This book is an introduction to the mathematical description of information in science and engineering. The necessary mathematical theory will be treated in a more vivid way than in the usual theorem-proof structure. This enables the reader to develop an idea of the connections between different information measures and to understand the trains of thoughts in their derivation. As there exist a great number of different possible ways to describe information, these measures are presented in a coherent manner. Some examples of the information measures examined are: Shannon information, applied in coding theory; Akaike information criterion, used in system identification to determine auto-regressive models and in neural networks to identify the number of neu-rons; and Cramer-Rao bound or Fisher information, describing the minimal variances achieved by unbiased estimators. This softcover edition addresses researchers and students in electrical engineering, particularly in control and communications, physics, and applied mathematics.

Klappentext

From the reviews: "Bioinformaticians are facing the challenge of how to handle immense amounts of raw data, [...] and render them accessible to scientists working on a wide variety of problems. [This book] can be such a tool." IEEE Engineering in Medicine and Biology

Inhalt

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.- 2.1.1 Unit and reference scale.- 2.1.2 Information and the unit element.- 2.2 Application of the information measure (Shannon's information).- 2.2.1 Summary.- 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.- 3.1.1 Samuel F. B. Morse 1837.- 3.1.2 Thomas Edison 1874.- 3.1.3 Nyquist 1924.- 3.1.4 Optimal number of characters of the alphabet used for the coding.- 3.2 Development of information functions.- 3.2.1 Hartley 1928.- 3.2.2 Dennis Gabor 1946.- 3.2.3 Shannon 1948.- 3.2.3.1 Validity of the postulates for Shannon's Information.- 3.2.3.2 Shannon's information (another possibility of a derivation).- 3.2.3.3 Properties of Shannon's information, entropy.- 3.2.3.4 Shannon's entropy or Shannon's information.- 3.2.3.5 The Kraft inequality.- Kraft's inequality:.- Proof of Kraft's inequality:.- 3.2.3.6 Limits of the optimal length of codewords.- 3.2.3.6.1 Shannon's coding theorem.- 3.2.3.6.2 A sequence of n symbols (elements).- 3.2.3.6.3 Application of the previous results.- 3.2.3.7 Information and utility (coding, porfolio analysis).- 4 The concept of entropy in physics.- The laws of thermodynamics:.- 4.1 Macroscopic entropy.- 4.1.1 Sadi Carnot 1824.- 4.1.2 Clausius's entropy 1850.- 4.1.3 Increase of entropy in a closed system.- 4.1.4 Prigogine's entropy.- 4.1.5 Entropy balance equation.- 4.1.6 Gibbs's free energy and the quality of the energy.- 4.1.7 Considerations on the macroscopic entropy.- 4.1.7.1 Irreversible transformations.- 4.1.7.2 Perpetuum mobile and transfer of heat.- 4.2 Statistical entropy.- 4.2.1 Boltzmann's entropy.- 4.2.2 Derivation of Boltzmann's entropy.- 4.2.2.1 Variation, permutation and the formula of Stirling.- 4.2.2.2 Special case: Two states.- 4.2.2.3 Example: Lottery.- 4.2.3 The Boltzmann factor.- 4.2.4 Maximum entropy in equilibrium.- 4.2.5 Statistical interpretation of entropy.- 4.2.6 Examples regarding statistical entropy.- 4.2.6.1 Energy and fluctuation.- 4.2.6.2 Quantized oscillator.- 4.2.7 Brillouin-Schrödinger negentropy.- 4.2.7.1 Brillouin: Precise definition of information.- 4.2.7.2 Negentropy as a generalization of Carnot's principle.- Maxwell's demon.- 4.2.8 Information measures of Hartley and Boltzmann.- 4.2.8.1 Examples.- 4.2.9 Shannon's entropy.- 4.3 Dynamic entropy.- 4.3.1 Eddington and the arrow of time.- 4.3.2 Kolmogorov's entropy.- 4.3.3 Rényi's entropy.- 5 Extension of Shannon's information.- 5.1 Rényi's Information 1960.- 5.1.1 Properties of Rényi's entropy.- 5.1.2 Limits in the interval 0 ? ?< ?.- 5.1.3 Nonnegativity for discrete events.- 5.1.4 Additivity and a connection to Minkowski's norm.- 5.1.5 The meaning of S?(A) for ? 1.- 5.1.6 Graphical presentations of Rényi's information.- 5.2 Another generalized entropy (logical expansion).- 5.3 Gain of information via conditional probabilities.- 5.4 Other entropy or information measures.- 5.4.1 Daroczy's entropy.- 5.4.2 Quadratic entropy.- 5.4.3 R-norm entropy.- 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.- 7.1.1 Rényi's ?-information.- 7.1.2 Rényi's G-divergence.- 7.2 Shannon's information.- 8 Shannon's information of discrete probability distributions.- 8.1 Continuous and discrete random variables.- 8.1.1 Summary.- 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.- 8.4.1 The resolution or the precision of the measurements.- 8.4.2 The uncertainty relation of the Fourier transformation.- 8.5 Maximization of the entropy of a continuous random variable.- 9 Information functions for gaussian distributions part II.- 9.1 Kullback's information.- 9.1.1 G1 for gaussian distribution densites.- 9.2 Kullback's divergence.- 9.2.1 Jensen's inequality for G1.- 9.3 Kolmogorov's information.- 9.4 Transformation of the coordinate system and the effects on the information.- 9.4.1 S?-information.- 9.4.2 G-divergence.- 9.4.3 S-information.- 9.4.3.1 Example.- 9.4.4 Discrimination information.- 9.4.5 Kolmogorov's information.- 9.4.6 Prerequisites for the transformations.- 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.- 10.1.1 Fisher's information for gaussian distribution densities.- 10.1.2 Fisher's information and Kullback's information.- 10.1.3 Fisher's information and the metric tensor.- 10.1.4 Fisher's information and the stochastic observability.- 10.1.4.1 Fisher's information and the Matrix-Riccati equation.- 10.1.5 Fisher's information and maximum likelihood estimation.- 10.1.6 Fisher's information and weighted least-squares estimation.- 10.1.7 The availability of the Cramér-Rao bound.- 10.1.8 Efficiency, asymptotic efficiency, consistency, bias.- 10.1.8.1 Unbiased estimator.- 10.1.8.2 Consistency.- 10.1.8.3 Efficiency.- 10.1.9 Summary.- 10.2 Chapman-Robbins bound.- 10.2.1 Cramér-Rao bound versus Chapman-Robbins bound.- 10.3 Bhattacharrya bound.- Remark:.- Remark.- 10.3.1 Bhattacharrya bound and Cramér-Rao bound.- 10.3.2 Bhattacharrya's bound for gaussian distribution densities.- 10.4 Barankin bound.- 10.5 Other bounds.- Fraser-Guttman bound.- Kiefer bound.- Extended Fraser-Guttman bound.- 10.6 Summary.- 10.7 Biased estimator.- 10.7.1 Biased estimator versus unbiased 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.- 11.3.1 Maximum likelihood estimation = minimum Kullback estimation = maximum ambiguity estimation = minimum variance estimation.- 11.3.2 Maximum likelihood estimation.- 11.3.2.1 Application: Discriminator (Demodulation).- 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.- 12.1.1 Least-…