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6.2. Adapting consensus functions to soft cluster ensembles Oλ = Iλ T Iλ = ⎛ 0 1 ⎞ 0 ⎜ 0 ⎜ 0 ⎜ ⎜1 ⎜ ⎜1 ⎜ ⎜1 ⎜ ⎜0 ⎝0 1 1 0 0 0 0 0 0 ⎟ 0 ⎟ ⎛ 0 ⎟ 0 0⎟⎝ ⎟ 1 0 ⎟ 0 1 ⎟ 1⎠ 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 ⎞ 0 0⎠ 1 0 ⎛ 1 0 1 1 1 0 0 0 0 0 ⎞ 0 ⎜ 1 ⎜ 1 ⎜ ⎜0 = ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎝0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 ⎟ 0 ⎟ 0 ⎟ 0 ⎟ 0 ⎟ 1 ⎟ 1⎠ 0 0 0 0 0 0 1 1 1 (6.9) In a fuzzy clustering scenario, the object co-association matrix can easily be derived by simply multiplying the transpose clustering matrix Λ by itself. Resorting again to our toy clustering example, the resulting object co-association matrix (denoted as OΛ) ispresented in equation (6.10). It is easy to see that OΛ is indeed a fuzzy adjacency matrix, as the statistical independence of the probabilities of assigning objects i and j to the same cluster allows to interpret its (i,j)th entry as the joint probability that objects i and j are placed in the same cluster by the clusterer. However, statistical independence does not hold for the elements of the diagonal of OΛ, as each object is always “co-clustered” with itself, which would require making the elements on the diagonal of OΛ equal to 1. 168
⎛ OΛ = ΛT ⎜ Λ = ⎜ ⎝ ⎛ ⎜ = ⎜ ⎝ Chapter 6. Voting based consensus functions for soft cluster ensembles 0.054 0.921 0.025 0.026 0.932 0.042 0.057 0.905 0.038 0.969 0.025 0.006 0.976 0.019 0.005 0.959 0.030 0.011 0.009 0.014 0.976 0.016 0.055 0.929 0.010 0.017 0.972 ⎞ ⎟ ⎛ ⎟ ⎝ ⎟ ⎠ 0.054 0.026 0.057 0.969 0.976 0.959 0.009 0.016 0.010 0.921 0.932 0.905 0.025 0.019 0.030 0.014 0.055 0.017 0.025 0.042 0.038 0.006 0.005 0.011 0.976 0.929 0.972 0.852 0.861 0.838 0.076 0.070 0.080 0.038 0.075 0.040 0.861 0.871 0.847 0.049 0.043 0.053 0.054 0.091 0.057 0.838 0.847 0.824 0.078 0.073 0.082 0.050 0.086 0.053 0.076 0.049 0.078 0.940 0.946 0.930 0.015 0.022 0.016 0.070 0.043 0.073 0.946 0.953 0.937 0.014 0.021 0.015 0.080 0.053 0.082 0.930 0.937 0.921 0.020 0.027 0.021 0.038 0.054 0.050 0.015 0.014 0.020 0.953 0.908 0.949 0.075 0.091 0.086 0.022 0.021 0.027 0.908 0.866 0.904 0.040 0.057 0.053 0.016 0.015 0.021 0.949 0.904 0.945 ⎞ ⎟ ⎠ ⎞ ⎠ (6.10) Quite obviously, consensus functions based on object co-association matrices do not operate on matrices Oλ or OΛ, as they are derived upon a single clustering solution. However, the computation of object co-association matrices can easily be extended to a set of clustering solutions compiled in either a hard or a soft cluster ensemble. We start with the derivation of the object co-association matrix upon a hard cluster ensemble E containing l clusterings, represented by means of a l × n integer valued matrix: ⎛ λ1 ⎜ ⎜λ2 ⎟ E = ⎜ ⎟ ⎝ . ⎠ λl ⎞ (6.11) In this case, the incidence matrix of the hard cluster ensemble, denoted as IEλ ,isa kl × n matrix resulting from the concatenation of the incidence matrices of the l clusterings that make up the ensemble (see equation (6.12)). IEλ = ⎛ ⎞ Iλ1 ⎜Iλ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ . ⎠ Iλl (6.12) As in the case of a single clustering, the object co-association matrix of the hard cluster ensemble, denoted as OEλ , can be derived upon IEλ by simple matrix multiplication, as presented in equation (6.13). OEλ T = IEλ IEλ 169 (6.13)
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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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Page 222 and 223: 6.4. Experiments Data set Soft clus
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Page 239 and 240: References Ben-Hur, A., D. Horn, H.
Page 241 and 242: References Deerwester, S., S.-T. Du
Page 243 and 244: References Fred, A. and A.K. Jain.
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Page 247 and 248: References Li, S.Z. and G. GuoDong.
Page 249 and 250: References Sebastiani, F. 2002. Mac
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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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