Reinhard Viertl

Statistical Methods for Fuzzy Data (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 272
• Erscheinungstermin: 26. November 2010
• Englisch
• ISBN-13: 9780470974421
• Artikelnr.: 37301325

Autorenporträt

Reinhard Viertl is Professor of Applied Statistics at Vienna University of Technology. Professor Viertl has been working on statistical analysis of fuzzy data for about 20 years. He is the author of various publications including 5 books and more than 100 papers.

Inhaltsangabe

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 Estimations. 11.1 Estimations 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 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. References. Part
VI REGRESSION ANALYSIS AND FUZZYINFORMATION. 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
Generalized estimators for linear regression models based on the extension
principle. 20.2 Generalized confidence regions for parameters. 20.3
Prediction in fuzzy regression models. 20.4 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. 24.2.1 Linear filtering. 24.2.2 Nonlinear
filters. 24.3 Exponential smoothing. 24.4 Components model. 24.4.1 Model
without seasonal component. 24.4.2 Model with seasonal component. 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. A3 Glossary. A4 Related literature. References. Index.