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.

• 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.

"I recommend this book to anyone interested in exploring new approaches to the extraction of information from novel data sources." (International Statistical Review,

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.

• 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.

"I recommend this book to anyone interested in exploring new approaches to the extraction of information from novel data sources." (International Statistical Review,

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.

• 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.

"I recommend this book to anyone interested in exploring new approaches to the extraction of information from novel data sources." (International Statistical Review,