Information Measures

Information and its Description in Science and Engineering
Softcover Nachdruck of the original 1. Auflage 2001
ISBN: 978-3-540-40855-0
Verlag: Springer Berlin Heidelberg

Buch, Englisch, 548 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 855 g

Reihe: Signals and Communication Technology

This book is intended to be an introduction to the mathematical description of information in science. The necessary mathematical theory of this introduction will be treated in a more vivid way than in the usual theorem-proof structure. This, however, enables us to develop an idea of the connections between different information measures and to understand the trains of thought in their derivation, which is a crucial point for correct applications. It is therefore our intention in the mathematical descriptions to evolve the important ideas of the derivations, so that we obtain the resulting functions as well as the main thoughts and the conditions for the validity of the result. This simplifies the handling of the information measures, which are sometimes hard to classify without any additional background information. Though the mathematical descriptions are the exact formulations of the measures examined, we do not restrict ourselves to rigorous mathematical considerations, but we will also integrate the different measures into the structure and context of possible information measures. Nevertheless the mathematical approach is unavoidable when we are looking for an objective description and for possible applications in optimization.

Zielgruppe

Research

Autoren/Hrsg.

Arndt, Christoph

Fachgebiete

• Interdisziplinäres Wissenschaften Wissenschaften: Allgemeines
• Mathematik | Informatik EDV | Informatik Angewandte Informatik Computeranwendungen in Wissenschaft & Technologie
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Informationstheorie, Kodierungstheorie
• Technische Wissenschaften Technik Allgemein Technik: Allgemeines
• Technische Wissenschaften Technik Allgemein Computeranwendungen in der Technik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik EDV | Informatik Daten / Datenbanken Informationstheorie, Kodierungstheorie
• Mathematik | Informatik EDV | Informatik Professionelle Anwendung Computer-Aided Design (CAD)
• Naturwissenschaften Physik Angewandte Physik Statistische Physik, Dynamische Systeme
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
• Technische Wissenschaften Elektronik | Nachrichtentechnik Nachrichten- und Kommunikationstechnik Funktechnik

Inhaltsverzeichnis

Abstract.- Structure and Structuring.- 1 Introduction.- Science and information.- Man as control loop.- Information, complexity and typical sequences.- Concepts of information.- Information, its technical dimension and the meaning of a message.- Information as a central concept.- 2 Basic considerations.- 2.1 Formal derivation of information.- 2.2 Application of the information measure (Shannon’s information).- 2.3 The law of Weber and Fechner.- 2.4 Information of discrete random variables.- 3 Historic development of information theory.- 3.1 Development of information transmission.- 3.2 Development of information functions.- 4 The concept of entropy in physics.- The laws of thermodynamics:.- 4.1 Macroscopic entropy.- 4.2 Statistical entropy.- 4.3 Dynamic entropy.- 5 Extension of Shannon’s information.- 5.1 Rényi’s Information 1960.- 5.2 Another generalized entropy (logical expansion).- 5.3 Gain of information via conditional probabilities.- 5.4 Other entropy or information measures.- 6 Generalized entropy measures.- 6.1 The corresponding measures of divergence.- 6.2 Weighted entropies and expectation values of entropies.- 7 Information functions and gaussian distributions.- 7.1 Rényi’s information of a gaussian distributed random variable.- 7.2 Shannon’s information.- 8 Shannon’s information of discrete probability distributions.- 8.1 Continuous and discrete random variables.- 8.2 Shannon’s information of a gaussian distribution.- 8.3 Shannon’s information as the possible gain of information in an observation.- 8.4 Limits of the information, limitations of the resolution.- 8.5 Maximization of the entropy of a continuous random variable.- 9 Information functions for gaussian distributions part II.- 9.1 Kullback’s information.- 9.2 Kullback’sdivergence.- 9.3 Kolmogorov’s information.- 9.4 Transformation of the coordinate system and the effects on the information.- 9.5 Transformation, discrete and continuous measures of entropy.- 9.6 Summary of the information functions.- 10 Bounds of the variance.- 10.1 Cramér-Rao bound.- 10.2 Chapman-Robbins bound.- 10.3 Bhattacharrya bound.- 10.4 Barankin bound.- 10.5 Other bounds.- 10.6 Summary.- 10.7 Biased estimator.- 11 Ambiguity function.- 11.1 The ambiguity function and Kullback’s information.- 11.2 Connection between ambiguity function and Fisher’s information.- 11.3 Maximum likelihood estimation and the ambiguity function.- 11.4 The ML estimation is asymptotically efficient.- 11.5 Transition to the Akaike information criterion.- 12 Akaike’s information criterion.- 12.1 Akaike’s information criterion and regression.- 12.2 BIC, SC or HQ.- 13 Channel information.- 13.1 Redundancy.- 13.2 Rate of transmission and equivocation.- 13.3 Hadamard’s inequality and Gibbs’s second theorem.- 13.4 Kolmogorov’s information.- 13.5 Kullbacks divergence.- 13.6 An example of a transmission.- 13.7 Communication channel and information processing.- 13.8 Shannon’s bound.- 13.9 Example of the channel capacity.- 14 ‘Deterministic’ and stochastic information.- 14.1 Information in state space models.- 14.2 The observation equation.- 14.3 Transmission faster than light.- 14.4 Information about state space variables.- 15 Maximum entropy estimation.- 15.1 The difference between maximum entropy and minimum variance.- 15.2 The difference from bootstrap or resampling methods.- 15.3 A maximum entropy example.- 15.4 Maximum entropy: The method.- 15.5 Maximum entropy and minimum discrimination information.- 15.6 Generation of generalized entropy measures.- 16 Concludingremarks.- 16.1 Information, entropy and self-organization.- 16.2 Complexity theory.- 16.3 Data reduction.- 16.4 Cryptology.- 16.5 Concluding considerations.- 16.6 Information.- A.1 Inequality for Kullback’s information.- A.2 The log-sum inequality.- A.3 Generalized entropy, divergence and distance measures.- A.3.1 Entropy measures.- A.3.2 Generalized measures of distance.- A.3.3 Generalized measures of the directed divergence.- A.3.4 Generalized measures of divergence.- A.3.4.1 Information radius and the J-divergence.- A.3.4.2 Generalization of the R-divergence.- A.3.4.3 Generalization of the J-divergence.- A.4 A short introduction to probability theory.- A.4.1 Axiomatic definition of probability.- A.4.1.1 Events, elementary events, sample space.- A.4.1.2 Classes of subsets, fields.- A.4.1.3 Axiomatic definition of probability according to Kolmogorov.- Probability space.- A.4.1.4 Random variables.- A.4.1.5 Probability distribution.- A.4.1.6 Probability space, sample space, realization space.- A.4.1.7 Probability distribution and distribution density function.- A.4.1.8 Probability distribution density function (PDF).- A.5 The regularity conditions.- A.6 State space description.