. Ou
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References Ben-Hur, A., D. Horn, H.
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References Deerwester, S., S.-T. Du
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References Fred, A. and A.K. Jain.
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References Ingaramo, D., D. Pinto,
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References Li, S.Z. and G. GuoDong.
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References Sebastiani, F. 2002. Mac
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References Topchy, A., A.K. Jain, a
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Appendix A Experimental setup A.1 T
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Appendix A. Experimental setup g. s
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A.2.1 Unimodal data sets Appendix A
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A.2.2 Multimodal data sets Appendix
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Appendix A. Experimental setup gene
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Appendix A. Experimental setup orig
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Appendix A. Experimental setup Data
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Appendix A. Experimental setup or N
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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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