Trees and proximity representations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Chichester [u.a.]
John Wiley & Sons
1991
|
| Series: | Wiley-Interscience Series in Discrete Mathematics and Optimization
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 232 |
| Physical Description: | XVI, 238 Seiten graph. Darst. |
| ISBN: | 0471922633 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface xi
Introduction xv
1 Trees 1
1.1 Terminology of graph theory 1
1.1.1 Graphs 1
1.1.2 Trees 3
1.1.3 Coding and representation of trees 6
1.1.4 Minimal spanning tree in a valued graph 9
1.1.5 Construction of a minimal spanning tree 10
1.2 The concept of X tree 12
1.2.1 The concept of X trees. Some examples 12
1.2.2 Definition of an X tree 15
1.2.3 Enumeration of X trees 16
1.2.4 Random selection of X trees 19
1.2.5 Rooted X trees and hierarchical trees 22
1.2.6 Valued X trees and dendrograms 23
1.2.7 Tree drawing 23
1.3 Bibliographical notes 31
1.3.1 Graphs and trees 31
1.3.2 X trees and hierarchical trees 33
1.3.3 Some remarks on X trees 34
1.4 References 35
2 Tree distances 39
2.1 Proximity measures and their representations 40
2.1.1 Remarks on proximity measures 40
2.1.2 Pseudo distances and distances. Representations 41
2.2 Our data and their acquisition 45
2.2.1 The experimental data 45
2.2.2 Acquisition of data 46
2.3 Tree distances 49
2.3.1 Additive distances on trees and tree distances 49
2.3.2 The four point condition 52
2.3.3 Tree distances and X trees. The uniqueness problem 54
2.4 Coding of tree distances and degrees of freedom 56
viii Contents
2.4.1 Posing the problem 56
2.4.2 The median of three tree vertices 57
2.4.3 The diagonal order and the optimal diagonal code 58
2.5 Approximation by a tree distance: least squares 60
2.5.1 Mathematical programming methods 60
2.5.2 Quadratic approximation on a support tree 62
2.6 Three heuristic methods 66
2.6.1 Quadruple reduction method 66
2.6.2 Method of closest predecessors 70
2.6.3 Method of dispersion 73
2.6.4 Applications to Henley s data 75
2.7 Bibliographical notes 76
2.7.1 Proximity measures and Euclidean representations 76
2.7.2 Tree distances 77
2.7.3 Concerning the diagonal code 82
2.8 References 83
3 Ultrametric and centroid distances 87
3.1 Centroid distances 87
3.1.1 Definition and characterisation 87
3.1.2 Some properties of centroid distances 89
3.1.3 Approximation by a centroid distance 89
3.2 Ultrametrics 90
3.2.1 Definition and characterisation of ultrametrics 91
3.2.2 Ultrametrics and hierarchical classifications 93
3.2.3 Drawing a dendrogram to represent an ultrametric 95
3.2.4 Ultrametrics and Euclidean distances 96
3.2.5 Minimal spanning trees and ultrametrics 97
3.2.6 Connected components of threshold graphs 99
3.2.7 Algorithms for constructing ultrametrics: Single, complete,
average linkages 101
3.2.8 Program for ascending hierarchical classification 104
3.2.9 Approximation of a distance index by an ultrametric 109
3.3 Tree distances, centroid distances and ultrametrics 112
3.3.1 Decomposition of a tree distance into the sum of a centroid
distance and an ultrametric 112
3.3.2 Some remarks 116
3.3.3 Use of (r,^ decompositions to construct the valued tree
representing a tree distance (Brossier) 118
3.4 Approximation by a tree distance using (r,^ decompositions 119
3.4.1 Least squares method 119
3.4.2 A heuristic method: method of decomposition 119
3.4.3 Henley s data 121
3.5 Bibliographical notes 122
Contents ix
3.5.1 Centroid distances 122
3.5.2 Ultrametrics 123
3.5.3 Ascending hierarchical classification algorithms (AHC) 125
3.5.4 Approximation by an ultrametric 125
3.5.5 Centroid distances, ultrametrics, tree distances and
Euclidean distances 126
3.5.6 Decompositions of a tree distance 126
3.5.7 The complexity of the problem of approximating a
dissimilarity by an ultrametric or by a centroid distance 127
3.6 References 129
4 X tree topology; scores and groupings 133
4.1 Trees and relations 134
4.1.1 The fundamental relation in an X tree 134
4.1.2 The case of free, separated X trees 137
4.1.3 Equivalence of X trees 138
4.2 Scores and groupings 140
4.2.1 Groupings and pregroupings relative to a 2 relation 140
4.2.2 Properties of groupings 141
4.2.3 The concept of score 142
4.2.4 Reconstructing a tree 143
4.3 Proximities and 2 relations. Grouping algorithms 147
4.3.1 The 2 relation associated with a pseudo distance index 147
4.3.2 Regrouping algorithms 149
4.3.3 The ADDTREE method of Sattah and Tversky (1977) 149
4.3.4 The method of groupings 151
4.3.5 Henley s data 156
4.4 The scores theorem 157
4.4.1 Reinterpretation of the concept of score and a generalisation 157
4.4.2 Arithmetic of scores 161
4.4.3 The scores theorem 164
4.5 Bibliographical notes 166
4.5.1 The fundamental relation in an X tree 166
4.5.2 The concept of grouping 167
4.5.3 The scores theorem 168
4.6 References 168
5 Combinatorial description of X trees 171
5.1 Global description. Buneman s theory 171
5.1.1 Trees and splits 171
5.1.2 The graph associated with a complemented family 173
5.1.3 Some examples 176
5.1.4 Buneman families 179
5.1.5 The equivalence of an X tree and a Buneman family 181
x Contents
5.1.6 Binary variables 183
5.1.7 Archaeological data 187
5.1.8 Enumeration of localisations, construction of G(E) 188
5.2 Some applications of Buneman s construction 192
5.2.1 Reinterpretation of tree distances 192
5.2.2 Every tree distance is the square of a Euclidean distance 194
5.2.3 An ordering of .Y trees. The concept of compatibility 194
5.3 Tree analysis of a set of binary variables 196
5.3.1 Recognition of Buneman families and tree decomposition 196
5.3.2 Tree analysis algorithms 200
5.4 Local study: four point configurations 205
5.4.1 Local study of X trees 205
5.4.2 Restrictions and configurations 206
5.4.3 The 2 relation associated with a complemented family 208
5.4.4 Local characterisation of Buneman families 209
5.5 Qualitative invariance 211
5.5.1 Preliminary remarks 211
5.5.2 Invariance and stability 212
5.5.3 The case of tree pseudo distances 213
5.6 Bibliographical notes 216
5.6.1 Buneman theory 216
5.6.2 Consequences of Buneman s theory 217
5.6.3 Analysis of 0/1 variables 217
5.6.4 Local description: meaningfulness: non metric algorithms 218
5.7 References 219
6 Instead of an ending 221
6.1 Comparison of algorithms 221
6.1.1 Comparison criteria 222
6.1.2 Results 224
6.1.3 Commentary 225
6.2 Rectangular array data 227
6.2.1 Problem formulation 227
6.2.2 Characterisation of rectangular tree pseudo distances 228
6.2.3 Algorithms 228
6.3 Some other types of data and representations 229
6.3.1 Asymmetric proximities 229
6.3.2 Cubical proximities 230
6.3.3 Generalised trees 231
6.4 References 231
Index 233
|
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| spelling | Barthélemy, Jean-Pierre Verfasser aut Les arbres et les représentations des proximités Trees and proximity representations Jean-Pierre Barthélemy and Alain Guénoche Chichester [u.a.] John Wiley & Sons 1991 XVI, 238 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley-Interscience Series in Discrete Mathematics and Optimization Literaturverzeichnis Seite 232 Combinatieleer gtt Grafentheorie gtt Combinatorial enumeration problems Mathematical statistics Trees (Graph theory) Baum Mathematik (DE-588)4004849-4 gnd rswk-swf Wissensrepräsentation (DE-588)4049534-6 gnd rswk-swf Abzählende Kombinatorik (DE-588)4132720-2 gnd rswk-swf Wissensrepräsentation (DE-588)4049534-6 s Baum Mathematik (DE-588)4004849-4 s DE-604 Abzählende Kombinatorik (DE-588)4132720-2 s Guénoche, Alain Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003392365&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Barthélemy, Jean-Pierre Guénoche, Alain Trees and proximity representations Combinatieleer gtt Grafentheorie gtt Combinatorial enumeration problems Mathematical statistics Trees (Graph theory) Baum Mathematik (DE-588)4004849-4 gnd Wissensrepräsentation (DE-588)4049534-6 gnd Abzählende Kombinatorik (DE-588)4132720-2 gnd |
| subject_GND | (DE-588)4004849-4 (DE-588)4049534-6 (DE-588)4132720-2 |
| title | Trees and proximity representations |
| title_alt | Les arbres et les représentations des proximités |
| title_auth | Trees and proximity representations |
| title_exact_search | Trees and proximity representations |
| title_full | Trees and proximity representations Jean-Pierre Barthélemy and Alain Guénoche |
| title_fullStr | Trees and proximity representations Jean-Pierre Barthélemy and Alain Guénoche |
| title_full_unstemmed | Trees and proximity representations Jean-Pierre Barthélemy and Alain Guénoche |
| title_short | Trees and proximity representations |
| title_sort | trees and proximity representations |
| topic | Combinatieleer gtt Grafentheorie gtt Combinatorial enumeration problems Mathematical statistics Trees (Graph theory) Baum Mathematik (DE-588)4004849-4 gnd Wissensrepräsentation (DE-588)4049534-6 gnd Abzählende Kombinatorik (DE-588)4132720-2 gnd |
| topic_facet | Combinatieleer Grafentheorie Combinatorial enumeration problems Mathematical statistics Trees (Graph theory) Baum Mathematik Wissensrepräsentation Abzählende Kombinatorik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003392365&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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