TESI DOCTORAL - La Salle

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3.4. Flat vs. hierarchical consensus of the consensus function employed, and that such differences are, in all cases, statistically significant. As far as the hierarchical consensus architectures are concerned, notice that the fastest DHCA variant (DRA) is more computationally efficient than its RHCA counterpart (which has s = 2 stages and mini-ensembles of size b = 28), except when consensus processes are conducted by means of the EAC and SLSAD consensus functions —in this case, statistically equivalent running times are attained by both HCA. If these results are contrasted with the predicted computationally optimal consensus architectures presented in tables 3.4 and 3.10, a single prediction error is detected (flat consensus turns out to be faster than the DRA DHCA variant when consensus is conducted by MCLA), which reinforces the notion that the proposed computationally optimal consensus architecture prediction methodology performs pretty well. Figure 3.15(b) presents the running times of the fully parallel optimal hierarhical consensus architectures and flat consensus. As in the serial case, it can be noticed that flat consensus tends to be more efficient than RHCA and DHCA. The only exception occurs when consensus is conducted by means of the MCLA consensus function —which is due to the fact that it is the only combiner the computational complexity of which increases quadratically with the size of the cluster ensemble. <strong>La</strong>st but not least, it is to note that, as opposed to what was observed in the serial implementation, the fastest RHCA is less time consuming than the most efficient DHCA variant, and the running time differences between them are statistically significant. The reason for this lies in the fact that this specific RHCA variant has s = 2 stages and consensus is conducted on mini-ensembles of size b = 7, whereas the DHCA variant consists of three consensus stages, in one of which consensus are built on larger mini-ensembles of size |dfD| = 14, which is responsible for the higher computational cost of parallel DHCA in this case. Diversity scenario |df A| =10 The results corresponding to the experiments conducted in the second diversity scenario (i.e. cluster ensembles generated by the compilation of the clusterings output by |dfA| =10 randomly selected clustering algorithms, giving rise to cluster ensembles of size l = 570) are presented in figure 3.16. In particular, figure 3.16(a) depicts the execution time boxplots of the serial implementation of hierarchical consensus architectures. The first noticeable situation is that, in contrast to what was observed in the lowest diversity scenario, the computationally optimal RHCA variant (s =2andb =20orb = 285 depending on the consensus function employed) is faster than its DHCA counterpart (DAR). This is due to the fact that the rise of the algorithmic diversity factor (from |dfA| =1to|dfA| = 10) entails an increase of the computational cost of one of the DHCA stages that exceeds the increment of the complexity of the RHCA caused by the same factor. Meanwhile, regarding the computational efficiency of flat consensus, two opposed behaviours are observed depending on the consensus function employed: while being faster than any hierarchical architecture when consensus are built using the CSPA and EAC consensus functions, one-step consensus is slower when the remaining clustering combiners are employed. Moreover, the differences between the running times of these consensus architectures is statistically significant at the 5% significance level in all cases. Figure 3.16(b) presents the results corresponding to the fully parallel implementation of consensus architectures. In this case, flat consensus is at least four times more computationally costly than the DHCA and RHCA variants. Moreover, the optimal RHCA 90
CPU time (sec.) CPU time (sec.) 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.5 1 0.5 CSPA RHCA DHCA flat CSPA RHCA DHCA flat CPU time (sec.) CPU time (sec.) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.25 0.2 0.15 0.1 0.05 EAC RHCA DHCA flat EAC RHCA DHCA flat CPU time (sec.) CPU time (sec.) 3.4 3.2 3 2.8 2.6 2.4 2.2 HGPA RHCA DHCA flat CPU time (sec.) Chapter 3. Hierarchical consensus architectures 25 20 15 10 5 MCLA RHCA DHCA flat CPU time (sec.) 2 1.5 1 0.5 ALSAD RHCA DHCA flat (a) Serial implementation running time 2.5 2 1.5 1 0.5 0 HGPA RHCA DHCA flat CPU time (sec.) 25 20 15 10 5 0 MCLA RHCA DHCA flat CPU time (sec.) 2 1.5 1 0.5 0 ALSAD RHCA DHCA flat (b) Parallel implementation running time CPU time (sec.) CPU time (sec.) 2.5 2 1.5 1 2.5 2 1.5 1 0.5 0 KMSAD RHCA DHCA flat KMSAD RHCA DHCA flat CPU time (sec.) CPU time (sec.) 2 1.5 1 0.5 2 1.5 1 0.5 0 SLSAD RHCA DHCA flat SLSAD RHCA DHCA flat Figure 3.16: Running times of the computationally optimal RHCA, DHCA and flat consensus architectures on the Zoo data collection for the diversity scenario corresponding to a cluster ensemble of size l = 570. is faster than its DHCA counterpart (except for the MCLA consensus function), although they both attain very similar execution times, their differences being statistically significant little below the 5% significance level. Diversity scenario |df A| =19 The running times of the consensus architectures corresponding to the experiments conducted on the third diversity scenario (i.e. cluster ensembles of size l = 1083) are presented. Figure 3.17(a) depicts the execution time boxplots of the serially implemented consensus architectures. In most cases, hierarchical consensus architectures are faster than their flat counterpart —the only exception occurs when consensus are built using the EAC consensus function, a trend that was already observed in sections 3.2 and 3.3. Notice, moreover, that flat consensus is not executable when MCLA is the consensus function employed for creating the consensus clustering solutions. When the entirely parallel implementation of hierarchical consensus architectures is evaluated from a computational viewpoint, the optimal RHCA and DHCA variants (see tables 91
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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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5.3. Multimodal consensus clusterin
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5.4. Discussion Data set Relative
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5.5. Related publications modes joi
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Chapter 6. Voting based consensus f
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6.2. Adapting consensus functions t
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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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