Statistical data are not always precise numbers, or vectors, or categories. Real data are frequently what is called fuzzy. Examples where this fuzziness is obvious are quality of life data, environmental, biological, medical, sociological and economics data. Also the results of measurements can be best described by using fuzzy numbers and fuzzy vectors respectively.Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information.Key Features:Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data.Describes methods of increasing importance with applications in areas such as environmental statistics and social science.Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples.Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data.This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.
Author(s): Reinhard Viertl
Edition: 1
Publisher: John Wiley & Sons
Year: 2011
Language: English
Pages: 270
Tags: Информатика и вычислительная техника;Искусственный интеллект;Интеллектуальный анализ данных;
Title
......Page 5
Contents......Page 7
Preface......Page 13
Part I FUZZY INFORMATION......Page 15
1.1 One-dimensional fuzzy data......Page 17
1.4 Fuzziness and errors......Page 18
1.5 Problems......Page 19
2.1 Fuzzy numbers and characterizing functions......Page 21
2.2 Vectors of fuzzy numbers and fuzzy vectors......Page 28
2.3 Triangular norms......Page 30
2.4 Problems......Page 32
3.1 Functions of fuzzy variables......Page 35
3.2 Addition of fuzzy numbers......Page 37
3.4 Mean value of fuzzy numbers......Page 39
3.6 Fuzzy valued functions......Page 41
3.7 Problems......Page 42
Part II DESCRIPTIVE STATISTICS FOR FUZZY DATA......Page 45
4.2 Maximum of fuzzy data......Page 47
4.4 Problems......Page 48
5.1 Fuzzy frequency of a fixed class......Page 51
5.2 Fuzzy frequency distributions......Page 52
5.3 Axonometric diagram of the fuzzy histogram......Page 54
5.4 Problems......Page 55
6.1 Fuzzy valued empirical distribution function......Page 57
6.3 Smoothed empirical distribution function......Page 59
6.4 Problems......Page 61
7.1 Fuzzy empirical correlation coefficient......Page 63
7.2 Problems......Page 66
Part III FOUNDATIONS OF STATISTICAL INFERENCE WITH FUZZY DATA......Page 67
8.1 Fuzzy probability densities......Page 69
8.2 Probabilities based on fuzzy probability densities......Page 70
8.3 General fuzzy probability distributions......Page 71
8.4 Problems......Page 72
9.1 Fuzzy random variables......Page 73
9.2 Fuzzy probability distributions induced by fuzzy random variables......Page 75
9.3 Sequences of fuzzy random variables......Page 76
9.4 Law of large numbers for fuzzy random variables......Page 77
9.5 Problems......Page 78
10.1 Observation space and sample space......Page 79
10.3 Statistics of fuzzy data......Page 80
10.4 Problems......Page 81
Part IV CLASSICAL STATISTICAL INFERENCE FOR FUZZY DATA......Page 83
11.1 Estimators based on fuzzy samples......Page 85
11.2 Sample moments......Page 87
11.3 Problems......Page 88
12.1 Confidence functions......Page 89
12.2 Fuzzy confidence regions......Page 90
12.3 Problems......Page 93
13.1 Test statistics and fuzzy data......Page 95
13.2 Fuzzy p-values......Page 96
13.3 Problems......Page 100
Part V BAYESIAN INFERENCE AND FUZZY INFORMATION......Page 101
14.1 Fuzzy a priori distributions......Page 105
14.2 Updating fuzzy a priori distributions......Page 106
14.3 Problems......Page 110
15.2 Bayes’ theorem for fuzzy a priori distribution and fuzzy data......Page 111
15.3 Problems......Page 115
16.1 Bayesian confidence regions based on fuzzy data......Page 117
16.2 Fuzzy HPD-regions......Page 118
16.3 Problems......Page 120
17.1 Discrete case......Page 121
17.2 Discrete models with continuous parameter space......Page 122
17.3 Continuous case......Page 124
17.4 Problems......Page 125
18.1 Bayesian decisions......Page 127
18.2 Fuzzy utility......Page 128
18.3 Discrete state space......Page 129
18.4 Continuous state space......Page 130
18.5 Problems......Page 131
Part VI REGRESSION ANALYSIS AND FUZZY INFORMATION......Page 133
19.1 Regression models......Page 135
19.2 Linear regression models with Gaussian dependent variables......Page 139
19.3 General linear models......Page 142
19.4 Nonidentical variances......Page 145
19.5 Problems......Page 146
20 Regression models and fuzzy data......Page 147
20.1 Generalized estimators for linear regression models based on the extension principle......Page 148
20.3 Prediction in fuzzy regression models......Page 152
20.4 Problems......Page 153
21.1 Calculation of a posteriori distributions......Page 155
21.4 Predictive distributions......Page 156
21.5 A posteriori Bayes estimators for regression parameters......Page 157
21.7 Problems......Page 158
22 Bayesian regression analysis and fuzzy information......Page 161
22.1 Fuzzy estimators of regression parameters......Page 162
22.3 Fuzzy predictive distributions......Page 164
22.4 Problems......Page 165
Part VII FUZZY TIME SERIES......Page 167
23.1 Support functions of fuzzy quantities......Page 169
23.2 Distances of fuzzy quantities......Page 170
23.3 Generalized Hukuhara difference......Page 175
24.1 Moving averages......Page 181
24.2.1 Linear filtering......Page 183
24.2.2 Nonlinear filters......Page 187
24.3 Exponential smoothing......Page 189
24.4 Components model......Page 190
24.4.2 Model with seasonal component......Page 191
24.5 Difference filters......Page 196
24.6 Generalized Holt–Winter method......Page 198
24.7 Presentation in the frequency domain......Page 200
25.1 Basics......Page 203
25.2 Expectation and variance of fuzzy random variables......Page 204
25.3 Covariance and correlation......Page 207
25.4 Further results......Page 210
26.1 Linear approximation and prediction......Page 213
26.2 Remarks concerning Kalman filtering......Page 226
Part VIII APPENDICES......Page 229
A1 List of symbols and abbreviations......Page 231
A2 Solutions to the problems......Page 237
A3 Glossary......Page 259
A4 Related literature......Page 261
References......Page 265
Index......Page 267
