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Entropy Theory and its Application in Environmental and WaterEngineering responds to the need for a book that deals withbasic concepts of entropy theory from a hydrologic and waterengineering perspective and then for a book that deals withapplications of these concepts to a range of water engineeringproblems. The range of applications of entropy is constantlyexpanding and new areas finding a use for the theory arecontinually emerging. The applications of concepts and techniquesvary across different subject areas and this book aims to relatethem directly to practical problems of environmental…mehr

Produktbeschreibung
Entropy Theory and its Application in Environmental and WaterEngineering responds to the need for a book that deals withbasic concepts of entropy theory from a hydrologic and waterengineering perspective and then for a book that deals withapplications of these concepts to a range of water engineeringproblems. The range of applications of entropy is constantlyexpanding and new areas finding a use for the theory arecontinually emerging. The applications of concepts and techniquesvary across different subject areas and this book aims to relatethem directly to practical problems of environmental and waterengineering. The book presents and explains the Principle of Maximum Entropy(POME) and the Principle of Minimum Cross Entropy (POMCE) and theirapplications to different types of probability distributions.Spatial and inverse spatial entropy are important for urbanplanning and are presented with clarity. Maximum entropy spectralanalysis and minimum cross entropy spectral analysis are powerfultechniques for addressing a variety of problems faced byenvironmental and water scientists and engineers and are describedhere with illustrative examples. Giving a thorough introduction to the use of entropy to measurethe unpredictability in environmental and water systems this bookwill add an essential statistical method to the toolkit ofpostgraduates, researchers and academic hydrologists, waterresource managers, environmental scientists and engineers. Itwill also offer a valuable resource for professionals in the sameareas, governmental organizations, private companies as well asstudents in earth sciences, civil and agricultural engineering, andagricultural and rangeland sciences. This book: * Provides a thorough introduction to entropy for beginners andmore experienced users * Uses numerous examples to illustrate the applications of thetheoretical principles * Allows the reader to apply entropy theory to the solution ofpractical problems * Assumes minimal existing mathematical knowledge * Discusses the theory and its various aspects in both univariateand bivariate cases * Covers newly expanding areas including neural networks from anentropy perspective and future developments.

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  • Produktdetails
  • Verlag: John Wiley & Sons
  • Seitenzahl: 664
  • Erscheinungstermin: 2. Januar 2013
  • Englisch
  • ISBN-13: 9781118428597
  • Artikelnr.: 37349805
Autorenporträt
Vijay P. Singh, Texas A & M University, USA
Inhaltsangabe
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
298 7.3 Continuous variable: general formulation
305 Questions
308 References
309 8 Parameter Estimation
310 8.1 Ordinary entropy-based parameter estimation method
310 8.2 Parameter-space expansion method
325 8.3 Contrast with method of maximum likelihood estimation (MLE)
329 8.4 Parameter estimation by numerical methods
331 Questions
332 References
333 Additional Reading
334 9 Spatial Entropy
335 9.1 Organization of spatial data
336 9.2 Spatial entropy statistics
339 9.3 One dimensional aggregation
353 9.4 Another approach to spatial representation
360 9.5 Two-dimensional aggregation
363 9.6 Entropy maximization for modeling spatial phenomena
376 9.7 Cluster analysis by entropy maximization
380 9.8 Spatial visualization and mapping
384 9.9 Scale and entropy
386 9.10 Spatial probability distributions
388 9.11 Scaling: rank size rule and Zipf's law
391 Questions
393 References
394 Further Reading
395 10 Inverse Spatial Entropy
398 10.1 Definition
398 10.2 Principle of entropy decomposition
402 10.3 Measures of information gain
405 10.4 Aggregation properties
417 10.5 Spatial interpretations
420 10.6 Hierarchical decomposition
426 10.7 Comparative measures of spatial decomposition
428 Questions
433 References
435 11 Entropy Spectral Analyses
436 11.1 Characteristics of time series
436 11.2 Spectral analysis
446 11.3 Spectral analysis using maximum entropy
464 11.4 Spectral estimation using configurational entropy
483 11.5 Spectral estimation by mutual information principle
486 References
490 Additional Reading
490 12 Minimum Cross Entropy Spectral Analysis
492 12.1 Cross-entropy
492 12.2 Minimum cross-entropy spectral analysis (MCESA)
493 12.3 Minimum cross-entropy power spectrum given auto-correlation
503 12.4 Cross-entropy between input and output of linear filter
509 12.5 Comparison
512 12.6 Towards efficient algorithms
514 12.7 General method for minimum cross-entropy spectral estimation
515 References
515 Additional References
516 13 Evaluation and Design of Sampling and Measurement Networks
517 13.1 Design considerations
517 13.2 Information-related approaches
518 13.3 Entropy measures
521 13.4 Directional information transfer index
530 13.5 Total correlation
537 13.6 Maximum information minimum redundancy (MIMR)
539 Questions
553 References
554 Additional Reading
556 14 Selection of Variables and Models
559 14.1 Methods for selection
559 14.2 Kullback-Leibler (KL) distance
560 14.3 Variable selection
560 14.4 Transitivity
561 14.5 Logit model
561 14.6 Risk and vulnerability assessment
574 Questions
578 References
579 Additional Reading
580 15 Neural Networks
581 15.1 Single neuron
581 15.2 Neural network training
585 15.3 Principle of maximum information preservation
588 15.4 A single neuron corrupted by processing noise
589 15.5 A single neuron corrupted by additive input noise
592 15.6 Redundancy and diversity
596 15.7 Decision trees and entropy nets
598 Questions
602 References
603 16 System Complexity
605 16.1 Ferdinand's measure of complexity
605 16.2 Kapur's complexity analysis
618 16.3 Cornacchio's generalized complexity measures
620 16.4 Kapur's simplification
627 16.5 Kapur's measure
627 16.6 Hypothesis testing
628 16.7 Other complexity measures
628 Questions
631 References
631 Additional References
632 Author Index
633 Subject Index
639