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Statistical.Methods.for.Fuzzy.Data
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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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