Entropy Theory and its Application in Environmental and Water Engineering

Vijay P. Singh,  Texas A & M University, USA

• Preface, xv Acknowledgments, xix1 Introduction, 11.1 Systems and their characteristics, 11.1.1 Classes of systems, 11.1.2 System states, 11.1.3 Change of state, 21.1.4 Thermodynamic entropy, 31.1.5 Evolutive connotation of entropy, 51.1.6 Statistical mechanical entropy, 51.2 Informational entropies, 71.2.1 Types of entropies, 81.2.2 Shannon entropy, 91.2.3 Information gain function, 121.2.4 Boltzmann, Gibbs and Shannon entropies, 141.2.5 Negentropy, 151.2.6 Exponential entropy, 161.2.7 Tsallis entropy, 181.2.8 Renyi entropy, 191.3 Entropy, information, and uncertainty, 211.3.1 Information, 221.3.2 Uncertainty and surprise, 241.4 Types of uncertainty, 251.5 Entropy and related concepts, 271.5.1 Information content of data, 271.5.2 Criteria for model selection, 281.5.3 Hypothesis testing, 291.5.4 Risk assessment, 29Questions, 29References, 31Additional References, 322 Entropy Theory, 332.1 Formulation of entropy, 332.2 Shannon entropy, 392.3 Connotations of information and entropy, 422.3.1 Amount of information, 422.3.2 Measure of information, 432.3.3 Source of information, 432.3.4 Removal of uncertainty, 442.3.5 Equivocation, 452.3.6 Average amount of information, 452.3.7 Measurement system, 462.3.8 Information and organization, 462.4 Discrete entropy: univariate case and marginal entropy, 462.5 Discrete entropy: bivariate case, 522.5.1 Joint entropy, 532.5.2 Conditional entropy, 532.5.3 Transinformation, 572.6 Dimensionless entropies, 792.7 Bayes theorem, 802.8 Informational correlation coefficient, 882.9 Coefficient of nontransferred information, 902.10 Discrete entropy: multidimensional case, 922.11 Continuous entropy, 932.11.1 Univariate case, 942.11.2 Differential entropy of continuous variables, 972.11.3 Variable transformation and entropy, 992.11.4 Bivariate case, 1002.11.5 Multivariate case, 1052.12 Stochastic processes and entropy, 1052.13 Effect of proportional class interval, 1072.14 Effect of the form of probability distribution, 1102.15 Data with zero values, 1112.16 Effect of measurement units, 1132.17 Effect of averaging data, 1152.18 Effect of measurement error, 1162.19 Entropy in frequency domain, 1182.20 Principle of maximum entropy, 1182.21 Concentration theorem, 1192.22 Principle of minimum cross entropy, 1222.23 Relation between entropy and error probability, 1232.24 Various interpretations of entropy, 1252.24.1 Measure of randomness or disorder, 1252.24.2 Measure of unbiasedness or objectivity, 1252.24.3 Measure of equality, 1252.24.4 Measure of diversity, 1262.24.5 Measure of lack of concentration, 1262.24.6 Measure of flexibility, 1262.24.7 Measure of complexity, 1262.24.8 Measure of departure from uniform distribution, 1272.24.9 Measure of interdependence, 1272.24.10 Measure of dependence, 1282.24.11 Measure of interactivity, 1282.24.12 Measure of similarity, 1292.24.13 Measure of redundancy, 1292.24.14 Measure of organization, 1302.25 Relation between entropy and variance, 1332.26 Entropy power, 1352.27 Relative frequency, 1352.28 Application of entropy theory, 136Questions, 136References, 137Additional Reading, 1393 Principle of Maximum Entropy, 1423.1 Formulation, 1423.2 POME formalism for discrete variables, 1453.3 POME formalism for continuous variables, 1523.3.1 Entropy maximization using the method of Lagrange multipliers, 1523.3.2 Direct method for entropy maximization, 1573.4 POME formalism for two variables, 1583.5 Effect of constraints on entropy, 1653.6 Invariance of total entropy, 167Questions, 168References, 170Additional Reading, 1704 Derivation of Pome-Based Distributions, 1724.1 Discrete variable and discrete distributions, 1724.1.1 Constraint E[x] and the Maxwell-Boltzmann distribution, 1724.1.2 Two constraints and Bose-Einstein distribution, 1744.1.3 Two constraints and Fermi-Dirac distribution, 1774.1.4 Intermediate statistics distribution, 1784.1.5 Constraint: E[N]: Bernoulli distribution for a single trial, 1794.1.6 Binomial distribution for repeated trials, 1804.1.7 Geometric distribution: repeated trials, 1814.1.8 Negative binomial distribution: repeated trials, 1834.1.9 Constraint: E[N] = n: Poisson distribution, 1834.2 Continuous variable and continuous distributions, 1854.2.1 Finite interval [a, b], no constraint, and rectangular distribution, 1854.2.2 Finite interval [a, b], one constraint and truncated exponential distribution, 1864.2.3 Finite interval [0, 1], two constraints E[ln x] and E[ln(1 − x)] and beta distribution of first kind, 1884.2.4 Semi-infinite interval (0,∞), one constraint E[x] and exponential distribution, 1914.2.5 Semi-infinite interval, two constraints E[x] and E[ln x] and gamma distribution, 1924.2.6 Semi-infinite interval, two constraints E[ln x] and E[ln(1 + x)] and beta distribution of second kind, 1944.2.7 Infinite interval, two constraints E[x] and E[x2] and normal distribution, 1954.2.8 Semi-infinite interval, log-transformation Y = lnX, two constraints E[y] and E[y2] and log-normal distribution, 1974.2.9 Infinite and semi-infinite intervals: constraints and distributions, 199Questions, 203References, 208Additional Reading, 2085 Multivariate Probability Distributions, 2135.1 Multivariate normal distributions, 2135.1.1 One time lag serial dependence, 2135.1.2 Two-lag serial dependence, 2215.1.3 Multi-lag serial dependence, 2295.1.4 No serial dependence: bivariate case, 2345.1.5 Cross-correlation and serial dependence: bivariate case, 2385.1.6 Multivariate case: no serial dependence, 2445.1.7 Multi-lag serial dependence, 2455.2 Multivariate exponential distributions, 2455.2.1 Bivariate exponential distribution, 2455.2.2 Trivariate exponential distribution, 2545.2.3 Extension to Weibull distribution, 2575.3 Multivariate distributions using the entropy-copula method, 2585.3.1 Families of copula, 2595.3.2 Application, 2605.4 Copula entropy, 265Questions, 266References, 267Additional Reading, 2686 Principle of Minimum Cross-Entropy, 2706.1 Concept and formulation of POMCE, 2706.2 Properties of POMCE, 2716.3 POMCE formalism for discrete variables, 2756.4 POMCE formulation for continuous variables, 2796.5 Relation to POME, 2806.6 Relation to mutual information, 2816.7 Relation to variational distance, 2816.8 Lin’s directed divergence measure, 2826.9 Upper bounds for cross-entropy, 286Questions, 287References, 288Additional Reading, 2897 Derivation of POME-Based Distributions, 2907.1 Discrete variable and mean E[x] as a constraint, 2907.1.1 Uniform prior distribution, 2917.1.2 Arithmetic prior distribution, 2937.1.3 Geometric prior distribution, 2947.1.4 Binomial prior distribution, 2957.1.5 General prior distribution, 2977.2 Discrete variable taking on an infinite set of values, 2987.2.1 Improper prior probability distribution, 2987.2.2 A priori Poisson probability distribution, 3017.2.3 A priori negative binomial distribution, 3047.3 Continuous variable: general formulation, 3057.3.1 Uniform prior and mean constraint, 3077.3.2 Exponential prior and mean and mean log constraints, 308Questions, 308References, 3098 Parameter Estimation, 3108.1 Ordinary entropy-based parameter estimation method, 3108.1.1 Specification of constraints, 3118.1.2 Derivation of entropy-based distribution, 3118.1.3 Construction of zeroth Lagrange multiplier, 3118.1.4 Determination of Lagrange multipliers, 3128.1.5 Determination of distribution parameters, 3138.2 Parameter-space expansion method, 3258.3 Contrast with method of maximum likelihood estimation (MLE), 3298.4 Parameter estimation by numerical methods, 331Questions, 332References, 333Additional Reading, 3349 Spatial Entropy, 3359.1 Organization of spatial data, 3369.1.1 Distribution, density, and aggregation, 3379.2 Spatial entropy statistics, 3399.2.1 Redundancy, 3439.2.2 Information gain, 3459.2.3 Disutility entropy, 3529.3 One dimensional aggregation, 3539.4 Another approach to spatial representation, 3609.5 Two-dimensional aggregation, 3639.5.1 Probability density function and its resolution, 3729.5.2 Relation between spatial entropy and spatial disutility, 3759.6 Entropy maximization for modeling spatial phenomena, 3769.7 Cluster analysis by entropy maximization, 3809.8 Spatial visualization and mapping, 3849.9 Scale and entropy, 3869.10 Spatial probability distributions, 3889.11 Scaling: rank size rule and Zipf’s law, 3919.11.1 Exponential law, 3919.11.2 Log-normal law, 3919.11.3 Power law, 3929.11.4 Law of proportionate effect, 392Questions, 393References, 394Further Reading, 39510 Inverse Spatial Entropy, 39810.1 Definition, 39810.2 Principle of entropy decomposition, 40210.3 Measures of information gain, 40510.3.1 Bivariate measures, 40510.3.2 Map representation, 41010.3.3 Construction of spatial measures, 41210.4 Aggregation properties, 41710.5 Spatial interpretations, 42010.6 Hierarchical decomposition, 42610.7 Comparative measures of spatial decomposition, 428Questions, 433References, 43511 Entropy Spectral Analyses, 43611.1 Characteristics of time series, 43611.1.1 Mean, 43711.1.2 Variance, 43811.1.3 Covariance, 44011.1.4 Correlation, 44111.1.5 Stationarity, 44311.2 Spectral analysis, 44611.2.1 Fourier representation, 44811.2.2 Fourier transform, 45311.2.3 Periodogram, 45411.2.4 Power, 45711.2.5 Power spectrum, 46111.3 Spectral analysis using maximum entropy, 46411.3.1 Burg method, 46511.3.2 Kapur-Kesavan method, 47311.3.3 Maximization of entropy, 47311.3.4 Determination of Lagrange multipliers λk, 47611.3.5 Spectral density, 47911.3.6 Extrapolation of autocovariance functions, 48211.3.7 Entropy of power spectrum, 48211.4 Spectral estimation using configurational entropy, 48311.5 Spectral estimation by mutual information principle, 486References, 490Additional Reading, 49012 Minimum Cross Entropy Spectral Analysis, 49212.1 Cross-entropy, 49212.2 Minimum cross-entropy spectral analysis (MCESA), 49312.2.1 Power spectrum probability density function, 49312.2.2 Minimum cross-entropy-based probability density functions given total expected spectral powers at each frequency, 49812.2.3 Spectral probability density functions for white noise, 50112.3 Minimum cross-entropy power spectrum given auto-correlation, 50312.3.1 No prior power spectrum estimate is given, 50412.3.2 A prior power spectrum estimate is given, 50512.3.3 Given spectral powers: Tk = Gj, Gj = Pk, 50612.4 Cross-entropy between input and output of linear filter, 50912.4.1 Given input signal PDF, 50912.4.2 Given prior power spectrum, 51012.5 Comparison, 51212.6 Towards efficient algorithms, 51412.7 General method for minimum cross-entropy spectral estimation, 515References, 515Additional References, 51613 Evaluation and Design of Sampling and Measurement Networks, 51713.1 Design considerations, 51713.2 Information-related approaches, 51813.2.1 Information variance, 51813.2.2 Transfer function variance, 52013.2.3 Correlation, 52113.3 Entropy measures, 52113.3.1 Marginal entropy, joint entropy, conditional entropy and transinformation, 52113.3.2 Informational correlation coefficient, 52313.3.3 Isoinformation, 52413.3.4 Information transfer function, 52413.3.5 Information distance, 52513.3.6 Information area, 52513.3.7 Application to rainfall networks, 52513.4 Directional information transfer index, 53013.4.1 Kernel estimation, 53113.4.2 Application to groundwater quality networks, 53313.5 Total correlation, 53713.6 Maximum information minimum redundancy (MIMR), 53913.6.1 Optimization, 54113.6.2 Selection procedure, 542Questions, 553References, 554Additional Reading, 55614 Selection of Variables and Models, 55914.1 Methods for selection, 55914.2 Kullback-Leibler (KL) distance, 56014.3 Variable selection, 56014.4 Transitivity, 56114.5 Logit model, 56114.6 Risk and vulnerability assessment, 57414.6.1 Hazard assessment, 57614.6.2 Vulnerability assessment, 57714.6.3 Risk assessment and ranking, 578Questions, 578References, 579Additional Reading, 58015 Neural Networks, 58115.1 Single neuron, 58115.2 Neural network training, 58515.3 Principle of maximum information preservation, 58815.4 A single neuron corrupted by processing noise, 58915.5 A single neuron corrupted by additive input noise, 59215.6 Redundancy and diversity, 59615.7 Decision trees and entropy nets, 598Questions, 602References, 60316 System Complexity, 60516.1 Ferdinand’s measure of complexity, 60516.1.1 Specification of constraints, 60616.1.2 Maximization of entropy, 60616.1.3 Determination of Lagrange multipliers, 60616.1.4 Partition function, 60716.1.5 Analysis of complexity, 61016.1.6 Maximum entropy, 61416.1.7 Complexity as a function of N, 61616.2 Kapur’s complexity analysis, 61816.3 Cornacchio’s generalized complexity measures, 62016.3.1 Special case: R = 1, 62416.3.2 Analysis of complexity: non-unique K-transition points and conditional complexity, 62416.4 Kapur’s simplification, 62716.5 Kapur’s measure, 62716.6 Hypothesis testing, 62816.7 Other complexity measures, 628Questions, 631References, 631Additional References, 632Author Index, 633Subject Index, 639