Contents

Cover

Title Page

Copyright

Preface

Part I: Fuzzy Information

1: Fuzzy data

1.1 One-dimensional fuzzy data

1.2 Vector-valued fuzzy data

1.3 Fuzziness and variability

1.4 Fuzziness and errors

1.5 Problems

2: Fuzzy numbers and fuzzy vectors

2.1 Fuzzy numbers and characterizing functions

2.2 Vectors of fuzzy numbers and fuzzy vectors

2.3 Triangular norms

2.4 Problems

3: Mathematical operations for fuzzy quantities

3.1 Functions of fuzzy variables

3.2 Addition of fuzzy numbers

3.3 Multiplication of fuzzy numbers

3.4 Mean value of fuzzy numbers

3.5 Differences and quotients

3.6 Fuzzy valued functions

3.7 Problems

Part II: Descriptive Statistics for Fuzzy Data

4: Fuzzy samples

4.1 Minimum of fuzzy data

4.2 Maximum of fuzzy data

4.3 Cumulative sum for fuzzy data

4.4 Problems

5: Histograms for fuzzy data

5.1 Fuzzy frequency of a fixed class

5.2 Fuzzy frequency distributions

5.3 Axonometric diagram of the fuzzy histogram

5.4 Problems

6: Empirical distribution functions

6.1 Fuzzy valued empirical distribution function

6.2 Fuzzy empirical fractiles

6.3 Smoothed empirical distribution function

6.4 Problems

7: Empirical correlation for fuzzy data

7.1 Fuzzy empirical correlation coefficient

7.2 Problems

Part III: Foundations of Statistical Inference With Fuzzy Data

8: Fuzzy probability distributions

8.1 Fuzzy probability densities

8.2 Probabilities based on fuzzy probability densities

8.3 General fuzzy probability distributions

8.4 Problems

9: A law of large numbers

9.1 Fuzzy random variables

9.2 Fuzzy probability distributions induced by fuzzy random variables

9.3 Sequences of fuzzy random variables

9.4 Law of large numbers for fuzzy random variables

9.5 Problems

10: Combined fuzzy samples

10.1 Observation space and sample space

10.2 Combination of fuzzy samples

10.3 Statistics of fuzzy data

10.4 Problems

Part IV: Classical Statistical Inference for Fuzzy Data

11: Generalized point estimators

11.1 Estimators based on fuzzy samples

11.2 Sample moments

11.3 Problems

12: Generalized confidence regions

12.1 Confidence functions

12.2 Fuzzy confidence regions

12.3 Problems

13: Statistical tests for fuzzy data

13.1 Test statistics and fuzzy data

13.2 Fuzzy p-values

13.3 Problems

Part V: Bayesian Inference and Fuzzy Information

14: Bayes’ theorem and fuzzy information

14.1 Fuzzy a priori distributions

14.2 Updating fuzzy a priori distributions

14.3 Problems

15: Generalized Bayes’ theorem

15.1 Likelihood function for fuzzy data

15.2 Bayes’ theorem for fuzzy a priori distribution and fuzzy data

15.3 Problems

16: Bayesian confidence regions

16.1 Bayesian confidence regions based on fuzzy data

16.2 Fuzzy HPD-regions

16.3 Problems

17: Fuzzy predictive distributions

17.1 Discrete case

17.2 Discrete models with continuous parameter space

17.3 Continuous case

17.4 Problems

18: Bayesian decisions and fuzzy information

18.1 Bayesian decisions

18.2 Fuzzy utility

18.3 Discrete state space

18.4 Continuous state space

18.5 Problems

Part VI: Regression Analysis and Fuzzy Information

19: Classical regression analysis

19.1 Regression models

19.2 Linear regression models with Gaussian dependent variables

19.3 General linear models

19.4 Nonidentical variances

19.5 Problems

20: Regression models and fuzzy data

20.1 Regression Models and Fuzzy Data

20.2 Generalized estimators for linear regression models based on the extension principle

20.3 Generalized confidence regions for parameters

20.4 Prediction in fuzzy regression models

20.5 Problems

21: Bayesian regression analysis

21.1 Calculation of a posteriori distributions

21.2 Bayesian confidence regions

21.3 Probabilities of Hypotheses

21.4 Predictive distributions

21.5 A posteriori Bayes estimators for regression parameters

21.6 Bayesian regression with Gaussian distributions

21.7 Problems

22: Bayesian regression analysis and fuzzy information

22.1 Fuzzy estimators of regression parameters

22.2 Generalized Bayesian confidence regions

22.3 Fuzzy predictive distributions

22.4 Problems

Part VII: Fuzzy time series

23: Mathematical concepts

23.1 Support functions of fuzzy quantities

23.2 Distances of fuzzy quantities

23.3 Generalized Hukuhara difference

24: Descriptive methods for fuzzy time series

24.1 Moving averages

24.2 Filtering

24.3 Exponential smoothing

24.4 Components model

24.5 Difference filters

24.6 Generalized Holt–Winter method

24.7 Presentation in the frequency domain

25: More on fuzzy random variables and fuzzy random vectors

25.1 Basics

25.2 Expectation and variance of fuzzy random variables

25.3 Covariance and correlation

25.4 Further results

26: Stochastic methods in fuzzy time series analysis

26.1 Linear approximation and prediction

26.2 Remarks concerning Kalman filtering

Part VIII: Appendices

A1: List of symbols and abbreviations

A2: Solutions to the problems

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

A3: Glossary

A4: Related literature

References

Index