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F.9. Segmentation data set φ (NMI) 1 0.8 0.6 0.4 0.2 0 10 0 MINING CPU time (sec.) 10 2 CSPA EAC HGPA MCLA VMA BC CC PC SC Figure F.8: φ (NMI) vs CPU time mean ± 2-standard deviation regions of the soft consensus functions on the miniNG data collection. Indeed, the consensus clustering solutions they yield are statistically significantly better than those output by state-of-the-art consensus functions such as VMA (which is the least time consuming), CSPA or MCLA —see table F.8 for further details regarding the statistical significance of the differences between consensus functions. The fourth consensus function proposed (CC) also yields higher quality than VMA, CSPA and MCLA, but its time complexity is notably higher, due to the nature of the Condorcet voting method. CSPA EAC HGPA MCLA VMA BC CC PC SC CSPA ——— 0.0001 0.0001 × 0.0001 0.0015 0.0001 0.0114 0.0115 EAC 0.0001 ——— 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 HGPA 0.0001 0.0001 ——— 0.0041 0.0001 0.0001 0.0001 0.0001 0.0001 MCLA 0.0001 0.0001 0.0001 ——— 0.0001 0.0025 0.0001 0.0185 0.0187 VMA × 0.0001 0.0001 0.0001 ——— 0.0001 0.0001 0.0001 0.0001 BC 0.0051 0.0001 0.0001 0.0001 0.0038 ——— 0.0001 × × CC 0.008 0.0001 0.0001 0.0001 0.0061 × ——— 0.0001 0.0001 PC × 0.0001 0.0001 0.0001 × × × ——— × SC 0.0001 0.0001 0.0001 0.0001 0.0001 × × 0.0126 ——— Table F.8: Significance levels p corresponding to the pairwise comparison of soft consensus functions using a t-paired test on the miniNG data set. The upper and lower triangular sections of the table correspond to the comparison in terms of CPU time and φ (NMI) , respectively. Statistically non-significant differences (p >0.05) are denoted by the symbol ×. F.9 Segmentation data set The results of the application of the nine soft consensus functions upon the Segmentation data set are described next. Figure F.9 presents the φ (NMI) vs CPU time mean ± 2-standard deviation regions employed for comparing them. Again, VMA is the most computationally efficient consensus function. The proposed preference voting based clustering combiners (PC and SC), together with MCLA and 382
φ (NMI) 1 0.8 0.6 0.4 0.2 0 10 0 Appendix F. Experiments on soft consensus clustering SEGMENTATION CPU time (sec.) 10 2 CSPA EAC HGPA MCLA VMA BC CC PC SC Figure F.9: φ (NMI) vs CPU time mean ± 2-standard deviation regions of the soft consensus functions on the Segmentation data collection. HPGA, are the immediate followers. Between the two proposed consensus functions based on positional voting, BC is once more the most efficient (being comparable to CSPA in terms of execution CPU time), as Borda voting is less computationally demanding than Condorcet voting. CSPA EAC HGPA MCLA VMA BC CC PC SC CSPA ——— 0.0001 0.0001 0.0001 0.0001 × 0.0001 0.0001 0.0001 EAC 0.0001 ——— 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 HGPA 0.0001 0.0001 ——— × 0.0001 0.0002 0.0001 × × MCLA 0.0001 0.0001 0.0001 ——— 0.0001 0.006 0.0001 × × VMA × 0.0001 0.0001 0.0001 ——— 0.0001 0.0001 0.0001 0.0001 BC 0.0006 0.0001 0.0001 0.0001 0.0069 ——— 0.0001 0.0007 0.0007 CC 0.0006 0.0001 0.0001 0.0001 0.0069 × ——— 0.0001 0.0001 PC × 0.0001 0.0001 0.0001 × 0.02 0.02 ——— × SC × 0.0001 0.0001 0.0001 × 0.0307 0.0307 × ——— Table F.9: Significance levels p corresponding to the pairwise comparison of soft consensus functions using a t-paired test on the Segmentation data set. The upper and lower triangular sections of the table correspond to the comparison in terms of CPU time and φ (NMI) , respectively. Statistically non-significant differences (p >0.05) are denoted by the symbol ×. However, these two consensus functions (BC and CC) are the ones to obtain the highest quality consensus clustering solutions, and the difference between their φ (NMI) scores and those of the remaining clustering combiners is statistically significant, as the figures shown in table F.9 reveal. The quality of the consensus clusterings output by the other two voting based consensus functions (PC and SC) is, from a statistical standpoint, equivalent to that of the VMA and CSPA consensus functions. 383
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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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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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Page 396 and 397: E.1. CAL500 data set φ (NMI) 1 0.8
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Page 402 and 403: E.3. Corel data set φ (NMI) 1 0.8
Page 410 and 411: F.1. Iris data set φ (NMI) 1 0.8 0
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Page 422 and 423: F.11. PenDigits data set φ (NMI) 1
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