ISBN: 379081475X
TITLE: Towards Efficient Fuzzy Information Processing
AUTHOR: Huang
TOC:

Foreword vii
Preface xi
Part I: Principle of Information Diffusion
1. Introduction 3
1.1 Information Sciences 3
1.2 Fuzzy Information 7
1.2.1 Some basic notions of fuzzy set theory 7
1.2.2 Fuzzy information defined by fuzzy entropy 9
1.2.3 Traditional fuzzy information without reference to entropy 10
1.2.4 Fuzzy information due to an incomplete data set 11
1.2.5 Fuzzy information and its properties 13
1.2.6 Fuzzy information processing 14
1.3 Fuzzy function approximation 15
1.4 Summary 19
Referencess 20
2. Information Matrix 23
2.1 Small-Sample Problem 23
2.2 Information Matrix 26
2.3 Information Matrix on Crisp Intervals 29
2.4 Information Matrix on Fuzzy Intervals 32
2.5 Mechanism of Information Matrix 38
2.6 Some Approaches Describing or Producing Relationships 39
2.6.1 Equations of mathematical physics 39
2.6.2 Regression 41
2.6.3 Neural networks 43
2.6.4 Fuzzy graphs 45
2.7 Conclusion and Discussion 47
References 49
Appendix 2.A: Some Earthquake Data 51
3. Some Concepts From Probability and Statistics 55
3.1 Introduction 55
3.2 Probability 56
3.2.1 Sample spaces, outcomes, and events 56
3.2.2 Probability 57
3.2.3 Joint, marginal, and conditional probabilities 57
3.2.4 Random variables 58
3.2.5 Expectation value, variance, functions of random variables 59
3.2.6 Continuous random variables 60
3.2.7 Probability density function 60
3.2.8 Cumulative distribution function 61
3.3 Some Probability Density Functions 61
3.3.1 Uniform distribution 61
3.3.2 Normal distribution 62
3.3.3 Exponential distribution 63
3.3.4 Lognormal distribution 64
3.4 Statistics and Some Traditional Estimation Methods 64
3.4.1 Statistics 64
3.4.2 Maximum likelihood estimate 65
3.4.3 Histogram 67
3.4.4 Kernel method 70
3.5 Monte Carlo Methods 73
3.5.1 Pseudorandom numbers 74
3.5.2 Uniform random numbers 75
3.5.3 Normal random numbers 77
3.5.4 Exponential random numbers 78
3.5.5 Lognormal random numbers 78
References 80
4. Information Distribution 81
4.1 Introduction 81
4.2 Definition of Information Distribution 83
4.3 l-Dimension Linear Information Distribution 84
4.4 Demonstration of Benefit for Probability Estimation 89
4.4.1 Model description 89
4.4.2 Normal experiment 91
4.4.3 Exponential experiment 94
4.4.4 Lognormal experiment 96
4.4.5 Comparison with maximum likelihood estimate 99
4.4.6 Results 100
4.5 Non-Linear Distribution 100
4.6 r-Dimension Distribution 105
4.7 Fuzzy Relation Matrix from Information Distribution 110
4.7.1 R_f based on fuzzy concepts 111
4.7.2 R_m based on fuzzy implication theory 115
4.7.3 R_c based on conditional falling shadow 117
4.8 Approximate Inference Based on Information Distribution 120
4.8.1 Max-min inference for R_f 120
4.8.2 Similarity inference for R_f 121
4.8.3 Max-min inference for R_m 121
4.8.4 Total-falling-shadow inference for R_c 121
4.9 Conclusion and Discussion 122
References 123
Appendix 4.A: Linear Distribution Program 124
Appendix 4.B: Intensity Scale 127
5. Information Diffusion 129
5.1 Problems in Information Distribution 129
5.2 Definition of Incomplete-Data Set 131
5.2.1 Incompleteness 131
5.2.2 Correct-data set 133
5.2.3 Incomplete-data set 135
5.3 Fuzziness of a Given Sample 136
5.3.1 Fuzziness in terms of fuzzy sets 137
5.3.2 Fuzziness in terms of philosophy 138
5.3.3 Fuzziness of an incomplete sample 139
5.4 Information Diffusion 141
5.5 Random Sets and Covering Theory 143
5.5.1 Fuzzy logic and possibility theory 144
5.5.2 Random sets 145
5.5.3 Covering function 147
5.5.4 Set-valuedization of observation 148
5.6 Principle of Information Diffusion 150
5.6.1 Associated characteristic function and relationships 150
5.6.2 Allocation function 152
5.6.3 Diffusion estimate 153
5.6.4 Principle of Information Diffusion 154
5.7 Estimating Probability by Information Diffusion 156
5.7.1 Asymptotically unbiased property 159
5.7.2 Mean squared consistent property 160
5.7.3 Asymptotically property of mean square error 161
5.7.4 Empirical distribution function, histogram and diffusion estimate 163
5.8 Conclusion and Discussion 165
References 167
6. Quadratic Diffusion 169
6.1 Optimal Diffusion Function 169
6.2 Choosing Delta Based on Kernel Theory 173
6.2.1 Mean integrated square error 173
6.2.2 References to a standard distribution 174
6.2.3 Least-squares cross-validation 175
6.2.4 Discussion 181
6.3 Searching for Delta by Golden Section Method 181
6.4 Comparison with Other Estimates 185
6.5 Conclusion 188
References 189
7. Normal Diffusion 191
7.1 Introduction 191
7.2 Molecule Diffusion Theory 192
7.2.1 Diffusion 192
7.2.2 Diffusion equation 194
7.3 Information Diffusion Equation 196
7.3.1 Similarities of molecule diffusion and information diffusion 196
7.3.2 Partial differential equation of information diffusion 197
7.4 Nearby Criteria of Normal Diffusion 199
7.5 The 0.618 Algorithm for Getting h 200
7.6 Average Distance Model 204
7.7 Conclusion and Discussion 210
References 212
Part II: Applications
8. Estimation of Epicentral Intensity 215
8.1 Introduction 215
8.2 Classical Methods 217
8.2.1 Linear regression 217
8.2.2 Fuzzy inference based on normal assumption 219
8.3 Self-Study Discrete Regression 222
8.3.1 Discrete regression 223
8.3.2 r-dimension diffusion 226
8.3.3 Self-study discrete regression 227
8.4 Linear Distribution Self-Study 228
8.5 Normal Diffusion Self-Study 230
8.6 Conclusion and Discussion 234
References 236
Appendix 8.A: Real and Estimated Epicentral Intensities 237
Appendix 8.B: Program of NDSS 241
9. Estimation of Isoseismal Area 247
9.1 Introduction 247
9.2 Some Methods for Constructing Fuzzy Relationships 249
9.2.1 Fuzzy relation and fuzzy relationship 249
9.2.2 Multivalued logical-implication operator 251
9.2.3 Fuzzy associative memories 251
9.2.4 Self-study discrete regression 253
9.3 Multitude Relationships Given by Information Diffusion 254
9.4 Patterns Smoothening 256
9.5 Learning Relationships by BP Neural Networks 257
9.6 Calculation 261
9.7 Conclusion and Discussion 268
References 270
10. Fuzzy Risk Analysis 273
10.1 Introduction 273
10.2 Risk Recognition and Management for Environment, Health, and Safety 275
10.3 A Survey of Fuzzy Risk Analysis 276
10.4 Risk Essence and Fuzzy Risk 278
10.5 Some Classical Models 281
10.5.1 Histogram 281
10.5.2 Maximum likelihood method 281
10.5.3 Kernel estimation 282
10.6 Model of Risk Assessment by Diffusion Estimate 283
10.7 Application in Risk Assessment of Flood Disaster 284
10.7.1 Normalized normal-diffusion estimate 286
10.7.2 Histogram estimate 288
10.7.3 Soft histogram estimate 289
10.7.4 Maximum likelihood estimate 290
10.7.5 Gaussian kernel estimate 291
10.7.6 Comparison 292
10.8 Conclusion and Discussion 292
References 294
11. System Analytic Model for Natural Disasters 297
11.1 Classical System Model for Risk Assessment of Natural Disasters 297
11.1.1 Risk assessment of hazard 298
11.1.2 From magnitude to site intensity 300
11.1.3 Damage risk 302
11.1.4 Loss risk 303
11.2 Fuzzy Model for Hazard Analysis 304
11.2.1 Calculating primary information distribution 306
11.2.2 Calculating exceeding frequency distribution 307
11.2.3 Calculating fuzzy relationship between magnitude and probability 307
11.3 Fuzzy Systems Analytic Model 308
11.3.1 Fuzzy attenuation relationship 308
11.3.2 Fuzzy dose-response relationship 310
11.3.3 Fuzzy loss risk 310
11.4 Application in Risk Assessment of Earthquake Disaster 312
11.4.1 Fuzzy relationship between magnitude and probability 312
11.4.2 Intensity risk 316
11.4.3 Earthquake damage risk 321
11.4.4 Earthquake loss risk 322
11.5 Conclusion and Discussion 323
References 324
12. Fuzzy Risk Calculation 325
12.1 Introduction 325
12.1.1 Fuzziness and probability 326
12.1.2 Possibility-probability distribution 327
12.2 Interior-outer-set Model 328
12.2.1 Model description 328
12.2.2 Calculation case 331
12.2.3 Algorithm and Fortran program 333
12.3 Ranking Alternatives Based on a PPD 337
12.3.1 Classical model of ranking alternatives 337
12.3.2 Fuzzy expected value 339
12.3.3 Center of gravity of a fuzzy expected value 343
12.3.4 Ranking alternatives by FEV 346
12.4 Application in Risk Management of Flood Disaster 348
12.4.1 Outline of Huarong county 348
12.4.2 PPD of flood in Huarong county 348
12.4.3 Benefit-output functions of farming alternatives 350
12.4.4 Ranking farming alternative based on the PPD 353
12.4.5 Comparing with the traditional probability method 354
12.5 Conclusion and Discussion 355
References 356
Appendix 12.A: Algorithm Program for Interior-outer-set Model 357
Subject Index 365
List of Special Symbols 371
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