Entropy Theory and Its Application in Environmental and Water - Engineerin

Informationen zum Autor Vijay P. Singh,  Texas A & M University, USA Klappentext Entropy Theory and its Application in Environmental and Water Engineering responds to the need for a book that deals with basic concepts of entropy theory from a hydrologic and water engineering perspective and then for a book that deals with applications of these concepts to a range of water engineering problems. The range of applications of entropy is constantly expanding and new areas finding a use for the theory are continually emerging. The applications of concepts and techniques vary across different subject areas and this book aims to relate them directly to practical problems of environmental and water engineering.The book presents and explains the Principle of Maximum Entropy (POME) and the Principle of Minimum Cross Entropy (POMCE) and their applications to different types of probability distributions. Spatial and inverse spatial entropy are important for urban planning and are presented with clarity. Maximum entropy spectral analysis and minimum cross entropy spectral analysis are powerful techniques for addressing a variety of problems faced by environmental and water scientists and engineers and are described here with illustrative examples.Giving a thorough introduction to the use of entropy to measure the unpredictability in environmental and water systems this book will add an essential statistical method to the toolkit of postgraduates, researchers and academic hydrologists, water resource managers, environmental scientists and engineers. It will also offer a valuable resource for professionals in the same areas, governmental organizations, private companies as well as students in earth sciences, civil and agricultural engineering, and agricultural and rangeland sciences.This book:* Provides a thorough introduction to entropy for beginners and more experienced users* Uses numerous examples to illustrate the applications of the theoretical principles* Allows the reader to apply entropy theory to the solution of practical problems* Assumes minimal existing mathematical knowledge* Discusses the theory and its various aspects in both univariate and bivariate cases* Covers newly expanding areas including neural networks from an entropy perspective and future developments. Zusammenfassung Entropy Theory and its Application in Environmental and Water Engineering responds to the need for a book that deals with basic concepts of entropy theory from a hydrologic and water engineering perspective and then for a book that deals with applications of these concepts to a range of water engineering problems. Inhaltsverzeichnis Preface, xv Acknowledgments, xix 1 Introduction, 1 1.1 Systems and their characteristics, 1 1.1.1 Classes of systems, 1 1.1.2 System states, 1 1.1.3 Change of state, 2 1.1.4 Thermodynamic entropy, 3 1.1.5 Evolutive connotation of entropy, 5 1.1.6 Statistical mechanical entropy, 5 1.2 Informational entropies, 7 1.2.1 Types of entropies, 8 1.2.2 Shannon entropy, 9 1.2.3 Information gain function, 12 1.2.4 Boltzmann, Gibbs and Shannon entropies, 14 1.2.5 Negentropy, 15 1.2.6 Exponential entropy, 16 1.2.7 Tsallis entropy, 18 1.2.8 Renyi entropy, 19 1.3 Entropy, information, and uncertainty, 21 1.3.1 Information, 22 1.3.2 Uncertainty and surprise, 24 1.4 Types of uncertainty, 25 1.5 Entropy and related concepts, 27 1.5.1 Information content of data, 27 1.5.2 Criteria for model selection, 28 1.5.3 Hypothesis testing, 29 1.5.4 Risk assessment, 29 Questions, 29 References, 31 Additional References, 32 2 Entropy Theory, 33 2.1 Formulation of entropy, 33 2.2 Shannon entropy, 39 2.3 Connotations of information and entropy, 42 2.3.1 A...

Table des matières

Preface, xv
 
Acknowledgments, xix
 
1 Introduction, 1
 
1.1 Systems and their characteristics, 1
 
1.2 Informational entropies, 7
 
1.3 Entropy, information, and uncertainty, 21
 
1.4 Types of uncertainty, 25
 
1.5 Entropy and related concepts, 27
 
Questions, 29
 
References, 31
 
Additional References, 32
 
2 Entropy Theory, 33
 
2.1 Formulation of entropy, 33
 
2.2 Shannon entropy, 39
 
2.3 Connotations of information and entropy, 42
 
2.4 Discrete entropy: univariate case and marginal entropy, 46
 
2.5 Discrete entropy: bivariate case, 52
 
2.6 Dimensionless entropies, 79
 
2.7 Bayes theorem, 80
 
2.8 Informational correlation coefficient, 88
 
2.9 Coefficient of nontransferred information, 90
 
2.10 Discrete entropy: multidimensional case, 92
 
2.11 Continuous entropy, 93
 
2.12 Stochastic processes and entropy, 105
 
2.13 Effect of proportional class interval, 107
 
2.14 Effect of the form of probability distribution, 110
 
2.15 Data with zero values, 111
 
2.16 Effect of measurement units, 113
 
2.17 Effect of averaging data, 115
 
2.18 Effect of measurement error, 116
 
2.19 Entropy in frequency domain, 118
 
2.20 Principle of maximum entropy, 118
 
2.21 Concentration theorem, 119
 
2.22 Principle of minimum cross entropy, 122
 
2.23 Relation between entropy and error probability, 123
 
2.24 Various interpretations of entropy, 125
 
2.25 Relation between entropy and variance, 133
 
2.26 Entropy power, 135
 
2.27 Relative frequency, 135
 
2.28 Application of entropy theory, 136
 
Questions, 136
 
References, 137
 
Additional Reading, 139
 
3 Principle of Maximum Entropy, 142
 
3.1 Formulation, 142
 
3.2 POME formalism for discrete variables, 145
 
3.3 POME formalism for continuous variables, 152
 
3.4 POME formalism for two variables, 158
 
3.5 Effect of constraints on entropy, 165
 
3.6 Invariance of total entropy, 167
 
Questions, 168
 
References, 170
 
Additional Reading, 170
 
4 Derivation of Pome-Based Distributions, 172
 
4.1 Discrete variable and discrete distributions, 172
 
4.2 Continuous variable and continuous distributions, 185
 
Questions, 203
 
References, 208
 
Additional Reading, 208
 
5 Multivariate Probability Distributions, 213
 
5.1 Multivariate normal distributions, 213
 
5.2 Multivariate exponential distributions, 245
 
5.3 Multivariate distributions using the entropy-copula method, 258
 
5.4 Copula entropy, 265
 
Questions, 266
 
References, 267
 
Additional Reading, 268
 
6 Principle of Minimum Cross-Entropy, 270
 
6.1 Concept and formulation of POMCE, 270
 
6.2 Properties of POMCE, 271
 
6.3 POMCE formalism for discrete variables, 275
 
6.4 POMCE formulation for continuous variables, 279
 
6.5 Relation to POME, 280
 
6.6 Relation to mutual information, 281
 
6.7 Relation to variational distance, 281
 
6.8 Lin's directed divergence measure, 282
 
6.9 Upper bounds for cross-entropy, 286
 
Questions, 287
 
References, 288
 
Additional Reading, 289
 
7 Derivation of POME-Based Distributions, 290
 
7.1 Discrete variable and mean E[x] as a constraint, 290
 
7.2 Discrete variable taking on an infinite set of values,

Détails du produit

• Auteurs: V. P. Singh, Vijay P Singh, Vijay P. Singh, Vijay P. (Texas a & M University) Singh, Vp Singh, Singh Vijay P.
• Edition: Wiley, John and Sons Ltd
• Langues: Anglais
• Format d'édition: Livre Relié
• Sortie: 01.02.2013
• EAN: 9781119976561
• ISBN: 978-1-119-97656-1
• Pages: 662
• Catégories: Sciences naturelles, médecine, informatique, technique > Sciences de la Terre > Général, dictionnaires; Hydrologie, Geowissenschaften, Umweltwissenschaften, Earth Sciences, Bauingenieur- u. Bauwesen, Civil Engineering & Construction, Hydraulic/Water Engineering, Hydraulik, Wasserbau, Environmental Geoscience, Umweltgeowissenschaften, Hydrological Sciences