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[资源] Statistical.Methods.for.Fuzzy.Data

Contents
Preface xi
Part I FUZZY INFORMATION 1
1 Fuzzy data 3
1.1 One-dimensional fuzzy data 3
1.2 Vector-valued fuzzy data 4
1.3 Fuzziness and variability 4
1.4 Fuzziness and errors 4
1.5 Problems 5
2 Fuzzy numbers and fuzzy vectors 7
2.1 Fuzzy numbers and characterizing functions 7
2.2 Vectors of fuzzy numbers and fuzzy vectors 14
2.3 Triangular norms 16
2.4 Problems 18
3 Mathematical operations for fuzzy quantities 21
3.1 Functions of fuzzy variables 21
3.2 Addition of fuzzy numbers 23
3.3 Multiplication of fuzzy numbers 25
3.4 Mean value of fuzzy numbers 25
3.5 Differences and quotients 27
3.6 Fuzzy valued functions 27
3.7 Problems 28
Part II DESCRIPTIVE STATISTICS FOR FUZZY DATA 31
4 Fuzzy samples 33
4.1 Minimum of fuzzy data 33
4.2 Maximum of fuzzy data 33
4.3 Cumulative sum for fuzzy data 34
4.4 Problems 34
vi CONTENTS
5 Histograms for fuzzy data 37
5.1 Fuzzy frequency of a fixed class 37
5.2 Fuzzy frequency distributions 38
5.3 Axonometric diagram of the fuzzy histogram 40
5.4 Problems 41
6 Empirical distribution functions 43
6.1 Fuzzy valued empirical distribution function 43
6.2 Fuzzy empirical fractiles 45
6.3 Smoothed empirical distribution function 45
6.4 Problems 47
7 Empirical correlation for fuzzy data 49
7.1 Fuzzy empirical correlation coefficient 49
7.2 Problems 52
Part III FOUNDATIONS OF STATISTICAL
INFERENCE WITH FUZZY DATA 53
8 Fuzzy probability distributions 55
8.1 Fuzzy probability densities 55
8.2 Probabilities based on fuzzy probability densities 56
8.3 General fuzzy probability distributions 57
8.4 Problems 58
9 A law of large numbers 59
9.1 Fuzzy random variables 59
9.2 Fuzzy probability distributions induced by fuzzy random variables 61
9.3 Sequences of fuzzy random variables 62
9.4 Law of large numbers for fuzzy random variables 63
9.5 Problems 64
10 Combined fuzzy samples 65
10.1 Observation space and sample space 65
10.2 Combination of fuzzy samples 66
10.3 Statistics of fuzzy data 66
10.4 Problems 67
Part IV CLASSICAL STATISTICAL INFERENCE
FOR FUZZY DATA 69
11 Generalized point estimators 71
11.1 Estimators based on fuzzy samples 71
CONTENTS vii
11.2 Sample moments 73
11.3 Problems 74
12 Generalized confidence regions 75
12.1 Confidence functions 75
12.2 Fuzzy confidence regions 76
12.3 Problems 79
13 Statistical tests for fuzzy data 81
13.1 Test statistics and fuzzy data 81
13.2 Fuzzy p-values 82
13.3 Problems 86
Part V BAYESIAN INFERENCE AND FUZZY
INFORMATION 87
14 Bayes’ theorem and fuzzy information 91
14.1 Fuzzy a priori distributions 91
14.2 Updating fuzzy a priori distributions 92
14.3 Problems 96
15 Generalized Bayes’ theorem 97
15.1 Likelihood function for fuzzy data 97
15.2 Bayes’ theorem for fuzzy a priori distribution and fuzzy data 97
15.3 Problems 101
16 Bayesian confidence regions 103
16.1 Bayesian confidence regions based on fuzzy data 103
16.2 Fuzzy HPD-regions 104
16.3 Problems 106
17 Fuzzy predictive distributions 107
17.1 Discrete case 107
17.2 Discrete models with continuous parameter space 108
17.3 Continuous case 110
17.4 Problems 111
18 Bayesian decisions and fuzzy information 113
18.1 Bayesian decisions 113
18.2 Fuzzy utility 114
18.3 Discrete state space 115
18.4 Continuous state space 116
18.5 Problems 117
viii CONTENTS
Part VI REGRESSION ANALYSIS AND FUZZY
INFORMATION 119
19 Classical regression analysis 121
19.1 Regression models 121
19.2 Linear regression models with Gaussian dependent variables 125
19.3 General linear models 128
19.4 Nonidentical variances 131
19.5 Problems 132
20 Regression models and fuzzy data 133
20.1 Generalized estimators for linear regression models based on the
extension principle 134
20.2 Generalized confidence regions for parameters 138
20.3 Prediction in fuzzy regression models 138
20.4 Problems 139
21 Bayesian regression analysis 141
21.1 Calculation of a posteriori distributions 141
21.2 Bayesian confidence regions 142
21.3 Probabilities of hypotheses 142
21.4 Predictive distributions 142
21.5 A posteriori Bayes estimators for regression parameters 143
21.6 Bayesian regression with Gaussian distributions 144
21.7 Problems 144
22 Bayesian regression analysis and fuzzy information 147
22.1 Fuzzy estimators of regression parameters 148
22.2 Generalized Bayesian confidence regions 150
22.3 Fuzzy predictive distributions 150
22.4 Problems 151
Part VII FUZZY TIME SERIES 153
23 Mathematical concepts 155
23.1 Support functions of fuzzy quantities 155
23.2 Distances of fuzzy quantities 156
23.3 Generalized Hukuhara difference 161
24 Descriptive methods for fuzzy time series 167
24.1 Moving averages 167
24.2 Filtering 169
24.2.1 Linear filtering 169
24.2.2 Nonlinear filters 173
CONTENTS ix
24.3 Exponential smoothing 175
24.4 Components model 176
24.4.1 Model without seasonal component 177
24.4.2 Model with seasonal component 177
24.5 Difference filters 182
24.6 Generalized Holt–Winter method 184
24.7 Presentation in the frequency domain 186
25 More on fuzzy random variables and fuzzy random vectors 189
25.1 Basics 189
25.2 Expectation and variance of fuzzy random variables 190
25.3 Covariance and correlation 193
25.4 Further results 196
26 Stochastic methods in fuzzy time series analysis 199
26.1 Linear approximation and prediction 199
26.2 Remarks concerning Kalman filtering 212
Part VIII APPENDICES 215
A1 List of symbols and abbreviations 217
A2 Solutions to the problems 223
A3 Glossary 245
A4 Related literature 247
References 251
Index 253

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附件 1 : Statistical.Methods.for.Fuzzy.Data,.Reinhard.Viertl,.Wiley,.2011.pdf

2016-09-12 19:24:44, 2.25 M