Chapter 1: In this chapter a brief literature survey on measures of entropy and divergence measures is presented. It also outlines the basic concepts of fuzzy sets. A brief review on fuzzy information measures and fuzzy directed divergence are given here. The concept of multiple criteria decision making problem is also presented. In addition, a general overview of coding theory is given and summarizes the objectives with an overview of the work reported in later chapters.
Chapter 2: In Chapter 2 after reviewing some literature on measures of information for fuzzy sets, a new generalized fuzzy information measure involving two parameters of order α and type β has been introduced. The necessary properties of the proposed measure have been verified. Further, the monotonic nature of generalized fuzzy information measure with respect to the parameters is studied and the validity of the same is verified by constructing the computed tables and plots on taking different values of the parameters.
Chapter 3: Divergence is an important measure in information theory as well as in fuzzy set theory which has widely used by researchers in many application areas. Generalized divergence measures provide flexibility to the users and enhance their applicability range. This chapter proposes a new generalized fuzzy divergence measure. It may be remarked that the strength of a measure lies in its properties. The new measure has important properties proved in this chapter to enhance the employability of this measure. Special cases are also discussed for providing particular results. Chapter 3 deals with the introduction of a new generalized measure of fuzzy directed divergence involving two real parameters. The proposed measure satisfies all the necessary properties of being a measure. Some additional properties of the proposed measure have also been studied. Further, the monotonic nature of generalized fuzzy directed divergence measure with respect to the parameters is studied and validity of the same is checked by constructing the computed tables and plots on taking different fuzzy sets and different values of the parameters. Corresponding measures of total ambiguity and fuzzy information improvement have also been defined and studied.
Table of Contents
Chapter 1: Introduction
1.1 Concept of Information Theory
1.2 Concept of Fuzzy Sets
1.3 Measures of Entropy
1.4 Measures of Directed Divergence
1.5 Measure of Fuzzy Information
1.6 Concept of Fuzzy Entropy
1.7 Concept of Fuzzy Divergence
1.8 Concept of Decision Making
1.9 Concept of Noiseless Coding Theorem
Chapter 2: New Parametric Measure of Fuzzy Information Measure
2.1 Introduction
2.2 A New Parametric Measure of Fuzzy Information Measure Involving Two Parameters α And β
2.3 A New Parametric Measure of Fuzzy Information Measure Involving Three Parameters α, β and γ.
Chapter 3: On Some New Divergence Measure between Fuzzy Sets
3.1 Introduction
3.2 Generalized Measure of Fuzzy Directed Divergence
Chapter 4: Application in Decision Making
4.1 Introduction
4.2 Numerical Example Based on Fuzzy Information Measure
4.3 Numerical Example Based On Fuzzy Directed Divergence
Chapter 5: Fuzzy Noiseless Coding Theorem
5.1 Introduction
5.2. Fuzzy Noiseless Coding Theorem
Chapter 6: Fuzzy Noiseless Coding Theorem for Binary 1:1 Codes
6.1 Introduction
6.2 Fuzzy Noiseless Coding Theorem for 1:1 Codes
Chapter 7: Conclusion and Future Scope
Research Objective and Scope
The primary objective of this work is to generalize various fuzzy entropy and divergence measures and explore their practical applications through mathematical modeling. The research addresses the challenge of quantifying uncertainty and vagueness in information systems, particularly in decision-making and coding theory, by introducing new parametric measures, establishing their theoretical validity, and demonstrating their utility in real-world scenarios.
- Development of new generalized parametric measures for fuzzy information and fuzzy directed divergence.
- Theoretical verification of the monotonic behavior and validity of the proposed information measures.
- Application of developed fuzzy measures to solve Multiple Criteria Decision Making (MCDM) problems.
- Derivation and rigorous proof of new noiseless coding theorems specifically tailored for fuzzy information and binary 1:1 codes.
Auszug aus dem Buch
1.1 CONCEPT OF INFORMATION THEORY
The utmost outstanding facet of twentieth era has ensued to be the enlargement and utilization of novel transmission channels coexisting with the progression of tools of communicating and dealing information. The term information and exclaiming information is potency is extremely conventional and frequently confronted in our regular existence. Information is diffused via Human voice, Telephone, Radio, Television, Books and Newspaper etc.
Information had drawn attention wholly from the conception of telegraph and telephone. In 1844, American originator Sameul, F.B.Morse fabricated a telegraph line amongst Washington D.C. and Batlimore. Maryland Morse confronted several electrical obstructions when indications across dormant communication lines, however unaccountably he met rare problems.
This function was furthermost diligently related with the work of American electrical engineer Shannon (1948) who promulgated Information theory while examining the numerical composition of electrical transmission tools. He contributed an arithmetical method to concretize nonetheless this method has foundation back in 20th century when Hartley (1928) attempted to acquire a numerical quantity of information to retrieve the potentials to telecommunication system.
Summary of Chapters
Chapter 1: Introduction: This chapter presents a comprehensive literature survey on entropy and divergence measures, outlines basic fuzzy set concepts, and introduces the overall scope of coding theory and decision-making applications.
Chapter 2: New Parametric Measure of Fuzzy Information Measure: This chapter introduces a new generalized fuzzy information measure involving two parameters and verifies its monotonic properties and validity through numerical analysis.
Chapter 3: On Some New Divergence Measure between Fuzzy Sets: This chapter proposes a generalized measure of fuzzy directed divergence and analyzes its properties, including total ambiguity and information improvement.
Chapter 4: Application in Decision Making: This chapter applies the proposed fuzzy information and divergence measures to practical Multiple Criteria Decision Making (MCDM) problems, supported by various numerical examples.
Chapter 5: Fuzzy Noiseless Coding Theorem: This chapter investigates the application of Holder’s inequality to coding theory, establishing noiseless coding theorems based on the proposed fuzzy information measures.
Chapter 6: Fuzzy Noiseless Coding Theorem for Binary 1:1 Codes: This chapter introduces mean code word length for 1:1 codes and explores the relationships between average code word length and fuzzy information measures for binary systems.
Chapter 7: Conclusion and Future Scope: This chapter encapsulates the research findings and discusses potential directions for future study in the field of fuzzy information measures.
Keywords
Fuzzy Set Theory, Information Theory, Fuzzy Entropy, Directed Divergence, Noiseless Coding Theorem, Decision Making, Parametric Measures, Ambiguity, Vagueness, Fuzzy Logic, Coding Theory, Monotonicity, Optimization, Uncertainty, Membership Function.
Frequently Asked Questions
What is the primary focus of this research?
The research focuses on the mathematical generalization of fuzzy entropy and divergence measures and their application to decision-making processes and information coding theory.
What are the central thematic areas?
The core thematic areas include Fuzzy Set Theory, Information Theory, Fuzzy Information Measures, Directed Divergence, Decision Making algorithms, and Coding Theorems.
What is the primary goal or research question?
The goal is to develop and validate new, robust parametric measures for fuzzy information and directed divergence that account for imprecision and vagueness, while demonstrating their effectiveness in decision-making and communications.
Which scientific methods are utilized?
The work employs mathematical derivation, analytical verification of properties (such as idempotency, monotonicity, and concavity), computational table construction, and graphical plotting to validate the performance of the proposed measures.
What topics are covered in the main body?
The main body covers the theoretical development of new fuzzy information and divergence measures, the study of their properties, their application to Multiple Criteria Decision Making (MCDM), and the derivation of fuzzy noiseless coding theorems.
Which keywords best characterize this work?
The essential keywords are Fuzzy Set Theory, Information Theory, Fuzzy Entropy, Directed Divergence, Noiseless Coding Theorem, Decision Making, and Parametric Measures.
How does this work contribute to Multiple Criteria Decision Making (MCDM)?
This work provides an algorithmic approach using fuzzy information and divergence measures to rank alternatives under vague or imprecise conditions, offering a more flexible framework than traditional crisp methods.
Why is the Fuzzy Noiseless Coding Theorem critical?
It establishes crucial lower bounds on code word lengths in communication systems where input information contains fuzzy uncertainty, essentially bridging the gap between classical coding theory and fuzzy set theory.
- Quote paper
- Manu Banga (Author), Information Coding Using Fuzzy Set Theory, Munich, GRIN Verlag, https://www.grin.com/document/1281742
