Dimensions of Uncertainty in Communication Engineering

Dimensions of Uncertainty in Communication Engineering

1st Edition - July 1, 2022

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  • Author: Ezio Biglieri
  • Paperback ISBN: 9780323992756

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Description

Dimensions of Uncertainty in Communication Engineering is a comprehensive and self-contained introduction to the problems of nonaleatory uncertainty and the mathematical tools needed to solve them. The book gathers together tools derived from statistics, information theory, moment theory, interval analysis and probability boxes, dependence bounds, nonadditive measures, and Dempster–Shafer theory. While the book is mainly devoted to communication engineering, the techniques described are also of interest to other application areas, and commonalities to these are often alluded to through a number of references to books and research papers. This is an ideal supplementary book for courses in wireless communications, providing techniques for addressing epistemic uncertainty, as well as an important resource for researchers and industry engineers. Students and researchers in other fields such as statistics, financial mathematics, and transport theory will gain an overview and understanding on these methods relevant to their field.

Key Features

  • Uniquely brings together a variety of tools derived from statistics, information theory, moment theory, interval analysis and probability boxes, dependence bounds, nonadditive measures, and Dempster—Shafer theory
  • Focuses on the essentials of various, wide-ranging methods with references to journal articles where more detail can be found if required
  • Includes MIMO-related results throughout

Readership

Students, researchers and industry engineers in communication engineering; Students and researchers in financial mathematics and transport theory

Table of Contents

  • 1 Model selection
    1.1 Parametric models
    1.2 Wireless channel models
    1.2.1 Rayleigh fading
    1.2.2 Rice fading
    1.2.3 Nakagami-m fading
    1.3 Parameter estimation
    1.3.1 A word of caution
    1.4 Differential entropy and Kullback–Leibler divergence
    1.4.1 Differential entropy
    1.4.2 Kullback–Leibler divergence
    1.5 KKT optimality conditions
    1.6 Choosing the best model: The maximum-entropy principle
    1.6.1 Maximizing entropy with order-statistics constraints
    1.6.2 Spherically invariant processes
    1.7 Choosing the best model in a set: Akaike information criterion
    1.8 Choosing the best model in a set: Minimum description length criterion
    Sources and parerga

    2 Performance bounds from epistemic uncertainty
    2.1 Model robustness
    2.2 Performance optimization with divergence constraints
    2.3 Scenarios of uncertainty
    2.3.1 Defining the constraints
    2.4 Concentration-of-measure inequalities 
    2.5 Some applications
    2.5.1 P unknown, ℎ known
    2.5.2 A trivial bound
    2.5.3 Some moments of 𝑋 known, ℎ known
    2.5.4 McDiarmid constraint sets
    2.5.5 Hoeffding constraint sets
    2.6 The certification problem
    2.6.1 Empirical measures
    2.6.2 Certification under epistemic uncertainty
    Sources and parerga

    3 Moment bounds
    3.1 Some classical results
    3.2 The general moment-bound problem
    3.3 Calculation of moments
    3.3.1 Using multinomial expansions
    3.3.2 Using moment-generating functions
    3.3.3 Using recursive computations
    3.4 Moments of unimodal pdfs
    3.4.1 Moment transfer
    3.5 When is a sequence a valid moment sequence?
    3.5.1 Probability measures represented by moments
    3.6 Moments in a parallelepiped 
    3.7 Geometric bounds
    3.7.1 Two-dimensional case
    3.7.2 Application to spectrum sensing
    3.8 Quadrature-rule approximations and bounds
    3.8.1 The algebraic moment problem
    3.8.2 Cebyšev form of orthogonal polynomials
    3.8.3 Recursive generation of orthogonal polynomials
    3.8.4 Numerical generation of orthogonal polynomials
    3.9 Cebyšev systems and principal representations
    3.9.1 Principal and canonical representations as quadrature rules
    3.9.2 Moment bounds and quadrature rules
    3.9.3 Bounds on CDFs and quadrature rules
    3.9.4 Bounds on Laplace–Stieltjes transform
    3.9.5 Extension to nondifferentiable functions: The method of contact polynomials
    3.10 Moment problems as semidefinite programs
    3.10.1 Moment problem in a parallelepiped
    3.10.2 Generalized Cebyšev bounds
    3.11 Multidimensional moment bounds and approximations
    3.11.1 Multidimensional moment bounds
    Sources and parerga

    4 Interval analysis
    4.1 Some definitions
    4.2 Set operations on intervals
    4.3 Algebraic operations on interval numbers
    4.3.1 Properties that may not be shared with ordinary real algebra
    4.4 Interval vectors and matrices
    4.5 Interval functions
    4.5.1 The vertex method
    4.6 The interval dependence problem
    4.7 Integrals
    4.8 Choosing a representative in an interval
    Sources and parerga

    5 Probability boxes
    5.1 Interval probabilities
    5.2 Generating probability boxes
    5.3 Aggregating probability boxes
    5.4 Combining probability boxes of random variables
    5.5 Using probability boxes in performance evaluation
    Sources and parerga

    6 Dependence bounds
    6.1 Copulas
    6.1.1 Copulas and joint CDFs
    6.1.2 Some explicit copula families
    6.1.3 Fréchet–Hoeffding inequalities
    6.1.4 A version of Fréchet–Hoeffding bounds involving probabilites
    6.1.5 Dual of a copula and the survival copula
    6.1.6 𝑑-dimensional copulas
    6.2 Dependence p-boxes from copula bounds
    6.2.1 Special operations
    6.2.2 Using p-boxes of 𝐹𝑋 and 𝐹𝑌
    6.2.3 Operations on 𝑑 RVs
    6.2.4 Dependence bounds with order statistics
    6.3 Some examples of application
    6.4 Bivariate dependence bounds on expectations
    6.5 Bounds with monotone marginal densities
    6.5.1 Tail-monotone marginal densities
    6.6 Deriving tighter dependence bounds
    6.6.1 Degree of dependence
    6.6.2 Linear correlation
    6.6.3 Rank correlation: Kendall tau, Spearman rho, and Blomqvist beta
    6.6.4 Tail dependence
    6.6.5 Quadrant/orthant dependence
    Sources and parerga

    7 Beyond probability
    7.1 Decisions under uncertainty
    7.2 Epistemic vs. aleatory uncertainty
    7.2.1 The Principle of Insufficient Reason
    7.2.2 Objective vs. subjective probabilities
    7.2.3 Using interval probabilities
    7.3 Lotteries, prospects, and utility functions
    7.3.1 Prospects
    7.3.2 St. Petersburg paradox
    7.3.3 Risk-averse and risk-seeking decisions
    7.3.4 Reversing risk aversion: Framing effect
    7.4 Other paradoxes arising from EUT
    7.4.1 Ellsberg paradox
    7.4.2 Allais paradox
    7.4.3 Entropy-modified expected utility of gambling
    7.5 Upper and lower probabilities
    7.5.1 Conditional upper and lower probabilities
    7.5.2 Inference using upper and lower probabilities
    7.5.3 The dilation phenomenon
    7.6 Expected utility with interval probabilities
    7.7 Some applications to digital communication
    7.7.1 Optimum decisions: Bayes criterion
    7.7.2 A case in which error probability does not tell the whole story
    7.7.3 Neyman–Pearson criterion
    7.8 Going beyond probability
    7.9 Set functions and their properties
    7.10 Infinite sets
    7.11 Capacities and Choquet integral
    7.12 Expected values and Choquet integral
    7.13 Dempster–Shafer theory
    7.13.1 Bayesian belief functions
    7.13.2 Dempster rule of combination
    Sources and parerga

Product details

  • Language: English
  • Copyright: © Academic Press 2022
  • Published: July 1, 2022
  • Imprint: Academic Press
  • Paperback ISBN: 9780323992756

About the Author

Ezio Biglieri

Ezio Biglieri
Ezio Biglieri received his formal training in Electrical Engineering at Politecnico di Torino (Italy), where he received his Dr. Engr. degree in 1967. Before being an Honorary Professor at University Pompeu Fabra, he was a Professor at Università di Napoli (Italy), at Politecnico di Torino (Italy), and at UCLA (USA). He has held visiting positions with Bell Labs (USA), the École Nationale Supérieure des Télécommunications (Paris, France), the University of Sydney (Australia), the Yokohama National University (Japan), Princeton University (USA), the University of South Australia, the Munich Institute of Technology (Germany), the National University of Singapore, the National Taiwan University, the University of Cambridge (U.K.), ETH Zurich (Switzerland), and Monash University Melbourne (Australia). Among other honors, in 2000 he received the IEEE Third-Millennium Medal and the IEEE Donald G. Fink Prize Paper Award, in 2001 the IEEE Communications Society Edwin Howard Armstrong Achievement Award, in 2004, 2012, and 2015 the Journal of Communications and Networks Best Paper Award, in 2012 the IEEE Information Theory Society Aaron D. Wyner Distinguished Service Award, and in 2021 the IEEE Communications Society Heinrich Hertz Award. He is a Life Fellow of the IEEE.

Affiliations and Expertise

Universitat Pompeu Fabra, Barcelona, Spain.

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