Entropy Theory and its Application in Environmental and Water Engineering

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.

Table of Contents:
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 Amount of information, 42 2.3.2 Measure of information, 43 2.3.3 Source of information, 43 2.3.4 Removal of uncertainty, 44 2.3.5 Equivocation, 45 2.3.6 Average amount of information, 45 2.3.7 Measurement system, 46 2.3.8 Information and organization, 46 2.4 Discrete entropy: univariate case and marginal entropy, 46 2.5 Discrete entropy: bivariate case, 52 2.5.1 Joint entropy, 53 2.5.2 Conditional entropy, 53 2.5.3 Transinformation, 57 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 2.11.1 Univariate case, 94 2.11.2 Differential entropy of continuous variables, 97 2.11.3 Variable transformation and entropy, 99 2.11.4 Bivariate case, 100 2.11.5 Multivariate case, 105 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 2.24.1 Measure of randomness or disorder, 125 2.24.2 Measure of unbiasedness or objectivity, 125 2.24.3 Measure of equality, 125 2.24.4 Measure of diversity, 126 2.24.5 Measure of lack of concentration, 126 2.24.6 Measure of flexibility, 126 2.24.7 Measure of complexity, 126 2.24.8 Measure of departure from uniform distribution, 127 2.24.9 Measure of interdependence, 127 2.24.10 Measure of dependence, 128 2.24.11 Measure of interactivity, 128 2.24.12 Measure of similarity, 129 2.24.13 Measure of redundancy, 129 2.24.14 Measure of organization, 130 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 3.3.1 Entropy maximization using the method of Lagrange multipliers, 152 3.3.2 Direct method for entropy maximization, 157 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 4.1.1 Constraint E[x] and the Maxwell-Boltzmann distribution, 172 4.1.2 Two constraints and Bose-Einstein distribution, 174 4.1.3 Two constraints and Fermi-Dirac distribution, 177 4.1.4 Intermediate statistics distribution, 178 4.1.5 Constraint: E[N]: Bernoulli distribution for a single trial, 179 4.1.6 Binomial distribution for repeated trials, 180 4.1.7 Geometric distribution: repeated trials, 181 4.1.8 Negative binomial distribution: repeated trials, 183 4.1.9 Constraint: E[N] = n: Poisson distribution, 183 4.2 Continuous variable and continuous distributions, 185 4.2.1 Finite interval [a, b], no constraint, and rectangular distribution, 185 4.2.2 Finite interval [a, b], one constraint and truncated exponential distribution, 186 4.2.3 Finite interval [0, 1], two constraints E[ln x] and E[ln(1 − x)] and beta distribution of first kind, 188 4.2.4 Semi-infinite interval (0,∞), one constraint E[x] and exponential distribution, 191 4.2.5 Semi-infinite interval, two constraints E[x] and E[ln x] and gamma distribution, 192 4.2.6 Semi-infinite interval, two constraints E[ln x] and E[ln(1 + x)] and beta distribution of second kind, 194 4.2.7 Infinite interval, two constraints E[x] and E[x2] and normal distribution, 195 4.2.8 Semi-infinite interval, log-transformation Y = lnX, two constraints E[y] and E[y2] and log-normal distribution, 197 4.2.9 Infinite and semi-infinite intervals: constraints and distributions, 199 Questions, 203 References, 208 Additional Reading, 208 5 Multivariate Probability Distributions, 213 5.1 Multivariate normal distributions, 213 5.1.1 One time lag serial dependence, 213 5.1.2 Two-lag serial dependence, 221 5.1.3 Multi-lag serial dependence, 229 5.1.4 No serial dependence: bivariate case, 234 5.1.5 Cross-correlation and serial dependence: bivariate case, 238 5.1.6 Multivariate case: no serial dependence, 244 5.1.7 Multi-lag serial dependence, 245 5.2 Multivariate exponential distributions, 245 5.2.1 Bivariate exponential distribution, 245 5.2.2 Trivariate exponential distribution, 254 5.2.3 Extension to Weibull distribution, 257 5.3 Multivariate distributions using the entropy-copula method, 258 5.3.1 Families of copula, 259 5.3.2 Application, 260 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 7.1.1 Uniform prior distribution, 291 7.1.2 Arithmetic prior distribution, 293 7.1.3 Geometric prior distribution, 294 7.1.4 Binomial prior distribution, 295 7.1.5 General prior distribution, 297 7.2 Discrete variable taking on an infinite set of values, 298 7.2.1 Improper prior probability distribution, 298 7.2.2 A priori Poisson probability distribution, 301 7.2.3 A priori negative binomial distribution, 304 7.3 Continuous variable: general formulation, 305 7.3.1 Uniform prior and mean constraint, 307 7.3.2 Exponential prior and mean and mean log constraints, 308 Questions, 308 References, 309 8 Parameter Estimation, 310 8.1 Ordinary entropy-based parameter estimation method, 310 8.1.1 Specification of constraints, 311 8.1.2 Derivation of entropy-based distribution, 311 8.1.3 Construction of zeroth Lagrange multiplier, 311 8.1.4 Determination of Lagrange multipliers, 312 8.1.5 Determination of distribution parameters, 313 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 9.1.1 Distribution, density, and aggregation, 337 9.2 Spatial entropy statistics, 339 9.2.1 Redundancy, 343 9.2.2 Information gain, 345 9.2.3 Disutility entropy, 352 9.3 One dimensional aggregation, 353 9.4 Another approach to spatial representation, 360 9.5 Two-dimensional aggregation, 363 9.5.1 Probability density function and its resolution, 372 9.5.2 Relation between spatial entropy and spatial disutility, 375 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 9.11.1 Exponential law, 391 9.11.2 Log-normal law, 391 9.11.3 Power law, 392 9.11.4 Law of proportionate effect, 392 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 10.3.1 Bivariate measures, 405 10.3.2 Map representation, 410 10.3.3 Construction of spatial measures, 412 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 11.1.1 Mean, 437 11.1.2 Variance, 438 11.1.3 Covariance, 440 11.1.4 Correlation, 441 11.1.5 Stationarity, 443 11.2 Spectral analysis, 446 11.2.1 Fourier representation, 448 11.2.2 Fourier transform, 453 11.2.3 Periodogram, 454 11.2.4 Power, 457 11.2.5 Power spectrum, 461 11.3 Spectral analysis using maximum entropy, 464 11.3.1 Burg method, 465 11.3.2 Kapur-Kesavan method, 473 11.3.3 Maximization of entropy, 473 11.3.4 Determination of Lagrange multipliers λk, 476 11.3.5 Spectral density, 479 11.3.6 Extrapolation of autocovariance functions, 482 11.3.7 Entropy of power spectrum, 482 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 12.2.1 Power spectrum probability density function, 493 12.2.2 Minimum cross-entropy-based probability density functions given total expected spectral powers at each frequency, 498 12.2.3 Spectral probability density functions for white noise, 501 12.3 Minimum cross-entropy power spectrum given auto-correlation, 503 12.3.1 No prior power spectrum estimate is given, 504 12.3.2 A prior power spectrum estimate is given, 505 12.3.3 Given spectral powers: Tk = Gj, Gj = Pk, 506 12.4 Cross-entropy between input and output of linear filter, 509 12.4.1 Given input signal PDF, 509 12.4.2 Given prior power spectrum, 510 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 13.2.1 Information variance, 518 13.2.2 Transfer function variance, 520 13.2.3 Correlation, 521 13.3 Entropy measures, 521 13.3.1 Marginal entropy, joint entropy, conditional entropy and transinformation, 521 13.3.2 Informational correlation coefficient, 523 13.3.3 Isoinformation, 524 13.3.4 Information transfer function, 524 13.3.5 Information distance, 525 13.3.6 Information area, 525 13.3.7 Application to rainfall networks, 525 13.4 Directional information transfer index, 530 13.4.1 Kernel estimation, 531 13.4.2 Application to groundwater quality networks, 533 13.5 Total correlation, 537 13.6 Maximum information minimum redundancy (MIMR), 539 13.6.1 Optimization, 541 13.6.2 Selection procedure, 542 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 14.6.1 Hazard assessment, 576 14.6.2 Vulnerability assessment, 577 14.6.3 Risk assessment and ranking, 578 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 16.1.1 Specification of constraints, 606 16.1.2 Maximization of entropy, 606 16.1.3 Determination of Lagrange multipliers, 606 16.1.4 Partition function, 607 16.1.5 Analysis of complexity, 610 16.1.6 Maximum entropy, 614 16.1.7 Complexity as a function of N, 616 16.2 Kapur’s complexity analysis, 618 16.3 Cornacchio’s generalized complexity measures, 620 16.3.1 Special case: R = 1, 624 16.3.2 Analysis of complexity: non-unique K-transition points and conditional complexity, 624 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