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3.4. Flat vs. hierarchical consensus 3.4 Flat vs. hierarchical consensus In sections 3.2 and 3.3, two specific implementations of hierarchical consensus architectures have been proposed, alongside a methodology for determining aprioriwhich is the fastest (random or deterministic) HCA variant, and deciding whether it is computationally advantageous with respect to classic flat consensus. In this section, we present a direct twofold comparison between flat consensus and those DHCA and RHCA variants deemed as the most computationally efficient by the proposed running time estimation methodologies. Firstly, we compare them in terms of computational complexity. In fact, such comparison could be made upon the results presented in sections 3.2 and 3.3, but we think that a comparison considering only the allegedly best performing variants may simplify the process of drawing meaningful conclusions. And secondly, these least time consuming hierarhical consensus architecture variants will be compared with flat consensus in terms of the quality of the consensus clustering solutions they yield. By doing so, we intend to present a comprehensive picture of our hierarchical consensus architecture proposals in terms of the two main factors that condition robust clustering by consensus: time complexity and quality. 3.4.1 Running time comparison This section compares the real execution times of the allegedly fastest DHCA and RHCA variants and flat consensus. The experiments conducted follow the design outlined next. Experimental design – What do we want to measure? The time complexity of the allegedly fastest DHCA and RHCA variants and flat consensus. – How do we measure it? We measure the CPU time required for the execution of the aforementioned consensus architectures. – How are the experiments designed? Such comparison entails the running times of ten independent runs of each one of the compared consensus architectures. So as to evaluate their computational efficiency under distinct experimental conditions, the consensus processes involved have been conducted by means of the seven consensus functions for hard cluster ensembles employed in this work —see appendix A.5. Moreover, experiments have been replicated on the four diversity scenarios described in appendix A.4 —recall that they differ in the algorithmic diversity factor, as a set of |dfA| = {1, 10, 19, 28} randomly chosen clustering algorithms are employed for creating the cluster ensemble in each diversity scenario. – How are results presented? In formal terms, the measured execution times are presented by means of boxplot charts, so as to provide the reader with a notion of the degree of dispersion and asymmetry of the running times of each consensus architecture. When comparing boxplots, notice that non-overlapping boxes notches indicate that the medians of the compared running times differ at the 5% significance level, which allows a quick inference of the statistical significance of the results. – Which data sets are employed? A detailed description of the results of this comparison on the Zoo data collection is presented in the following paragraphs. Recall 88
CPU time (sec.) CPU time (sec.) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.3 0.25 0.2 0.15 0.1 0.05 CSPA RHCA DHCA flat CSPA RHCA DHCA flat CPU time (sec.) CPU time (sec.) 0.2 0.15 0.1 0.05 0 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 EAC RHCA DHCA flat EAC RHCA DHCA flat CPU time (sec.) CPU time (sec.) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 HGPA RHCA DHCA flat CPU time (sec.) Chapter 3. Hierarchical consensus architectures 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 MCLA RHCA DHCA flat CPU time (sec.) 0.15 0.1 0.05 0 ALSAD RHCA DHCA (a) Serial implementation running time 0.3 0.25 0.2 0.15 0.1 0.05 HGPA RHCA DHCA flat CPU time (sec.) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 MCLA RHCA DHCA flat CPU time (sec.) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 flat ALSAD RHCA DHCA (b) Parallel implementation running time flat CPU time (sec.) CPU time (sec.) 0.5 0.4 0.3 0.2 0.1 0 0.2 0.15 0.1 0.05 KMSAD RHCA DHCA flat KMSAD RHCA DHCA flat CPU time (sec.) CPU time (sec.) 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.12 0.1 0.08 0.06 0.04 0.02 SLSAD RHCA DHCA flat SLSAD RHCA DHCA flat Figure 3.15: 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 = 57. that, the cardinalities of the dimensional and representational diversity factors of this fata set are |dfD| =14and|dfR| = 5, respectively. For brevity reasons, the results obtained on eleven more unimodal data sets are described in detail in appendix C.4. However, at the end of this section, the running times of the three compared consensus architectures measured across the experiments conducted on the twelve unimodal data collections employed in this work are compiled and compared. The goal of such comparison is to analyze whether any of the consensus architecture is inherently faster than the rest. Diversity scenario |df A| =1 The running times of the estimated computationally optimal serial and parallel DHCA and RHCA implementations and flat consensus architectures in the lowest diversity scenario are presented in figure 3.15. As regards the fully serial implementation (figure 3.15(a)), it can be observed that flat consensus is 1.2 to 5 times faster than the fastest hierarchical consensus variant regardless 89
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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 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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2.2. Related work on consensus func
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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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6.4. 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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B.1. Clustering indeterminacies in
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C.1. Configuration of a random hier
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C.4. Computationally optimal RHCA,
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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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