Statistical Implicative Analysis: Theory and Applications

Preface......Page 6
Manuscript coordinator......Page 8
Acknowledgments......Page 9
Contents......Page 10
List of Contributors......Page 13
Introduction......Page 16
Structure of the book......Page 17
1 Introduction......Page 23
1.1 From didactics to data mining......Page 24
2.1 The basic situation......Page 25
2.3 Definitions......Page 26
2.4 Comparison with some classical measures......Page 28
2.5 Stability of the implication intensity......Page 29
3.1 Modal and frequential variables The basic situation......Page 30
3.3 Interval variables The basic situation......Page 31
3.4 The entropic version of the implication intensity The limits of the basic implication intensity for large datasets......Page 32
5.1 The basic situation......Page 35
5.3 A measure of cohesion of the R-rules......Page 36
6.2 Definitions......Page 38
6.3 Construction of an implicative hierarchy......Page 40
7.2 A criterium to determine the signi cative levels......Page 41
8.2 A representation space......Page 42
8.3 Implicative power of an individual on a class......Page 43
8.4 Individual and group typicalities Definition 12.......Page 44
8.5 Contribution......Page 45
9 Illustration......Page 46
References......Page 50
1 Introduction......Page 53
2 Variables......Page 55
3 How to eÿciently compute association rules......Page 58
4 Similarity and hierarchy tree......Page 59
5 Implication graph......Page 61
6 Other possibilities......Page 62
7 An illustration with interval variables and computation of typicality and contribution......Page 63
References......Page 64
1 Introduction......Page 66
2.1 Context......Page 68
2.2 Notations......Page 69
2.3 Sequential rules......Page 70
2.4 Random model......Page 72
3.1 Counter-example increase......Page 73
3.2 Sequence enlargement......Page 74
3.3 Sequence repetition......Page 75
3.4 Window enlargement......Page 77
References......Page 79
1 Introduction......Page 83
2 Interactive Learning Environments for Algebra......Page 84
3.1 Presentation......Page 86
3.2 Rules Diagnosis......Page 87
4.1 Diÿculties in De ning what Stable Behaviours Are......Page 88
4.2 Algebraic Context as Source of Behaviour......Page 89
5.1 Algebraic Context Variables......Page 90
5.2 Creation of the Algebraic Context Variables List......Page 91
5.4 Implications......Page 92
6 An Accurate Student’s Model: the Case of Factoring......Page 93
6.1 The Attributes The Actions.......Page 94
6.3 The Results......Page 95
7 Detection of a Task that is the Source of a Student’s Errors......Page 97
7.2 Some Results......Page 98
8.1 Behaviour Groups......Page 99
8.2 The Case of Collecting Like Terms......Page 100
9 Conclusion......Page 104
References......Page 105
1 The function and its graphic: its basis and its form......Page 107
2 A priori analysis and factorial analysis......Page 110
3 Ostensive use of the graph in teaching......Page 114
4 Similarity Analysis and Implicative Analysis......Page 115
5 Synthesis and conclusions......Page 119
References......Page 120
Appendix: rst and second questionnaires......Page 122
1 Introduction......Page 126
2 Processing data using CHIC software program......Page 127
3.1 Characteristics of the population under study......Page 128
3.2 Three networks at the 0.70 threshold......Page 129
3.3 The in uence of the additional “sex” and “gender” variables on the networks revealed by the......Page 132
4 Conclusion......Page 134
References......Page 135
Appendix......Page 136
1 Introduction......Page 138
2 Theoretical considerations: The role of representations on the understanding of functions......Page 139
4.1 Participants, instrument and variables......Page 141
4.2 Data analysis......Page 142
5.1 The outcomes of CFA......Page 146
5.2 The outcomes of the hierarchical clustering of variables and the implicative method of analysis......Page 150
6 Discussion......Page 159
References......Page 165
Appendix......Page 168
1 Introduction......Page 170
2 Teaching Experiment......Page 171
3 A-Priori Analysis of the Questionnaire......Page 172
4 Implications in Learning Outcomes......Page 175
5 Implicative hierarchy of learning outcomes......Page 179
6 Discussion......Page 183
References......Page 184
Appendix: Questionnaire......Page 186
1 Presentation of the study......Page 192
2.1 Theoretical premises......Page 193
2.3 The key problem “Charlotte and Marie”......Page 194
3.1 Towards a classification of the students’ answers......Page 196
3.2 A look at reasoning problems......Page 197
4.1 Limit of the didactic study......Page 198
4.2 The factorial approach......Page 199
4.3 The implicative approach......Page 201
4.4 Precision on the components of the geometrical working space......Page 202
4.5 An interpretation in terms of personal Geometrical Working Space......Page 203
5 Conclusion......Page 205
References......Page 206
Appendix 1 Student’s answers......Page 207
Appendix 2 Codification of the problem Charlotte and Marie......Page 208
1 Introduction......Page 210
2 Related work......Page 211
3.1 Definitions......Page 212
3.2 Numerical and computational considerations......Page 214
3.3 Comparison of association rules and correlation techniques......Page 215
4 Application to tumour discrimination......Page 217
4.1 Gene selection......Page 218
4.2 Tumour classification......Page 221
5 Analysis of gene association networks......Page 223
5.1 Gene representation based on implicative analysis......Page 224
5.2 Discovery of gene association using the intensity of implication......Page 225
6 Conclusion......Page 228
References......Page 229
1 Introduction......Page 231
2.1 Textual taxonomy matching......Page 233
2.2 Interestingness Measures......Page 236
3.1 Association rules discovery between hierarchies
......Page 239
3.2 Selection of significant rules......Page 240
4.1 Analysed data......Page 242
4.3 Distributions of IMs......Page 243
5 Conclusion......Page 246
References......Page 247
1 The research in didactics, some tools......Page 250
1.1 The data......Page 252
1.2 The correspondence factor analysis and the implicative analysis among variables in research in didactics of mathematics: an......Page 256
1.3 Experimental comparison between Statistic Implicative Analysis (SIA) and the Correspondence Factor Analysis (CFA)......Page 257
1.4 Conclusions......Page 258
2.2 The historical context of the research......Page 259
2.4 The experimentation......Page 260
2.5 The rst experiment and its analysis by CHIC......Page 261
2.8 Supplementary variables and pupils’ pro les......Page 262
2.9 The implicative graph......Page 264
2.11 Some nal observations......Page 265
3.1 Introduction......Page 266
3.2 A condensed theoretical framework......Page 267
3.3 Methodology of the research......Page 268
3.4 The a-priori analysis......Page 269
3.5 Statistic Implicative Analysis (ASI) Implicative graph......Page 271
3.6 The correspondence factor analysis (CFA)......Page 274
3.7 Conclusions......Page 275
References......Page 276
Appendix: Table of frequencies......Page 279
Didactics of Mathematics and Implicative Statistical Analysis......Page 280
1 Rules and Regulations......Page 281
2.1 Asymmetries of Rules Established and Chronology of Tasks......Page 282
2.2 Asymmetry of Rules and Representations of Subjects.......Page 284
2.3 Rules Interpreted as Traces of Skills.......Page 288
3.2 Characteristic Abilities......Page 293
3.3 Groups Characterized by Capacities......Page 294
4 Conclusion......Page 295
References......Page 296
Appendix......Page 297
1 Introduction......Page 302
2 Applicative Context......Page 304
3 The PerformanSe Echo Tool......Page 305
4 Problematic and Goals......Page 308
5.1 The Reference Population......Page 309
5.2 The Studied Population......Page 310
6 Why We Used the Statistical Implicative Analysis and CHIC......Page 311
7.2 Study of Deviations......Page 312
7.3 Analysis with Similarity Trees......Page 313
8.2 Analysis of the Subpopulations ASNSubpopulation......Page 315
8.3 ASN0 Subpopulation......Page 318
9 Results and Outlooks......Page 319
References......Page 320
1 Introduction......Page 323
2.2 Description of the Case......Page 325
3.1 Classification Analysis......Page 327
3.2 Implicative Analysis......Page 328
3.3 Cohesion Analysis......Page 330
4.1 Similarity Tree......Page 333
4.2 Cohesion Tree......Page 334
4.3 Implicative Graphs......Page 337
5 Conclusions......Page 338
References......Page 340
Appendix A Theoretical Introduction to Implicative Analysis and some Inequalities Regarding the Increment of the Intensity under......Page 342
B Questionnaire Driven in the Case Study......Page 345
1 Introduction......Page 348
2 Cognitive development, learning and obstacles......Page 350
2.1 Representations, concepts and conceptual development......Page 351
2.2 Obstacles to learning and conceptual development......Page 353
2.3 The moon phases: an object of study for astronomy......Page 354
2.5 Phases of the moon: an object of everyday learning......Page 355
2.6 Phases of the moon: an object of learning at school......Page 358
3.1 An initial situation problem focussed on the phases of the Moon......Page 360
3.2 Exploration of the mental representations of the phases of the moon......Page 362
4 Conclusion......Page 372
References......Page 373
Appendix 1......Page 375
Appendix 2......Page 378
Appendix 3......Page 379
Appendix 4......Page 380
1 Introduction......Page 381
2.1 Rule Discovery......Page 382
2.2 Example of the Objective Interestingness Measures for Rule Discovery......Page 383
3.1 Desiderata on Objective Interestingness Measures......Page 384
3.2 Cluster Analysis of Objective Interestingness Measures......Page 385
4.1 Rule Bias......Page 386
4.3 Expert Bias......Page 387
5.1 Respect the True Objective......Page 388
5.2 Show Experimental Results in an Objective Manner......Page 389
5.4 Be a Structuralist in Deriving Conclusions......Page 390
References......Page 391
1 Introduction......Page 394
2.1 Trees and Rules......Page 396
2.2 Counter-examples and Implication Index......Page 398
2.3 Implication Index and Residuals......Page 400
3 Individual Rule Relevance......Page 402
4 Adopting a Typical Profile Paradigm......Page 404
4.1 Maximal Implication Strength versus Majority Rule......Page 405
4.2 Growing Trees with Implication Strength Criteria......Page 406
5.1 Compared Behavior of the 4 Indexes......Page 407
5.2 Application on a Student Administrative Dataset......Page 409
5.4 Recall and Precision......Page 411
6 Conclusion......Page 413
References......Page 414
1 Introduction......Page 417
2.1 Characteristics of statistical and probabilistic measures......Page 421
2.2 Discriminating power of statistical measures......Page 423
2.3 Adaptation of the entropic intensity of implication......Page 426
3.2 Discriminant version......Page 428
4 Generalized statistical measures......Page 429
5.1 A o -centered version of the entropy......Page 433
5.2 Weighting approach......Page 435
5.3 Contextual approach......Page 437
6 Conclusion and perspectives......Page 438
References......Page 439
1 Introduction......Page 444
2.1 Computing the intensity of implication......Page 445
2.2 Practical computation of the intensity of implication......Page 446
3.1 Test value and normalized test value Test value......Page 448
3.2 TVpercent criterion on the counter-examples statistic......Page 450
3.3 Test value and index of implication......Page 451
4.1 Description of the experimentation Database......Page 452
4.2 Results and comments......Page 453
References......Page 456
1 Introduction......Page 458
2 Redundant functional dependencies......Page 460
2.1 Definitions Functional dependency and Armstrong’s axioms :......Page 461
3.1 Proposed algorithms......Page 462
4.1 Propositional logic......Page 464
4.2 Conceptual lattices......Page 465
5.1 Further propositions to circumvent these limits......Page 469
6 Conclusion......Page 471
References......Page 472
1 Introduction......Page 475
2.1 Definition of Fuzzy Partitions......Page 477
3 Choosing Statistical Implication Indexes......Page 478
4 Generalization of Statistical Indexes to Fuzzy Knowledge......Page 481
5 A Knowledge Extraction Algorithm......Page 483
6.1 Di erences between Fuzzy Rules and Statistical Implications......Page 484
6.2 Main Sets of Fuzzy Operators......Page 485
7.1 Our Method......Page 488
7.2 One Simple Example......Page 489
7.3 Results on real databases......Page 492
9 Operators for Fuzzy Rule Reduction......Page 494
9.1 A First Scheme of Rule Reduction......Page 495
9.2 A Second Scheme of Rule Reduction......Page 496
10 Conclusion......Page 497
References......Page 498
About the editors......Page 501
About the manuscript coordinator......Page 502
Index......Page 503