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3.2. Random hierarchical consensus architectures SERT RHCA (sec.) PERT RHCA (sec.) 10 2 10 1 10 0 10 1 10 0 10 −1 s : number of stages 10 7 6 5 4 4 3 3 2 2 1 2 3 4 5 6 10 11 32 33 798 1596 b : mini−ensemble size (a) Serial estimated running time s : number of stages 10 7 6 5 4 4 3 3 2 2 1 2 3 4 5 6 10 11 32 33 798 1596 b : mini−ensemble size (c) Parallel estimated running time CSPA EAC HGPA MCLA ALSAD KMSAD SLSAD CSPA EAC HGPA MCLA ALSAD KMSAD SLSAD SRT RHCA (sec.) PRT RHCA (sec.) 10 2 10 1 10 0 10 1 10 0 10 −1 s : number of stages 10 7 6 5 4 4 3 3 2 2 1 2 3 4 5 6 10 11 32 33 798 1596 b : mini−ensemble size (b) Serial real running time s : number of stages 10 7 6 5 4 4 3 3 2 2 1 2 3 4 5 6 10 11 32 33 798 1596 b : mini−ensemble size (d) Parallel real running time CSPA EAC HGPA MCLA ALSAD KMSAD SLSAD CSPA EAC HGPA MCLA ALSAD KMSAD SLSAD Figure 3.7: Estimated and real running times of the serial RHCA on the Zoo data collection in the diversity scenario corresponding to a cluster ensemble of size l = 1596. Conclusions regarding the computationally efficiency of RHCA The observation of the results obtained across the four diversity scenarios allow drawing several conclusions (together with the experiments presented in appendix C.2) as regards the computational efficiency of hierarchical and flat consensus architectures: – hierarchical consensus architectures can constitute a feasible way to obtain a consensus clustering solution in cases where one-step consensus is not affordable, which ultimately depends on the size of the cluster ensemble, the characteristics of the consensus function and the computational resources at hand. – as expected, parallel RHCA are highly efficient, being faster than flat consensus even in low diversity scenarios. – serial RHCA implementations become computationally competitive in medium to high diversity scenarios. – depending on the characteristics of the consensus function(s) employed for conducting clustering combination, large variations of the overall execution time of consensus architectures are observed. 62
Chapter 3. Hierarchical consensus architectures Serial RHCA Parallel RHCA Dataset % correct ΔRT % correct ΔRT predictions (sec.) (%) predictions (sec.) (%) Zoo 72.2 1.11 109.7 54.9 0.10 67.2 Iris 90.4 0.05 26.1 56.7 0.12 102.7 Wine 77.1 0.60 37.8 46.8 0.21 139.5 Glass 74.6 0.49 26.5 25.9 0.26 97.3 Ionosphere 73.1 2.63 16.6 67.8 0.77 110.5 WDBC 63.0 12.11 39.1 38.6 8.17 113.9 Balance 92.4 0.31 29.5 73.2 3.09 87.3 MFeat 83.4 7.02 27.7 76.3 14.41 50.3 average 78.3 3.04 39.1 55.0 3.39 96.1 Table 3.3: Evaluation of the minimum complexity RHCA variant estimation methodology in terms of the percentage of correct predictions and running time penalizations resulting from mistaken predictions. Conclusions regarding the optimal RHCA prediction methodology As far as the proposed running time estimation methodology is concerned, the following conclusions are drawn: – the computation of SERTRHCA and PERTRHCA constitutes a reasonable, simple and fast means for predicting whether flat or hierarchical consensus must be conducted. – the selection of computationally suboptimal consensus architectures caused by prediction errors of the proposed methodology entails a (usually assumable) execution time overhead. In order to provide the reader with a more quantitative analysis of the predictive power of the proposed running time estimation methodology, we have computed the percentage of experiments –considering the eight data sets over which they have been conducted– the minimum value of the estimated and real running times is obtained for the same consensus architecture. If, for a given experiment, both functions are simultaneously minimized, then our methodology suceeds in determining aprioriwhich is the fastest consensus architecture. If not, we compute the difference between the real running times of the truly (i.e. the one that minimizes SRTRHCA or PRTRHCA) and the allegedly (that is, the one minimizing SERTRHCA or PERTRHCA) computationally optimal consensus architectures, so as to provide a measure of the impact of choosing a suboptimal consensus configuration both in absolute and relative terms. Table 3.3 presents the percentage of experiments where the minima of SERTRHCA and PERTRHCA predict the most efficient consensus architecture correctly (expressed as ‘% correct predictions’). In this case, SERTRHCA and PERTRHCA have been estimated upon a single execution (c = 1) of a consensus process on the mini-ensembles of size bij, i.e. no statistical running time averaging is conducted. Moreover, the difference between the real running times (ΔRT, measured both in seconds and in relative percentage) of the truly and the allegedly computationally optimal consensus architectures is also presented, 63
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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 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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φ (NMI) φ (NMI) φ (NMI) φ (NMI)
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Chapter 7 Conclusions The contribut
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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 Data
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E.1. CAL500 data set φ (NMI) 1 0.8
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F.11. PenDigits data set φ (NMI) 1
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