Differential Evolution in Electromagnetics (Adaptation, Learning, and Optimization, 4)

Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics Adaptation, Learning, and Optimization, Vo...

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Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics

Adaptation, Learning, and Optimization, Volume 4 Series Editor-in-Chief Meng-Hiot Lim Nanyang Technological University, Singapore E-mail: [email protected] Yew-Soon Ong Nanyang Technological University, Singapore E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 1. Jingqiao Zhang and Arthur C. Sanderson Adaptive Differential Evolution, 2009 ISBN 978-3-642-01526-7 Vol. 2. Yoel Tenne and Chi-Keong Goh (Eds.) Computational Intelligence in Expensive Optimization Problems, 2010 ISBN 978-3-642-10700-9 Vol. 3. Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms, 2010 ISBN 978-3-642-12833-2 Vol. 4. Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics, 2010 ISBN 978-3-642-12868-4

Anyong Qing and Ching Kwang Lee

Differential Evolution in Electromagnetics

123

Dr. Anyong Qing Temasek Laboratories National University of Singapore 5 Sports Dr 2 Singapore 117508 E-mail: [email protected]

Dr. Ching Kwang Lee School of Electrical and Electronic Engineering Nanyang Technological University Division of Communication Engineering S1-B1a-10, Nanyang Avenue Singapore 639798 E-mail: [email protected]

ISBN 978-3-642-12868-4

e-ISBN 978-3-642-12869-1

DOI 10.1007/978-3-642-12869-1 Adaptation, Learning, and Optimization

ISSN 1867-4534

Library of Congress Control Number: 2010926029 c 2010 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Preface

1 Motivations After many years of development and applications, differential evolution has proven itself a very simple while very powerful stochastic global optimizer. Since its inception, it has been applied to solve problems in many scientific and engineering fields. Nowadays, our daily life relies heavily on electromagnetics. Differential evolution has played an essential role in many synthesis and design problems in electromagnetics. This book focuses on applications of differential evolution in electromagnetics to showcase the achievement of differential evolution and further boost its acceptance in electromagnetics community.

2 Layout This book is composed of two parts. Part one includes the first three chapters while the remaining five chapters belong to part two of this book. 2.1

Part One

This part focuses on a literature survey on differential evolution. As far as we know, it is by far the most extensive and exhaustive one. 2.1.1 Chapter 1 Chapter 1 gives details of the literature survey which covers publication collection, refining and analysis. It opens up with the purposes this literature survey aims to serve. Next, Platforms over which the literature survey is actually conducted are then discussed. Initial statistical results over these platforms are presented. After that, the refining process to remove irrelevant publications is discussed. Yearly outputs of formal publications with and without refining are presented. Result analysis, or publication classification, is then discussed. Topics according to which collected publications are clustered are suggested. In particular, theoretical studies on differential evolution are summarized. Finally, some future actions are discussed. We have noticed several misconceptions and misconducts on differential evolution through this literature survey. They are clearly pointed out at the end of this chapter.

VI

Preface

2.1.2 Chapter 2 Basics of differential evolution are presented in Chapter 2. It also serves the second part of this book so that repetition of description of differential evolution is avoided. A short history of differential evolution is first discussed. It covers its inception, early years until 1998 and years from 1998 onwards. Major events in early years and key milestones in years from 1998 onwards are highlighted. The basic framework of differential evolution is then explained by revisiting the originators’ inventory publication, followed by a description of the more generic classic differential evolution. Some prominent variants of the two fundamental evolutionary operations in differential evolution are presented. Next, dynamic differential evolution which was misunderstood and seriously underestimated before is briefly mentioned. Finally, essential features of differential evolution including both advantages and disadvantages are highlighted. It has to be pointed out that a state of the art review of differential evolution is not presented in this book due to tight time limit. Such a review will be part of a forming up encyclopedia of differential evolution. 2.1.3 Chapter 3 A retrospection of applications of differential evolution in electromagnetics in and before 2008 is presented in this Chapter. The coverage of the retrospection is clearly specified right at the beginning of this Chapter. The pioneering works of applications of differential evolution in electromagnetics are highlighted. Statistical results by both publication year and subject are presented. Detailed discussion of applications of differential evolution in specific subject is then given. Involved subjects include electromagnetic inverse problems, antenna arrays, microwave & RF engineering, antennas, electromagnetic structures, electromagnetic composite materials, frequency planning, radio network design, MIMO, radar, computational electromagnetics and electromagnetic compatibility. An outlook of applications of differential evolution in electromagnetics is also presented at the end of this Chapter. 2.2

Part Two

This part presents five new applications of differential evolution in differential evolution by different research groups. 2.2.1 Chapter 4 Reconstruction of two-dimensional dielectric cylinders by using differential evolution is presented in this Chapter. The efficiency of differential evolution has been numerically shown through various examples. In addition, the impact of initial guess on differential evolution is presented. The multiple signal classification is used to determine the number of cylinders, their approximate centers and approximate geometric dimensions while a least squares based method is used to generate an estimate of the permittivity of the cylinders. It has been shown that a proper choice of the initial guess can speed up the convergence of the optimization significantly.

Preface

VII

2.2.2 Chapter 5 Inspection of penetrable objects by using differential evolution together with a recently proposed iterative multiscaling approach is discussed in this Chapter. The solving procedure starts from a fixed test area and successively focuses on one or more "regions of interest" in order to determine the approximate shapes of the unknown objects. At each step of the minimization process, differential evolution is used to retrieve this support by minimizing a proper functional, which relates the measured scattered field data to the data numerically produced, at any iteration, by the current solution. Several new results are included concerning the reconstruction of inhomogeneous targets under various imaging conditions. The combined strategy has been proved to be quite effective in reconstructing complex dielectric cylinders such as hollow and E-shape cylinders in noisy environment. 2.2.3 Chapter 6 In this Chapter a flexible method for prediction of far-field radiated emissions is presented. It is a promising computational alternative to the expensive large semianechoic chambers necessary to perform electromagnetic compatibility far-field radiated emission measurements. In this method, the equipment under test is replaced by an equivalent set of infinitesimal dipoles (both electric and magnetic) distributed inside the volume occupied by the equipment under test which is determined from near-field measurements at a short distance of the equipment under test. A memetic metaheuristic technique combing genetic algorithms, differential evolution and downhill simplex method is used to determine the type, position, orientation and excitation current of each dipole of the equivalent set of dipoles. The information obtained from the equivalent dipole set is used to determine the radiation at the far-field, as well as to identify the radiating parts of the equipment. 2.2.4 Chapter 7 Differential evolution with Pareto tournaments (DEPT) was applied to address the multi-objective optimization of frequency assignment problem in two real-world GSM networks in this Chapter. Two performance indicators, hypervolume and coverage relation, are implemented to analyze results. Results are compared with those by other multi-objective metaheuristics. Final results show that fine-tuned DEPT outperforms both MO-VNS and MO-SVNS while performs worse than both GMO-SVNS and GMO-VNS, among which GMO-SVNS performs best. 2.2.5 Chapter 8 In this Chapter, differential evolution is combined with particle swarm optimization (PSO) and another evolutionary algorithm (EA) to create a novel hybrid algorithm, the PSO-EA-DEPSO. The alteration between PSO, PSO-EA, and DEPSO provides additional diversity to counteract premature convergence. This hybrid algorithm is then shown to outperform PSO, PSO-EA, and DEPSO when applied to wireless MIMO channel prediction.

VIII

Preface

3 Readership As its name indicates, this book is specially prepared for electromagnetic researchers facing optimization problems. It will be particularly attractive to researchers who have been frustrated by other optimization algorithms. This book is a premium resource for differential evolution community. People in this community will have a better understanding on differential evolution and its huge application potential. This book is also an ideal resource for evolutionary computation community. People in this community may find it helpful in presenting a more appropriate approach to conduct concerned literature survey and providing real engineering application examples.

Acknowledgement

First of all, I would take this opportunity to thank Prof. Hock Lim and Mr. Joseph Sing Kwong Ting, directors of Temasek Laboratories, National University of Singapore, for their support and encouragement of my study on differential evolution. The financial funding from Defence Science & Technology Agency, Singapore is greatly appreciated too. I would also like to thank Prof. Meng-Hiot Lim, series editor on evolutionary learning and optimization, and Dr. Thomas Ditzinger, senior editor within Springer Verlag responsible for this series, for their strong recommendation to publish this book. Thanks also go to Mr. Heather King for his carefulness and patience.

Contents

Contents 1

A Literature Survey on Differential Evolution…………………………...1 1.1 Motivations .............................................................................................1 1.1.1 Eliminating Inconsistencies .........................................................1 1.1.2 Crediting Original Contributions .................................................1 1.1.3 Knowing the State of the Art .......................................................1 1.1.4 Gaining Insight ............................................................................2 1.2 Platforms.................................................................................................2 1.2.1 Starting Point ...............................................................................2 1.2.2 Databases .....................................................................................3 1.2.3 Informal Online Resources and Tools .........................................4 1.3 Result Refining .......................................................................................5 1.3.1 Books ...........................................................................................5 1.3.2 Book Chapters .............................................................................6 1.3.3 Other Formal Publications ...........................................................6 1.3.4 Informal Notes .............................................................................7 1.4 Result Analysis .......................................................................................7 1.4.1 Theory of Differential Evolution .................................................7 1.4.2 Fundamentals of Differential Evolution ......................................8 1.4.3 Intrinsic Control Parameters ........................................................9 1.4.4 Evaluation of Differential Evolution............................................9 1.4.5 Applications of Differential Evolution ........................................9 1.4.6 Hybridization ...............................................................................9 1.5 Future Actions ......................................................................................10 1.5.1 Open Access ..............................................................................10 1.5.2 Future Update ............................................................................10 1.6 Misconceptions and Misconducts on Differential Evolution................10 References .............................................................................................................10

2

Basics of Differential Evolution…………………………………………..19 2.1 A Short History.....................................................................................19 2.1.1 Inception ....................................................................................19 2.1.2 Early Years ................................................................................20 2.1.2.1 Assessment..................................................................20 2.1.2.2 Reputation Building ....................................................20

XII

Contents

2.2

2.3

2.4 2.5 2.6

2.1.2.3 Applications ................................................................20 2.1.2.4 Promotion....................................................................21 2.1.2.5 Practical Advice ..........................................................21 2.1.2.6 Standardization ...........................................................22 2.1.2.7 More Adventures ........................................................22 2.1.3 Key Milestones in and after 1998 ..............................................22 The Foundational Differential Evolution Strategies .............................23 2.2.1 Notations....................................................................................23 2.2.2 Strategy Framework...................................................................24 2.2.2.1 Pseudo-code ................................................................24 2.2.2.2 Initialization ................................................................25 2.2.2.3 Differential Mutation ..................................................25 2.2.2.4 Crossover ....................................................................26 2.2.2.5 Selection......................................................................27 2.2.2.6 Termination Conditions ..............................................27 2.2.3 Intrinsic Control Parameters ......................................................27 Classic Differential Evolution ..............................................................28 2.3.1 Initialization ...............................................................................28 2.3.2 Differential Mutation .................................................................28 2.3.2.1 Current ........................................................................29 2.3.2.2 Best .............................................................................29 2.3.2.3 Better...........................................................................29 2.3.2.4 Random.......................................................................29 2.3.2.5 Mean ...........................................................................29 2.3.2.6 Best of Random ..........................................................29 2.3.2.7 Arithmetic Best ...........................................................29 2.3.2.8 Arithmetic Better ........................................................29 2.3.2.9 Arithmetic Random.....................................................29 2.3.2.10 Trigonometric .............................................................30 2.3.2.11 Directed.......................................................................30 2.3.3 Crossover ...................................................................................30 2.3.3.1 Binary Crossover ........................................................31 2.3.3.2 One-Point Crossover ...................................................32 2.3.3.3 Multi-point Crossover .................................................32 2.3.3.4 Arithmetic Crossover ..................................................32 2.3.3.5 Arithmetic One-Point Crossover.................................33 2.3.3.6 Arithmetic Multi-point Crossover...............................33 2.3.3.7 Arithmetic Binomial Crossover ..................................34 2.3.3.8 Arithmetic Exponential Crossover..............................35 Dynamic Differential Evolution ...........................................................36 State of the Art of Differential Evolution .............................................36 Essential Features of Differential Evolution.........................................37 2.6.1 Advantages ................................................................................37 2.6.1.1 Reliability....................................................................37 2.6.1.2 Efficiency....................................................................37

Contents

XIII

2.6.1.3 Simplicity....................................................................37 2.6.1.4 Robustness ..................................................................38 2.6.2 Disadvantages ............................................................................38 2.6.2.1 Efficiency....................................................................38 2.6.2.2 Incapability for Epistatic and Noisy Problems............38 References .............................................................................................................38 3

A Retrospective of Differential Evolution in Electromagnetics……….43 3.1 Introduction ..........................................................................................43 3.1.1 Coverage ....................................................................................43 3.1.2 Pioneering Works ......................................................................44 3.1.3 An Overview of Applications of Differential Evolution in Electromagnetics........................................................................44 3.1.3.1 Yearly Output .............................................................44 3.1.3.2 Output by Subject .......................................................44 3.2 Electromagnetic Inverse Problems .......................................................45 3.2.1 A Bird’s Eye View.....................................................................45 3.2.2 Further Classification.................................................................45 3.2.2.1 One-Dimensional Electromagnetic Inverse Problems .....................................................................46 3.2.2.2 Two-Dimensional Electromagnetic Inverse Problems .....................................................................46 3.2.2.3 Three-Dimensional Electromagnetic Inverse Problems .....................................................................47 3.3 Antenna Arrays.....................................................................................48 3.3.1 Conventional Antenna Arrays....................................................48 3.3.1.1 Ideal Antenna Arrays ..................................................48 3.3.1.2 Practical Antenna Arrays ............................................48 3.3.1.3 Phased Arrays .............................................................49 3.3.2 Time-Modulated Antenna Arrays ..............................................49 3.3.2.1 Ideal Antenna Arrays with Time Modulation .............49 3.3.2.2 Practical Antenna Arrays with Time Modulation .......49 3.3.2.3 Phased Antenna Arrays with Time Modulation..........50 3.3.3 Moving Phase Center Antenna Arrays.......................................50 3.4 Microwave and RF Engineering ...........................................................50 3.4.1 Design of Microwave and RF Devices ......................................50 3.4.1.1 Designing Microwave and RF Devices Using Differential Evolution .................................................51 3.4.1.2 Extracting Empirical Synthesis Formulas Using Differential Evolution .................................................51 3.4.2 Characterization of Microwave and RF Devices .......................51 3.4.2.1 Calibration of Measuring System for Characterizing Microwave and RF Devices ........................................51 3.4.2.2 Modeling of Microwave and RF Devices ...................52 3.5 Antennas ...............................................................................................52 3.5.1 Design of Antennas....................................................................52

XIV

Contents

3.5.1.1 Designing Antennas Using Differential Evolution .....52 3.5.1.2 Extracting Empirical Formulas for Synthesizing Antennas .....................................................................52 3.5.2 Measurement of Antennas .........................................................53 3.6 Electromagnetic Structures ...................................................................53 3.6.1 Plain Electromagnetic Structures ...............................................54 3.6.2 Frequency Selective Surfaces ....................................................55 3.7 Electromagnetic Composite Materials ..................................................56 3.7.1 Modeling of Electromagnetic Composite Materials ..................56 3.7.2 Retrieval of Effective Permittivity Tensor.................................56 3.8 Frequency Planning ..............................................................................57 3.9 Radio Network Design..........................................................................58 3.10 MIMO...................................................................................................58 3.11 Radar.....................................................................................................59 3.12 Computational Electromagnetics ..........................................................59 3.13 Electromagnetic Compatibility .............................................................60 3.14 Miscellaneous Applications ..................................................................60 3.15 An Outlook to Future Applications of Differentia Evolution in Electromagnetics...................................................................................60 References .............................................................................................................61

4

Application of Differential Evolution to a Two-Dimensional Inverse Scattering Problem …………………………......………………………...73 4.1 Introduction ..........................................................................................73 4.2 General Description of the Problem .....................................................74 4.2.1 Experimental Setup....................................................................74 4.2.2 The Optimization Problem.........................................................76 4.3 Mathematical Nature of the Optimization Problem and Differential Evolution ..............................................................................................76 4.4 Initial Guess ..........................................................................................77 4.4.1 Foldy-Lax Model of Scattering..................................................78 4.4.2 Multiple Signal Classification for Estimating the Scatterer Support.......................................................................................79 4.4.3 Least Square Based Method for Generating Initial Guess for the Relative Permittivity ..................................................................80 4.5 Numerical Results.................................................................................81 4.5.1 Measurement Setup....................................................................81 4.5.2 Control Parameters ....................................................................81 4.5.3 Numerical Example 1: A Single Cylinder .................................82 4.5.4 Numerical Example 2: Two Identical Cylinders........................87 4.5.5 Numerical Example 3: Two Different Cylinders .......................90 4.5.6 Numerical Example 4: Two Closely Located Identical Cylinders....................................................................................94 4.5.7 Numerical Example 5: Kite Cross-Section Cylinder .................97

Contents

XV

4.6 Conclusions ........................................................................................101 References ...........................................................................................................102 5

The Use of Differential Evolution for the Solution of Electromagnetic Inverse Scattering Problems…………………………………………….107 5.1 Introduction ........................................................................................107 5.2 Problem Formulation ..........................................................................108 5.2.1 The Inverse Scattering Formulation.........................................108 5.2.2 Discrete Setting........................................................................109 5.2.3 The Inverse Scattering Problem as an Optimization Problem....................................................................................110 5.3 The Iterative Multiscaling Approach ..................................................110 5.4 Numerical Results...............................................................................112 5.4.1 Off-Centered Dielectric Cylinder ............................................112 5.4.2 Off-Centered Dielectric Hollow Cylinder................................117 5.4.3 Centered Stratified Dielectric Square Cylinder........................121 5.4.4 Centered E-Shape Dielectric Cylinder .....................................126 5.5 Conclusions ........................................................................................129 References ...........................................................................................................129

6

Modeling of Electrically Large Equipment with Distributed Dipoles Using Metaheuristic Methods …………………………………………..133 6.1 Introduction ........................................................................................133 6.1.1 Near-Field to Far-Field Transformation ..................................133 6.1.2 Radiating Equipment Modeling with Prefixed Position Dipoles.....................................................................................134 6.1.3 Present Work ...........................................................................135 6.2 Electromagnetic Modeling of a Radiating Equipment with Distributed Infinitesimal Dipoles ..........................................................................135 6.2.1 Integral Equations for the Radiation of Electronic Equipment................................................................................136 6.2.2 Point-Matching Method with Dirac Delta Basis Functions .....137 6.2.3 Ground Plane in Semi-anechoic Chambers..............................137 6.3 Proposed Method for Near-Field to Far-Field Transformation...........138 6.3.1 Description of the Method .......................................................138 6.3.2 Optimization Problem..............................................................140 6.3.3 Source Identification................................................................140 6.4 Electromagnetic Optimization by Genetic Algorithms.......................140 6.4.1 EMOGA v1.0: Genetic Algorithm...........................................141 6.4.2 EMOGA 2.0: Metaheuristic Method .......................................142 6.4.2.1 Current Scaling .........................................................142 6.4.2.2 Correlation between Dipoles.....................................143 6.4.2.3 Memetization ............................................................143 6.5 Numerical Results...............................................................................144 6.5.1 Measurement Systems .............................................................144 6.5.1.1 General Measurement System ..................................144

XVI

Contents

6.5.1.2 Near-Field Measurement System..............................145 6.5.1.3 Far-Field Measurement System ................................147 6.5.2 Near-Field Results ...................................................................147 6.5.3 Far-Field Prediction ............................................................... `149 6.6 Conclusions ........................................................................................150 References ...........................................................................................................151 7

Application of Differential Evolution to a Multi-Objective Real-World Frequency Assignment Problem………………………………………...155 7.1 Introduction ........................................................................................155 7.2 Multi-objective FAP in a GSM Network............................................156 7.2.1 GSM Components and Frequency Planning ............................156 7.2.2 Interference Cost......................................................................157 7.2.3 Separation Cost ........................................................................158 7.3 Multi-objective Differential Evolution with Pareto Tournaments ......159 7.3.1 Algorithm Structure .................................................................159 7.3.2 Pareto Tournament...................................................................159 7.3.3 Problem Domain Knowledge...................................................160 7.4 Multi-objective Variable Neighborhood Search .................................160 7.4.1 Variable Neighborhood Search................................................160 7.4.2 Multi-objective Variable Neighborhood Search ......................161 7.4.3 Greedy Mutation ......................................................................162 7.4.4 Multi-objective Skewed Variable Neighborhood Search.........162 7.5 Experiments and Results.....................................................................163 7.5.1 Experimental Setup..................................................................163 7.5.1.1 Used GSM Instances.................................................163 7.5.1.2 Encoding ...................................................................165 7.5.1.3 Computational Facilities ...........................................165 7.5.1.4 Termination Conditions and Process Monitoring .....165 7.5.1.5 Confidence Building .................................................165 7.5.2 Methodology and Metrics ........................................................166 7.5.2.1 Hypervolume ............................................................166 7.5.2.2 Coverage Relation.....................................................166 7.5.3 Tuning of the DEPT Parameters ..............................................166 7.5.3.1 Population Size .........................................................167 7.5.3.2 Crossover Probability................................................168 7.5.3.3 Mutation Intensity.....................................................169 7.5.3.4 DEPT Scheme...........................................................170 7.5.3.5 Findings ....................................................................172 7.5.4 Empirical Results.....................................................................173 7.6 Conclusions ........................................................................................174 References ...........................................................................................................175 8

RNN Based MIMO Channel Prediction……………..............................177 8.1 Introduction ........................................................................................177 8.2 Received Signal Model.......................................................................178

Contents

XVII

8.2.1 Received Signal Model ............................................................178 8.2.2 Optimization Problem..............................................................179 8.3 Hybrid PSO-ES-DEPSO Training Algorithm.....................................179 8.4 MIMO Channel/Beam-Forming Models ............................................180 8.4.1 Channel Model.........................................................................180 8.4.2 Channel Estimation Model ......................................................182 8.4.3 MIMO Beam-Forming.............................................................182 8.5 Recurrent Neural Network for Channel Prediction.............................184 8.6 Training Procedure .............................................................................185 8.7 Numerical Results...............................................................................187 8.7.1 Algorithm Comparison ............................................................187 8.7.2 Robustness of PSO-ES-DEPSO Algorithm .............................188 8.7.3 Linear and Nonlinear Predictors with PSO-EA-DEPSO Algorithm.................................................................................191 8.7.4 Non-convexity of the Solution Space ......................................192 8.8 Performance Measures of RNN Predictors.........................................193 8.9 Conclusions ........................................................................................203 References ...........................................................................................................204 Index……….…………………………………………………………………...207

Chapter 1

A Literature Survey on Differential Evolution Anyong Qing

1

1.1 Motivations 1.1.1 Eliminating Inconsistencies It has been observed since 2004 that there are many inconsistent or even false claims prevailing in the community of differential evolution [1]. Two measures have been taken to clarify them. The first is a system level parametric study on differential evolution [1]-[4]. The second is the large scale literature survey mentioned here. It is one of the foundation stones of this book.

1.1.2 Crediting Original Contributions The academic society nowadays has become more and more utilitarian and impetuous. Many researchers dream a shortcut to their academic success. They tend to accept established view points especially those from topical review articles by leading researchers. Original publications are neglected that insufficient credits are given to originality. In some cases, they may not be aware that the original contributions are cited incorrectly [1]. Academic misconducts such as multiple submissions, exaggerated claims, or even plagiarism are not rare. It is one of the objectives of this survey to promote good academic conducts by locating and appropriately crediting original contributions.

1.1.3 Knowing the State of the Art It has been more than ten years since the inception of differential evolution. However, as far as we know, nobody else has done any comprehensive literature Anyong Qing Temasek Laboratories, National University of Singapore 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected] 1

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 1–17. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

2

A. Qing

survey on differential evolution. The state of the art of differential evolution is therefore not precisely known to interested researchers. This literature survey aims to fill this gap. It also serves to reveal the popularity of differential evolution.

1.1.4 Gaining Insight The literature survey involves not only literature collection but also literature analysis among which the latter is more important. Through the analysis, the following questions will be answered (a) (b) (c)

What is differential evolution? When is differential evolution used and why is it useful? When will differential evolution fail and why does it fail?

Answers to the above questions are crucial for potential applications of differential evolution. Insights gained may lead to future improvements on differential evolution.

1.2 Platforms In general, there are two platforms to look for publications on differential evolution. Although conventional publications printed on paper still play an important role, digital resources electronically available have been increasingly more preferred by both researchers and publishers. We have seen a quick transition from paper platform to digital platform within the last decade. It is noticed that differential evolution was proposed when paper platform was dominating [1], [5]-[15]. However, the dominance of paper platform does not last long. Both researchers and publishers have quickly realized the advantages of digital platform and have not hesitated to turn their attention to it. In this regard, digital platform is chosen as the main platform to carry out the literature survey. However, it is not the sole platform. Paper platform is also implemented whenever possible to supplement the digital platform so that missing of publications is minimized.

1.2.1 Starting Point At the initial stage, the literature survey is selective. Attention was focused on a bibliography [16] compiled by Prof. J. A. Lampinen which was posted online for open access. The bibliography itself was downloaded. Each and every publication included was also downloaded or copied whenever possible. The bibliography was expanded to include missing relevant publications appearing as references in available publications in the bibliography. Unfortunately, the bibliography has not been updated since it was last updated on Oct. 14, 2002. Consequently, it is very incomplete and can not stand in line with the state of the art of differential evolution.

1 A Literature Survey on Differential Evolution

3

1.2.2 Databases Many organizations have established their own databases which have been subscribed by most major libraries. Publications covered are usually formal, in another word, peer-reviewed and printed on paper. Nowadays, these databases go electronic for more exposure. They are also more timely updated. A publication from the following databases is counted if the keyword, differential evolution, appears in any field (title, abstract, keywords, text, and references) in the publication. Number of hits from different databases is shown in Table 1.1. (a) (b) (c) (d) (e) (f) (g) (h) (i)

Chinese Electronic Periodical Services Engineering Village 2 (EI) IEEE Explore Institute of Scientific and Technical Information of China ISI Web of Science (SCI) National Knowledge Infrastructure Scopus & ScienceDirect SpringerLink Wiley InterScience Table 1.1 Number of Hits from Different Databases year

IEEE

SCI

EI

InterScience

SpringerLink

Scopus

1995

0

7

9

0

1

4

1996

4

3

10

2

0

56

1997

4

9

19

2

7

56

1998

4

5

12

0

7

51

1999

8

18

45

7

7

71

2000

10

25

43

8

10

93

2001

8

23

50

4

11

112

2002

18

37

56

3

14

130

2003

23

60

96

8

15

206

2004

32

76

157

10

30

302

2005

55

132

247

11

43

381

2006

88

156

343

12

131

542

2007

116

258

528

26

143

696

2008

209

384

768

35

195

1132

4

A. Qing

Please note that (a) The last search was conducted in October, 2009. To avoid any potential misleading to readers, partial search result for year 2009 is not presented here. (b) The search results may contain irrelevant publications on differential evolution equation, social differential evolution, cultural differential evolution, economical differential evolution, geological differential evolution, geographical differential evolution, genetic differential evolution, and so on. (c) Each and every database has its unique coverage. No database is exhaustive. (d) Usually, a database contains publications from the publisher owning the database. However, Scopus provided by Elsevier covers some non-Elsevier publications. Therefore, the number of hits on Scopus is the largest except for year 1995. In this sense, Scopus is more comprehensive. (e) The search is focused on publications in Chinese and English. Publications presented in other languages are not considered unless they are indexed in the above databases.

1.2.3 Informal Online Resources and Tools Besides the above electronic resources, many informal electronic resources scatter over the internet. Some of the covered publications are notes that have not been published in any formal publishing platforms such as books, journals, conference proceedings, technical reports, or theses. They are usually posted to the internet by either individual researchers or non-academic and/or non-profitable organizations for various purposes. Access is in general free. These resources can be reached with the help of free search engines such as Google, Yahoo, Microsoft Bing, or Ask (http://www.ask.com/). Alternatively, researchers may visit the websites where these resources are actually stored. Three of the most prominent websites are Google Scholar (http://scholar.google.com.sg/) provided by Google, Computer Science Bibliographies (http://liinwww.ira.uka.de/bibliography/index.html) maintained by AlfChristian Achilles and Paul Ortyl, and citeSeerX (http://citeseerx.ist.psu.edu/) provided by College of Information Sciences and Technology, Pennsylvania State University. The number of hits from these three websites is shown in Table 1.2. The search result is accurate as of October 22, 2009 and may similarly contain irrelevant publications. Likewise, partial search result for year 2009 is not presented here.


5

Table 1.2 Number of Hits from Google Scholar, CSB, and citeSeerX

Google Scholar

CSB

citeSeerX

1995

6

9

6

1996

5

10

6

1997

6

6

6

1998

7

6

8

1999

38

18

9

2000

10

20

12

2001

12

21

17

2002

17

27

20

2003

24

32

22

2004

43

80

24

2005

67

80

24

2006

92

103

19

2007

138

167

9

2008

206

220

7

Due to the vast number of publications available from the internet, the search for relevant publications can only be done in a very restrictive way. Collection of publications here is accordingly far less than exhaustive.

1.3 Result Refining As mentioned before, publications found by the search may be irrelevant. Therefore, refining the survey result to eliminate irrelevant publications is compelling. This is done through tediously reading the reachable part of each and every found publication. Qualified publications are classified into four major categories: books, book chapters, other formal publications, and informal publications.

1.3.1 Books Book is a very important form of publications. 5 monographs on differential evolution [1], [17]-[20] have been published by now, among which the first book by the originators is introductory and well circulated.

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1.3.2 Book Chapters Some of the publications are presented as chapters in edited books. By now, at least one chapter is dedicated to differential evolution in at least 86 books [21]-[106] before (excluding) 2009.

1.3.3 Other Formal Publications Publications covered here include journal papers, presentations in conferences, symposiums, and workshops, degree theses, and technical reports. The number of qualified publications under this category is shown graphically in Fig. 1.1 where the ordinate is the year of publication while the number of qualified publications under this category is given beside the corresponding bar. Books and book chapters are excluded. Steady and accelerating growth has been observed since 1995, in which differential evolution was proposed. 2008

1029

2007

744

2006

536

2005

363

2004

236

2003

145

2002

94

2001

79

2000

57

1999

54

1998

21

1997

16

1996

7

1995

4

Fig. 1.1 Formal Publications on Differential Evolution

It has been noted that some publications do not actually involve a case study on differential evolution. Such publications do not contribute anything to differential evolution and are therefore unimportant to the differential evolution community. No further analysis on these publications is necessary. Remaining publications is shown graphically in Fig. 1.2. Similarly, books and book chapters are excluded. Please note that some publications may be wrongly treated as those without case studies on differential evolution because of availability.


7

638

2008 507

2007 365

2006 257

2005 154

2004 99

2003 68

2002

62

2001 41

2000

48

1999 1998

17

1997

15

1996

6

1995

4

Fig. 1.2 Formal Publications with Case Study on Differential Evolution

1.3.4 Informal Notes All qualified publications outside the above three categories are assembled under this category. There are 6 publications in this category as of October 22, 2009. Most of them are preprints or PowerPoint presentation notes posted to the internet by individual researchers. Bibliographic record is incomplete.

1.4 Result Analysis To make differential evolution benefit more existing researchers from active application fields and attract hesitating researchers from promising application fields, analysis on collected publications has to be conducted. The main goal of the result analysis is to look for practical usage advice for future applications and gain insight to further improve differential evolution. The analysis assembles publications on a specific topic so that researchers interested in the topic will not waste their time on irrelevant publications. At present, the analysis is focused on the following topics.

1.4.1 Theory of Differential Evolution Building a precise mathematical model for differential evolution has been posed as a challenge to the community as soon as differential evolution was proposed

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[1]. Such a solid mathematical foundation, if any, may hint desired revolutionary upgrading of differential evolution for higher reliability, better efficiency, and more robustness. However, it is yet to establish. Theoretical treatments on differential evolution in the history of differential evolution are very rare. The condition in rigorous mathematics under which differential evolution is sure to converge, the most fundamental question facing the differential evolution community or even the whole evolutionary computation community, is still pending for answer. Mathematical models for involved evolutionary operations have not been built. Interaction between evolutionary operations, intrinsic and non-intrinsic control parameters, and problems features has not been disclosed either. There is still a long way to go before differential evolution is fully appreciated. Nevertheless, some valuable pioneering efforts have been made to have a deeper understanding on differential evolution and may lead to the eventual theory of differential evolution. Studies by the originators are presented in their introductory monograph [17] and the latest book chapter by Price [19]. Other relevant studies are summarized in Table 1.3.

Table 1.3 Studies on Theory of Differential Evolution

focus

reference

evolution dynamics

[107]-[108]

stagnation

[109]-[111]

differential mutation

[112]-[115]

crossover

[116]-[117]

selection

[116]

parameter adaptation

[118]-[119]

Termination conditions

[120]-[125]

1.4.2 Fundamentals of Differential Evolution It is very critical for an applicant of differential evolution to have sufficient knowledge on differential evolution so that he or she can choose the most suitable differential evolution strategy and its corresponding intrinsic control parameter values. Otherwise, he or she may be confused by the huge number of differential evolution strategies and the infinite possibilities of setting intrinsic control parameter values. In the worst scenario, he or she may even be misled by past


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inappropriate usage of differential evolution and get an unnecessarily negative impression on differential evolution. An analysis on collected publications regarding fundamentals of differential evolution is in progress. Subjects in mind include evolution mechanism, encoding and decoding, initialization, differential mutation, crossover, selection, termination conditions, constraint handling, co-evolution, and so on.

1.4.3 Intrinsic Control Parameters Pleasant usage of differential evolution comes with not only the most suitable strategy but also appropriate setting of intrinsic control parameters. It has been well known that intrinsic control parameters of differential evolution play an essential role. Publications focusing on studying intrinsic control parameters will be assembled separately.

1.4.4 Evaluation of Differential Evolution It is a common practice to find out the advantages and disadvantages of an optimization algorithm through evaluation and comparison. Differential evolution has been evaluated by many researchers over various test bed and has earned its reputation in many comparative evaluations. Through specific evaluations, we may figure out a clearer picture on concerned component of differential evolution.

1.4.5 Applications of Differential Evolution Differential evolution will eventually be applied to solve practical application problems. Past applications provides valuable experience on usage of differential evolution. A preliminary attempt to classify qualified publications has been made to identify the fields in which differential evolution has been applied [1]. Such a preliminary classification did not cover publications in and after year 2008 and may be imprecise due to limited personal knowledge.

1.4.6 Hybridization It has been a well known fact that “for any algorithm, any elevated performance over one class of problems is offset by performance over another class” [126]. In another word, an optimization algorithm may outperform its counterparts over a specific class of problems. There is no exception for differential evolution. In accordance, many researchers have tried to hybridize different differential evolution strategies or differential evolution with other optimization algorithms. Analyzing different hybridization approaches is now in progress.

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1.5 Future Actions 1.5.1 Open Access We are currently managing a personal library covering different fields, among which differential evolution is one of the essential components. The personal library will be posted to internet for open access at appropriate time. It has to be seriously pointed out that although we have taken every possible measure to minimize missing publications, exhaustiveness of collection can not be claimed now and will not be claimed at any time in the future. Feedback from readers about any missing publications, be them formal or informal, is always welcome and appreciated. Missing publications notified by informants will be immediately integrated into the library. Moreover, this author will never claim absolute accuracy for the collection and analysis results. Besides limited personal knowledge, availability of collected publications may be the prime culprit. By chance, some title-only publications are collected during the search process. This author sincerely appeals to publishers, authors, and readers having such publications to share them as much as possible among the differential evolution community.

1.5.2 Future Update Finalizing for year 2009 will be carried out in early 2010. Update for years 2010 onwards is expected to take place on a yearly basis in order to make the survey result more accurate.

1.6 Misconceptions and Misconducts on Differential Evolution Differential evolution is one of the essential members of evolutionary algorithms. It shares some evolutionary operations and/or essential features with other evolutionary algorithms. However, it does not mean that it can be used interchangeably with other evolutionary algorithms. It is fundamentally different with other evolutionary algorithms in terms of evolution mechanism and evolutionary operations even if some evolutionary operations in differential evolution are identically named for historical reasons. It has been noticed that differential evolution has been mistermed as differential genetic algorithm [127] and has been regarded as variations of genetic algorithms [128]-[129], evolution strategies [130]. Such terms and/or classification are misleading.

References [1] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [2] Qing, A.: Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 44(1), 116– 125 (2006)


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[3] Qing, A.: A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem. In: 2007 IEEE Congress Evolutionary Computation, Singapore, September 25-28, pp. 1904–1909 (2007) [4] Qing, A.: A study on base vector for differential evolution. In: 2008 IEEE World Congress Computational Intelligence/2008 IEEE Congress Evolutionary Computation, Hong Kong, June 1-6, pp. 550–556 (2008) [5] Storn, R.: Modeling and Optimization of PET-Redundancy Assignment for MPEGSequences, Technical Report TR-95-018, International Computer Science Institute (May 1995) [6] Storn, R.: Differential Evolution Design of an IIR-Filter with Requirements for Magnitude and Group Delay, Technical Report TR-95-026, International Computer Science Institute (June 1995) [7] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (Mar 1995) [8] Price, K.V.: Differential evolution: a fast and simple numerical optimizer. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 524–527 (1996) [9] Storn, R.: Differential evolution design of an IIR-filter. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 268–273 (1996) [10] Storn, R.: On the usage of differential evolution for function optimization. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 519–523 (1996) [11] Storn, R.: System Design by Constraint Adaptation and Differential Evolution, Technical Report TR-96-039, International Computer Science Institute (November 1996) [12] Storn, R., Price, K.V.: Minimizing the real functions of the ICEC’96 contest by differential evolution. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 842–844 (1996) [13] Price, K.V.: Differential evolution vs. the functions of the 2nd ICEO. In: 1997 IEEE Int. Conf. Evolutionary Computation, Indianapolis, IN, April 13-16, pp. 153–157 (1997) [14] Price, K., Storn, R.: Differential evolution: a simple evolution strategy for fast optimization. Dr. Dobb’s J. 22(4), 18–24, 78 (1997) [15] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [16] Lampinen, J.: A bibliography on differential evolution algorithm, Technical Report, Lappeenranta University of Technology, Department of Information Technology, Laboratory of Information Processing (2001) (last updated on October 14, 2002) available via internet, http://www2.lut.fi/~jlampine/debiblio.htm (accessed on October 12, 2009) [17] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [18] Feoktistov, V.: Differential Evolution: in Search of Solutions. Springer, Berlin (2006) [19] Chakraborty, U.K. (ed.): Advances in Differential Evolution. Springer, Berlin (2008)

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[20] Onwubolu, G.C., Davendra, D.: Differential Evolution: A Handbook for Global Permutation-Based Combinatorial Optimization-Studies in Computational Intelligence, vol. 175. Springer, Heidelberg (2009) [21] Corn, D., Dorigo, M., Glover, F. (eds.): New Ideas in Optimization. McGraw-Hill, London (1999) [22] Mastorakis, N.E. (ed.): Recent Advances in Circuits and Systems. World Scientific, Singapore (1998) [23] Topping, B.H.V. (ed.): Developments in computational mechanics with high performance computing. Civil-Comp Press, Edinburgh (1999) [24] Sincak, P., Vascak, J., Kvasnicka, V., Pospichal, J. (eds.): Intelligent Technologies Theory and Applications. IOS Press, Amsterdam (2002) [25] Huijsing, J.H., Steyaert, M., van Roermund, A. (eds.): Analog Circuit Design: Scalable Analog Circuit Design, High Speed D/A Converters, RF Power Amplifiers. Kluwer Academic Publishers, New York (2003) [26] Sarker, R., Mohammadian, M., Yao, X. (eds.): Evolutionary Optimization. Kluwer Academic Publishers, New York (2003) [27] Johnston, R.L. (ed.): Applications of Evolutionary Computation in ChemistryStructure & Bonding, vol. 110. Springer, Berlin (2004) [28] Onwubolu, G.C., Babu, B.V.: New Optimization Techniques in Engineering. Studies in Fuzziness and Soft Computing, vol. 141. Springer, Berlin (2004) [29] Zhong, J.J. (ed.): Biomanufacturing-Advances in Biochemical Engineering/Biotechnology, vol. 87. Springer, Berlin (2004) [30] Grigoras, D., Nicolau, A. (eds.): Concurrent information processing and computing. NATO science series, series III, Computer and systems sciences, vol. 195. IOS Press, Amsterdam (May 2005) [31] Hart, W.E., Krasnogor, N., Smith, J.E. (eds.): Recent Advances in Memetic Algorithms. Studies in Fuzziness and Soft Computing, vol. 166. Springer, Berlin (2005) [32] Hoffmann, F., Köppen, M., Klawonn, F., Roy, R. (eds.): Soft Computing: Methodologies and Applications-Advances in Soft Computing, vol. 32. Springer, Berlin (2005) [33] Palit, A.K., Popovic, D.: Computational Intelligence in Time Series Forecasting: Theory and Engineering Applications. Springer, Berlin (2005) [34] Pieruci, S. (ed.): Computer-Aided Chemical Engineering. Elsevier, Amsterdam (2005) [35] Tan, K.C., Khor, E.F., Lee, T.H.: Multiobjective Evolutionary Algorithms and Applications. Springer, Berlin (2005) [36] Abraham, A., de Baets, B., Köppen, M., Nickolay, B. (eds.): Applied Soft Computing Technologies: The Challenge of Complexity-Applied Soft Computing, vol. 34. Springer, Berlin (2006) [37] Abraham, A., Grosan, C., Ramos, V. (eds.): Stigmergic Optimization-Studies in Computational Intelligence, vol. 31. Springer, Berlin (2006) [38] Abraham, A., Grosan, C., Ramos, V. (eds.): Swarm Intelligence in Data Mining. Studies in Computational Intelligence, vol. 34. Springer, Berlin (2006) [39] Alba, E., Marti, R.: Metaheuristic Procedures for Training Neutral NetworksOperations Research/Computer Science Interfaces Series, vol. 36. Springer, Berlin (2006) [40] Brabazon, A., O’Neill, M.: Biologically Inspired Algorithms for Financial Modelling. Spriinger, Berlin (2006)


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[41] Burke, E.K., Kendall, G. (eds.): Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques. Springer, Berlin (2006) [42] Caiti, A., Chapman, N.R., Hermand, J.P., Jesus, S.M. (eds.): Acoustic Sensing Techniques for the Shallow Water Environment: Inversion Methods and Experiments. Springer, Berlin (2006) [43] Castro-López, R., Fernández, F.V., Guerra-Vinuesa, O., Rodríguez-Vázquez, Á.: Reuse-Based Methodologies and Tools in the Design of Analog and Mixed-Signal Integrated Circuits. Springer, Berlin (2006) [44] Dzemyda, G., Šsltenis, V., Žilinskas, A. (eds.): Stochastic and Global Optimization. Springer, Berlin (2006) [45] Jin, Y. (ed.): Multi-Objective Machine Learning. Studies in Computational Intelligence, vol. 16. Springer, Berlin (2006) [46] Li, Z., Halang, W.A., Chen, G.: Integration of Fuzzy Logic and Chaos. Theory. Studies in Fuzziness and Soft Computing, vol. 187. Springer, Berlin (2006) [47] Liberti, L., Maculan, N. (eds.): Global Optimization: from Theory to Implementation-Nonconvex Optimization and Its Applications, vol. 84. Springer, Berlin (2006) [48] Liu, J., Jin, X., Tsui, K.C.: Autonomy Oriented Computing. Kluwer Academic Publishers, Bonston (2006) [49] Nedjah, N., Alba, E., de Macedo Mourelle, L. (eds.): Parallel Evolutionary Computations. Studies in Computational Intelligence, vol. 22. Springer, Berlin (2006) [50] Nedjah, N., de Macedo Mourelle, L. (eds.): Swarm Intelligent Systems. Studies in Computational Intelligence. Springer, Berlin (2006) [51] Pintér, J.D. (ed.): Global Optimization: Scientific and Engineering Case StudiesNonconvex Optimization and Its Applications, vol. 85. Springer, Berlin (2006) [52] Steyaert, M., van Roermund, A.H.M., Huijsing, J.H. (eds.): Analog Circuit Design. Springer, Berlin (2006) [53] Tiwari, A., Knowles, J., Avineri, E., Dahal, K., Roy, R. (eds.): Applications of Soft Computing: Recent Trends. Springer, Heidelberg (2006) [54] Wiak, S., Krawczyk, A., Trlep, M. (eds.): Computer Engineering in Applied Electromagnetism. Springer, Berlin (2006) [55] Zhang, H., Liu, D.: Fuzzy Modeling and Fuzzy Control. Birkhäuser, Boston (2006) [56] Zhang, G.Q., Van Driel, W.D., Fan, X.J. (eds.): Mechanics of Microelectronics. Springer, Berlin (2006) [57] Zomaya, A.Y. (ed.): Handbook of Nature-Inspired and Innovative Computing. Springer, Berlin (2006) [58] Zomaya, A.Y.: Parallel computing for bioinformatics and computational biology; models, enabling technologies, and case studies. John Wiley, New York (2006) [59] Chahl, J.S., Jain, L.C., Mizutani, A., Sato-Ilic, M. (eds.): Innovations in Intelligent Machines, vol. 1. Springer, Berlin (2007) [60] Cios, K.J., Pedrycz, W., Swiniarski, R.W., Kurgan, L.A.: Data Mining: A Knowledge Discovery Approach. Springer, Berlin (2007) [61] Corchado, E., Corchado, J.M., Abraham, A. (eds.): Innovations in Hybrid Intelligent Systems-Advances in Soft Computing, vol. 44. Springer, Berlin (2007) [62] Ebashi, S., Ohtsuki, I.: Regulatory Mechanisms of Striated Muscle ContractionAdvances in Experimental Medicine and Biology, vol. 592. Springer, Berlin (2007) [63] Grosan, C., Abraham, A., Ishibuchi, H. (eds.): Hybrid Evolutionary Algorithms. Studies in Computational Intelligence, vol. 75. Springer, Berlin (2007)

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[64] Jain, L.C., Palade, V., Srinivasan, D.: Advances in Evolutionary Computing for System Design. Studies in Computational Intelligence, vol. 66. Springer, Heidelberg (2007) [65] Kaburlasos, V.G., Ritter, G.X.: Computational Intelligence Based on Lattice Theory. Studies in Computational Intelligence, vol. 67. Springer, Berlin (2007) [66] Lobo, F.G., Lima, C.F., Michalewicz, Z. (eds.): Parameter Setting in Evolutionary Algorithms. Studies in Computational Intelligence, vol. 54. Springer, Berlin (2007) [67] Melin, P., Castillo, O., Ramírez, E.G., Kacprzyk, J., Pedrycz, W. (eds.): Analysis and Design of Intelligent Systems using Soft Computing Techniques-Advances in Soft Computing, vol. 41. Springer, Berlin (2007) [68] Nedjah, N., Abraham, A., de Macedo Mourelle, L. (eds.): Computational Intelligence in Information Assurance and Security. Studies in Computational Intelligence, vol. 57. Springer, Berlin (2007) [69] Nedjah, N., dos Santos Coelho, L., de Macedo Mourelle, L.: Mobile Robots: The Evolutionary Approach. Studies in Computational Intelligence, vol. 50. Springer, Berlin (2007) [70] Saad, A., Avineri, E., Dahal, K., Sarfraz, M., Roy, R.: Soft Computing in Industrial Applications: Recent and Emerging Methods and Techniques-Advances in Soft Computing, vol. 39. Springer, Berlin (2007) [71] Sobh, T., Elleithy, K., Mahmood, A., Karim, M.: Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications. Springer, Berlin (2007) [72] Suri, J.S., Farag, A.A.: Deformable Models: Biomedical and Clinical Applications. Springer, Berlin (2007) [73] Törn, A., Žilinskas, J.: Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday. Springer, Berlin (2007) [74] Valavanis, K.P. (ed.): Advances in Unmanned Aerial Vehicles: State of the Art and the Road to Autonomy. Springer, Berlin (2007) [75] Welcker, K.: Evolutionäre Algorithmen, Teubner (2007) [76] Yang, S., Ong, Y.S., Jin, Y.: Evolutionary Computation in Dynamic and Uncertain Environments. Studies in Computational Intelligence, vol. 51. Springer, Berlin (2007) [77] Abraham, A., Grosan, C., Pedrycz, W. (eds.): Engineering Evolutionary Intelligent Systems. Studies in Computational Intelligence, vol. 82. Springer, Berlin (2008) [78] Ao, S.I., Riger, B., Chen, S.S. (eds.): Advances in Computational Algorithms and Data Analysis. Lecture Notes Electrical Engineering, vol. 14. Springer, Berlin (2008) [79] Brabazon, A., O’Neill, M. (eds.): Natural Computing in Computational Finance. Studies in Computational Intelligence, vol. 100. Springer, Berlin (2008) [80] Castillo, O., Xu, L., Ao, S.I. (eds.): Trends in Intelligent Systems and Computer Engineering. Lecture Notes Electrical Engineering, vol. 6. Springer, Berlin (2008) [81] Chaturvedi, D.K.: Soft Computing: Techniques and Its Applications in Electrical Engineering. Studies in Computational Intelligence, vol. 103. Springer, Berlin (2008) [82] Cotta, C., Reich, S., Schaefer, R., Ligęza, A. (eds.): Knowledge-Driven Computing. Studies in Computational Intelligence, vol. 102. Springer, Berlin (2008) [83] Cotta, C., Seraux, M., Sörensen, K. (eds.): Adaptive and Multilevel MetaheuristicsStudies in Computational Intelligence, vol. 136. Springer, Heidelberg (2008)


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[84] Cotta, C., van Hemert, J. (eds.): Recent Advances in Evolutionary Computation for Combinatorial Optimization. Studies in Computational Intelligence, vol. 153. Springer, Berlin (2008) [85] Fulcher, J., Jain, L.C. (eds.): Computational Intelligence: a Compendium. Studies in Computational Intelligence, vol. 115. Springer, Berlin (2008) [86] Ghosh, A., Dehuri, S., Ghosh, S. (eds.): Multi-objective Evolutionary Algorithms for Knowledge Discovery from Databases. Studies in Computational Intelligence, vol. 98. Springer, Berlin (2008) [87] Grosse, C.U., Ohtsu, M. (eds.): Acoustic Emission Testing. Springer, Berlin (2008) [88] Kelemen, A., Abraham, A., Chen, Y. (eds.): Computational Intelligence in Bioinformatics. Studies in Computational Intelligence, vol. 94. Springer, Berlin (2008) [89] Kontoghiorghes, E.J., Rustem, B., Winker, P. (eds.): Computational Methods in Financial Engineering: Essays in Honour of Manfred Gilli. Springer, Berlin (2008) [90] Kramer, O.: Self-Adaptive Heuristics for Evolutionary Computation. Studies in Computational Intelligence, vol. 147. Springer, Berlin (2008) [91] Krasnogor, N., Nicosia, G., Pavone, M., Pelta, D. (eds.): Nature Inspired Cooperative Strategies for Optimization. Studies in Computational Intelligence, vol. 129. Springer, Berlin (2008) [92] Lee, K.Y., El-Sharkawi, M.A. (eds.): Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems. Wiley-IEEE Press, New York (2008) [93] Liang, S. (ed.): Advances in Land Remote Sensing: System, Modeling, Inversion and Application. Springer, Berlin (2008) [94] Liu, Y., Sun, A., Loh, H.T., Lu, W.F., Lim, E.P. (eds.): Advances of Computational Intelligence in Industrial Systems. Studies in Computational Intelligence, vol. 116. Springer, Berlin (2008) [95] Prasad, B. (ed.): Soft Computing Applications in Industry. Studies in Fuzziness and Soft Computing, vol. 226. Springer, Berlin (2008) [96] Prasad, B. (ed.): Soft Computing Applications in Business. Studies in Fuzziness and Soft Computing, vol. 230. Springer, Berlin (2008) [97] Prokhorov, D. (ed.): Computational Intelligence in Automotive Applications. Studies in Computational Intelligence, vol. 132. Springer, Berlin (2008) [98] Riolo, R., Soule, T., Worzel, B. (eds.): Genetic Programming Theory and Practice, vol. 5. Springer, Berlin (2008) [99] Siarry, P., Michalewicz, Z. (eds.): Advances in Metaheuristics for Hard Optimization. Springer, Berlin (2008) [100] Smolinski, T.G., Milanova, M.G., Hassanien, A.E. (eds.): Applications of Computational Intelligence in Biology. Studies in Computational Intelligence, vol. 122. Springer, Berlin (2008) [101] Smolinski, T.G., Milanova, M.G.,, A.E.: Computational Intelligence in Biomedicine and Bioinformatics. Studies in Computational Intelligence, vol. 151. Springer, Berlin (2008) [102] Tizhoosh, H.R., Ventresca, M. (eds.): Oppositional Concepts in Computational Intelligence. Studies in Computational Intelligence, vol. 155. Springer, Berlin (2008) [103] Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems II. Springer, Berlin (2008) [104] Wiak, S., Krawczyk, A., Dolezel, I. (eds.): Intelligent Computer Techniques in Applied Electromagnetics. Studies in Computational Intelligence, vol. 119. Springer, Berlin (2008)

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[105] Xhafa, F., Abraham, A. (eds.): Metaheuristics for Scheduling in Industrial and Manufacturing Applications. Studies in Computational Intelligence, vol. 128. Springer, Berlin (2008) [106] Yang, A., Shan, Y., Bui, L.T.: Success in Evolutionary Computation. Studies in Computational Intelligence, vol. 92. Springer, Berlin (2008) [107] Zhang, J., Sanderson, A.C.: An approximate Gaussian model of differential evolution with spherical fitness function. In: 2007 IEEE Congress Evolutionary Computation, Singapore, september 25-28, pp. 2220–2228 (2007) [108] Montgomery, J.: Differential evolution: Difference vectors and movement in solution space. In: IEEE Congress Evolutionary Computation, Trondheim, Norway, May 18-21, pp. 2833–2840 (2009) [109] Lampinen, J., Zelinka, I.: On stagnation of the differential evolution algorithm. In: 6th Int. Mendel Conf. Soft Computing, Brno, Czech Republic, June 7-9, pp. 76–83 (2000) [110] Sukov, A., Borisov, A.: A study of search technique in differential evolution. In: 7th Int. MENDEL Conf. Soft Computing, Brno, Czech Republic, June 6-8, pp. 144–148 (2001) [111] Tomislav, Š.: Improving convergence properties of the differential evolution algorithm. In: 8th Int. MENDEL Conf. Soft Computing, Brno, Czech Republic, June 57, pp. 80–86 (2002) [112] Ali, M.M.: Differential evolution with preferential crossover. European J. Operational Research 181(3), 1137–1147 (2007) [113] Sutton, A.M., Lunacek, M., Whitley, L.D.: Differential evolution and nonseparability: using selective pressure to focus search. In: 2007 Genetic Evolutionary Computation Conf., London, UK, July 7-11, pp. 1428–1435 (2007) [114] Zaharie, D.: Statistical properties of differential evolution and related random search algorithms. In: 18th Symp. Computational Statistics, Oporto, Portugal, August 2429, pp. 473–485 (2008) [115] Dasguptu, S., Das, S., Biswas, A., Abraham, A.: On stability and convergence of the population-dynamics in differential evolution. AI Communications 22(1), 1–20 (2009) [116] Zielinski, K., Laur, R.: Variants of differential evolution for multi-objective optimization. In: 2007 IEEE Symp. Computational Intelligence Multicriteria Decision Making, Honolulu, HI, April 1-5, pp. 91–98 (2007) [117] Zaharie, D.: A comparative analysis of crossover variants in differential evolution. In: Int. Multiconference Computer Science Information Technology, pp. 171–181 (2007) [118] Zaharie, D.: Parameter adaption in differential evolution by controlling the population diversity. In: 4th Int. Workshop Symbolic Numeric Algorithms Scientific Computing, Timi¸ soara, Romania, October 9-12, pp. 385–397 (2002) [119] Zaharie, D.: Control of population diversity and adaptation in differential evolution algorithms. In: 9th Int. Mendel Conf. Soft Computing, Brno, Czech Republic, June 2003, pp. 41–46 (2003) [120] Hajji, O., Brisset, S., Brochet, P.: A stop criterion to accelerate magnetic optimization process using genetic algorithms and finite element analysis. IEEE Trans. Magnetics 39(3 I), 1297–1300 (2003)


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[121] Zielinski, K., Peters, D., Laur, R.: Run time analysis regarding stopping criteria for differential evolution and particle swarm optimization. In: 1st Int. Conf. Experiments/Process/System Modelling/Simulation/Optimization, Athens, Greece, July 6-9 (2005) [122] Zielinski, K., Peters, D., Laur, R.: Stopping criteria for single-objective optimization. In: 3rd Int. Conf. Computational Intelligence Robotics Autonomous Systems, Singapore, December 13-16 (2005) [123] Zielinski, K., Laur, R.: Stopping criteria for constrained optimization with particle swarms. In: 2nd Int. Conf. Bioinspired Optimization Methods Applications, Ljubljana, Slovenia, October 9-10, pp. 45–54 (2006) [124] Zielinski, K., Weitkemper, P., Laur, R., Kammeyer, K.D.: Examination of stopping criteria for differential evolution based on a power allocation problem. In: 10th Int. Conf. Optimization Electrical Electronic Equipment, Brasov, Romania, May 18-19, pp. 149–156 (2006) [125] Zielinski, K., Laur, R.: Stopping criteria for a constrained single-objective particle swarm optimization algorithm. Informatics (Ljubljana) 31(1), 51–59 (2007) [126] Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evolutionary Computation 1(1), 67–82 (1997) [127] Michael, C., McGraw, G.: Opportunism and Diversity in Automated Software Test Data Generation, Technical Report RSTR-003-97-13, version 1.3, RST Corporation, Sterling, VA, USA (December 8, 1997) [128] Masters, T., Land, W.: A new training algorithm for the general regression neural network. In: 1997 IEEE Int. Conf. Systems Man Cybernetics, Orlando, FL, October 12-15, vol. 3, pp. 1990–1994 (1997) [129] Engle, R.F., Manganelli, S.: CAViaR: conditional autoregressive value at risk by regression quantiles, UCSD Economics Discussion Paper 99-20, University of California, San Diego, Department of Economics (October 1999) [130] Cafolla, A.A.: A new stochastic optimisation strategy for quantitative analysis of core level photoemission data. Surface Science 402-404, 561–565 (1998)

Chapter 2

Basics of Differential Evolution Anyong Qing

1

2.1 A Short History 2.1.1 Inception Differential evolution was proposed by K.V. Price and R. Storn in 1995 [1]. At that time, Price was asked to solve the Chebyshev polynomial fitting problem [1]-[5] by Storn [2], [5]. Initially, he tried to solve it by using genetic annealing algorithm [6]. However, although he eventually found the solution to the 5-dimensional Chebyshev polynomial fitting problem by using genetic annealing algorithm, he was frustrated to notice that genetic annealing algorithm fails to fulfill the three requirements for a practical optimization technique: strong global search capability, fast convergence, and user friendliness. A breakthrough happened when Price came up with an innovative scheme for generating trial parameter vectors. In this scheme, a new parameter vector is generated by adding the weighted difference vector between two population members to a third member. Such a scheme was named as differential mutation and has been well known to be the crucial idea behind the success of differential evolution. The cornerstone for differential evolution was therefore laid. Price wrapped up his invention with other critical ideas: natural real code, arithmetic operations, mother-child competition and selection, and execution of evolutionary operations in the order of mutation-crossover-selection. Consequently, differential evolution, a very reliable, efficient, robust, and simple evolutionary algorithm was developed. Anyong Qing Temasek Laboratories, National University of Singapore 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected] 1

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 19–42. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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2.1.2 Early Years 2.1.2.1 Assessment Evaluation is one of the essential parts of algorithm development and has to be conducted right after an optimization algorithm is developed. It is the undeniable responsibility of the algorithm’s originator(s) to evaluate the proposed optimization algorithm. During the evaluation process, the originator(s) may gain more insights behind the concerned new optimization algorithm and make further efforts to improve it. The first evaluation of differential evolution was reported in the founding publication [1]. DE/rand/1/exp and DE/current-to-best/1/exp were evaluated over a test bed containing 7 unconstrained toy functions and 2 constrained toy functions. All intrinsic control parameters are fixed by trial and error approach. Comparison with annealed Nelder & Mead strategy and adaptive simulated annealing was also presented. Price and Storn, the originators, published two successive performance evaluation reports in 1996 [7] and 1997 [8]. Different strategies of differential evolution were evaluated over larger test beds. It is interesting to note that differential evolution quickly came to the attention of other researchers [9]. Evaluation of differential evolution over a test bed of 15 functions was carried out. Comparison with a variety of methods was also made. Differential evolution solves all functions successfully. It converges most rapidly while optimizing 11 of the 15 functions. 2.1.2.2 Reputation Building Differential evolution proved itself by winning in the two International Contests on Evolutionary Optimization (ICEO) [2], [3], [5], [10] in 1996 and 1997. More importantly, it was the best evolutionary algorithm among all entries since the first two places were won by non-evolutionary algorithms. It finished 3rd in the 1st International Contest on Evolutionary Optimization held in Nagoya, Japan from May 20, 1996 to May 22, 1996 [2], [5]. Differential evolution was the best among all qualified entries in the 2nd International Contests on Evolutionary Optimization [3] although the actual contest was cancelled due to lack of valid entries. 2.1.2.3 Applications Each and every practical optimization algorithm has to be picked up by people working in practical fields. More importantly, it will receive further evaluation through practical applications and gain more insights to improve it to better suit requirements arising from outstanding problems. There is no exception for differential evolution. It was soon applied by both the originators [11]-[15] and other application engineers [16]-[29] to solve various practical problems. Benefiting fields in these early applications of differential evolution are summarized in Table 2.1.

2 Basics of Differential Evolution

21

Table 2.1 Fields Benefiting from Early Application of Differential Evolution application

reference

Priority encoding transmission (PET)-redundancy assignment

[11]

Design of an IIR-filter

[12], [13]

Design of a howling remover

[14]

Redesign a switched capacitor (SC)-filter suffering from parasitic capacitances

[15]

multisensor fusion

[16], [19]-[21]

Parameter estimation of a batch bioprocess

[17]

Parallel optimization

[18]

Training of general regression neural network

[22]

Software test data generation

[23]

Scheduling of core blowers

[24]-[25]

Image registration

[27]

Temperature control of a chemical reactor system

[28]-[29]

Selection of control policy of a robotic arm

[29]

2.1.2.4 Promotion To boost the awareness of differential evolution and expand the community, Rainer established a website [5]. Latest contents include (a) (b) (c) (d) (e)

History, basics and practical advice Codes Demos Applications Useful relevant links

The website is still one of the prime resources for differential evolution. 2.1.2.5 Practical Advice Differential evolution involves intrinsic control parameters. It has been realized right from the beginning that differential evolution is dependent on proper setting of these intrinsic control parameters. To convenience differential evolution applicants, some practical advices to choose intrinsic control parameter values are recommended by the originators [3], [5], [8], [14], [30]. The recommendations are (a) Choose a population 2 [30], 5 [8], 10 [5], [14], [30] times problem dimension, or larger [3]. (b) Choose mutation intensity from [0.5, 1] [5], [14].

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(c) Choose crossover probability considerably lower than 1 if convergent, otherwise choose it from [0.8, 1] [5], [8], [14]. However, in [30], it is suggested to choose crossover probability “as large as possible without causing the population to become devoid of diversity”. (d) Choose lower mutation intensity for larger population size and vice versa [14]. (e) Adjust mutation intensity and crossover probability in parallel [3]. (f) Increase population size as crossover probability increases [3]. Some usage rules regarding generating initial population [14], formulating objective function [14], monitoring evolution [14], and strategy selection [3] are also recommended by the originators. 2.1.2.6 Standardization Strategies of differential evolution are denoted by DE/x/y/z where x indicates how the differential mutation base is chosen, y≥1 is the number of vector differences added to the base vector, and z is the law which the number of parameters donated by the mutant follows [3], [8]. The notation was inked by the originators in 1997 [8]. 2.1.2.7 More Adventures DE/rand/1/exp and DE/target-to-best/1/exp were proposed in the founding publication of differential evolution [1]. Later on, Price and Storn went on to explore better strategies. DE/best/2/bin [7], DE/random-to-best/1/exp [13]-[14] (corrected as DE/target-to-best/1/exp in [3]), DE/best/1/exp [14], DE/best/2/exp [10], [14], DE/current/1/exp [15], DE/rand/1/bin [8], [30], and other variants [2], were subsequently developed. It is very interesting to note that the inherent defect of differential evolutional, namely, slow convergence, was soon observed [17], [22]. Proposals to hybridize differential evolution with deterministic local optimizers were timely put forward.

2.1.3 Key Milestones in and after 1998 Differential evolution turned into fast track in 1998. Its potential has been increasingly realized by more and more researchers from ever growing number of fields. Researchers with different background have worked together to know more about its mathematical foundation, develop new strategies, or solve more challenging problems. Key events happening in and after 1998 in the short history of differential evolution are summarized in Table 2.2.


23

Table 2.2 Key Milestones in and after 1998 year

event

source

1998

first modification to initialization

[31]

1998

first modification to differential mutation

[31]

1998

first thesis on differential evolution

[32]

1999

first differential evolution for integer optimization parameters

[33]

1999

initial idea of dynamic differential evolution

[4], [34]-[35]

1999

Pareto differential evolution for multi-objective optimization

[36]-[37]

1999

first multi-population differential evolution

[38]

1999

first adaptive differential evolution

[39]

2000

first application in electromagnetics

[40]

2000

first empirical study on differential evolution

[41]-[42]

2000

first approximation of objective and constraint functions

[43]

2001

bibliography on differential evolution

[44]

2001

generalized differential evolution for multi-objective optimization

[45]

2005

first book on differential evolution

[3]

2005

non-dominated sorting differential evolution

[46]

2005

first strategy adaptation differential evolution

[47]

2006

first special session on differential evolution in IEEE Congress on Evolutionary Computation

2006

opposition-based differential evolution

[48]-[49]

2006

first system-level parametric study

[35]

2007

first thematic study on crossover in differential evolution

[50]

2009

evolutionary crimes

[4]

2009-

special issue on differential evolution in IEEE Transactions on Evolutionary

2010

Computation

2.2 The Foundational Differential Evolution Strategies Two differential evolution strategies, DE/rand/1/exp and DE/target-to-best/1/exp, were proposed in the founding publication of differential evolution [1] to minimize a single objective function f(x) with N-dimensional real optimization parameters x. Details of these two foundational strategies are given here to entertain readers with basic features of differential evolution.

2.2.1 Notations The notations applied in [4] are strictly followed here. However, for the convenience of readers and self-completeness of this book, primary notations are

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summarized in Table 2.3. These notations will be consistently applied throughout this book unless specified otherwise. Secondary notations will be explained when it is mentioned for the first time. Table 2.3 Primary Notations notation

legend

f(x)

objective function to minimize

notation

legend

x

N-dimensional vector of optimization parameters

xj

the jth optimization parameter

pc

crossover probability

b jL

lower bound of xj

bjU

upper bound of xj

P

population

Np

population size

P0

initial population

Pn

population of generation n

pi

the ith individual in P

pn,i

the ith individual in Pn

pbest

the best individual in P

pn,best

the best individual in Pn

pworst

the worst individual in P

pn,worst

the worst individual in Pn

x

i

i j

i

vector of optimization parameters of p i

x i

n,i n,i

vector of optimization parameters of pn,i the jth optimization parameter in xn,i of pn,i

x

the jth optimization parameter in x of p

xj

vi

mutant for pi

vn+1,i

mutant for pn,i

xv,i

vector of optimization parameters of vi

xn+1,v,i

vector of optimization parameters of vn+1,i

xj

v,i i

v,i

the jth optimization parameter in x of v

i

i

xj

n+1,v,i n,i

the jth optimization parameter in xn+1,v,i of vn+1,i base for vn+1,i

b

base for v

b

xb,i

vector of optimization parameters of bi

xn,b,i

vector of optimization parameters of bn,i

xjb,i

the jth optimization parameter in xb,i of bi

xjn,b,i

the jth optimization parameter in xn,b,i of bn,i

the ith child

cn+1,i

the ith child of generation n + 1

vector of optimization parameters of c

xn+1,c,i

vector of optimization parameters of cn+1,i

xjc,i

the jth optimization parameter in xc,i of ci

xjn+1,c,i

F

mutation intensity

Fy

the yth mutation intensity

p1

index for donor 1 (one vector difference case)

p1y

index for donor 1 for the yth vector difference

p2

index for donor 2 (one vector difference case)

p2y

index for donor 2 for the yth vector difference

ci x

c,i

i

the jth optimization parameter in xn+1,c,i of cn+1,i

2.2.2 Strategy Framework Differential evolution optimizes f(x) with a population of Np individuals. It involves two stages, namely, initialization and evolution. Initialization generates initial population P0. Then the population evolves from one generation (Pn) to the next (Pn+1) until termination conditions are satisfied. While evolving from Pn to Pn+1, the three evolutionary operations, namely, differential mutation, crossover and selection, are executed in sequence. 2.2.2.1 Pseudo-code The Fortran-style pseudo-code of differential evolution is shown in Fig. 2.1.


25

Initialization n=0 do i = 1, Np generate p0,i evaluate f(x0,i) end do Evolution do while termination conditions are not satisfied n=n+1 do i = 1, Np differential mutation to obtain mutant vn+1,i crossover mutant vn+1,i with pn,i to deliver child cn+1,i evaluate child f(xn+1,c, i) selection to get individual pn+1,i end do end do

Fig. 2.1 Fortran-style Pseudo-code of Foundational Differential Evolution Strategies

2.2.2.2 Initialization Initialization generates the initial population P0 which contains Np individuals p0,i.

(

x 0j , i = b Lj + α ij bUj − b Lj

)

1≤ i ≤ Np, 1≤ j ≤ N

(1)

where the real random number, αji , is usually but not necessarily uniform in [0,1]. Alternatively, “in case a preliminary solution is available, the initial population is often generated by adding normally distributed random deviations to the nominal solution” x0 [1].

x

0, i j

⎧⎪ x 0j =⎨ 0 i ⎪⎩ x j + σ j

i =1 2 ≤ i ≤ N p −1

1≤ j ≤ N

(2)

where the real random number, σji , is usually but not necessarily “normally distributed”. 2.2.2.3 Differential Mutation Differential mutation generates a mutant vn+1,i for pn,i as follows

(

)

x n+1,v ,i = x n,b ,i + F x n, p1 − x n , p2 , 1 ≤ i ≠ p1 ≠ p2 ≤ N p n,i

The differential mutation base b is chosen in two different ways in [1].

(3)

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A Random bn,i is randomly chosen from Pn and is different with both donors p n , p1 and p n , p2 . B Target-to-best bn,i is a point on a line between pn,i and pn,best. It is generated through the following arithmetic recombination

(

x n ,b ,i = x n ,i + λ x n ,best − x n ,i

)

(4)

where λ is the recombination coefficient. Alternatively, by looking at the general formulation of differential mutation shown in (2.2) in [1], this form of differential mutation, target-to-best/1, can be regarded as a biased form of current/2 in which the current individual pn,i serves as differential mutation base, the first vector difference is xn,best-xn,i weighted by λ, and the second vector difference is x n , p1 − x n , p2 weighted by F. 2.2.2.4 Crossover Crossover delivers a child cn+1,i through mating vn+1,i with pn,i. Exponential crossover is applied in [1]. The Fortran-style pseudo-code for exponential crossover is given in Fig. 2.2. A starting point r (1≤r≤N) is first randomly chosen. xrn+1,c,i of cn+1,i is taken from xrn+1,v,i of vn+1,i. Parameters of cn+1,i after (in cyclic sense) r depends on a series of Bernoulli experiments of probability pc, a constant in [0,1]. vn+1,i will keep donating its parameters to cn+1,i until the Bernoulli experiment is unsuccessful or the crossover length L, i.e., the number of parameters of the child donated by the mutant, is already N - 1. The remaining parameters of cn+1,i come from pn,i. do j = 1, N xjn+1,c,i = xjn,i end do r = N * rand(0, 1) + 1 k=r

xkn+1,c,i = xkn+1,v,i L=1

E = rand(0, 1) do while (E pc and L < N -1) L=L+1 k = 1 + mod(k, N) xkn+1,c,i = xkn+1,v,i

E = rand(0, 1) end do

Fig. 2.2 Fortran-style Pseudo-code of Exponential Crossover for Differential Evolution


27

A demonstrative example is shown in Fig. 2.3. N = 8, r = 7 and the crossover length is 3.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

1

0

15.23

35.09

Bernoulli experiments

xn+1,c,i

1 57.82

12.06

26.99

82.96

99.28

85.86

Fig. 2.3 Exponential Crossover

2.2.2.5 Selection The selection operation follows Darwin’s natural selection, or survival of the fittest [51]. Child cn+1,i competes with a predetermined individual in population Pn and replaces it if cn+1,i dominates its competitor. cn+1,i usually competes with pn,i [12] although it is not stated explicitly in [1]. Such a selection operation is mathematically expressed as

p

n +1, i

⎧⎪c n +1,i f (x n +1,c, i ) ≤ f (x n, i ) = ⎨ n ,i ⎪⎩p otherwise

(5)

Sometimes, the selection is conducted in a sense of stronger dominance [10], i.e.,

⎧⎪c n +1, i f (x n +1,c, i ) < f (x n, i ) p n +1,i = ⎨ n ,i ⎪⎩p otherwise

(6)

As far as we know, by now, there is no thematic study on the effect of these two selection schemes. 2.2.2.6 Termination Conditions Termination conditions are not specified in [1]. However, other early publications by the originators [8]-[10] suggest that limit of number of generations is implemented to terminate the evolution process. It is yet to clarify whether “objective met” [3] is implemented although it is mentioned in [1] that the best individual pn+1,best in the new population Pn+1 is updated at the end of each evolution loop.

2.2.3 Intrinsic Control Parameters The two differential evolution strategies in [1] share three intrinsic control parameters

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(1) Population size Np (2) Mutation intensity F (3) Crossover probability pc In general, DE/target-to-best/1/exp has one more intrinsic control parameter, the recombination coefficient λ. However, for simplicity, λ is usually chosen identical with F.

2.3 Classic Differential Evolution Besides the aforementioned foundational differential evolution strategies proposed in the originators’ founding publication on differential evolution [1], many more differential evolution strategies under the same umbrella of classic differential evolution have been put forward. All classic differential evolution strategies share the same framework as shown in Fig. 2.1 while distinguish themselves in terms of initialization, differential mutation, crossover, objective function evaluation, selection, and termination conditions.

2.3.1 Initialization The two general initialization schemes have already been described by the originators clearly in their founding publication on differential evolution [1]. However, Different differential evolution strategy may apply different probability distribution i i function to generate random numbers αj in (1) [52] or σj in (2) [31].

2.3.2 Differential Mutation According to the standard notation DE/x/y/z, the general formulation of differential mutation for classic differential evolution is [1]

(

x n +1,v ,i = x n ,b ,i + ∑ Fy x y ≥1

n , p1 y

−x

n , p2 y

),

1 ≤ i ≠ p1 y ≠ p2 y ≤ N p

(7)

It is interesting to note that Storn [53] uses normalized vector differences to generate mutant vn+1,i which is mathematically expressed as

x n+1,v ,i = x n,b ,i +

1 y

∑ F (x y

y ≥1

n , p1 y

−x

n , p2 y

),

1 ≤ i ≠ p1 y ≠ p2 y ≤ N p

(8)

Unless specified otherwise, we will stick to the general formulation shown in (7). Different differential mutation implements different differential mutation base n, p n, p bn,i and uses various number of vector differences x 1 y − x 2 y . Some of the most prominent differential mutation schemes are summarized here.


29

2.3.2.1 Current Individual pn,i serves as differential mutation base bn,i for vn+1,i. 2.3.2.2 Best Individual pn,best serves as differential mutation base bn,i for vn+1,i. 2.3.2.3 Better The differential mutation base bn,i for vn+1,i is randomly chosen from individuals dominating individual pn,i, i.e., d(bn,i, pn,i)=true where d(bn,i, pn,i) is the logic dominance function [4]. 2.3.2.4 Random The differential mutation base bn,i for vn+1,i is randomly chosen from Pn and is different with pn,i and all donors. 2.3.2.5 Mean The differential mutation base bn,i for vn+1,i is the geometrical center of Pn, i.e.,

x n ,b ,i =

Np

1 Np

∑x

n ,i

(9)

i =1

2.3.2.6 Best of Random The differential mutation base bn,i for vn+1,i is randomly chosen from Pn and dominates all donors which are also randomly chosen from Pn, i.e.,

(

d b n,i , p

n , p1 y

) = true

(

∩ d b n,i , p

n , p2 y

) = true

∀y

(10)

2.3.2.7 Arithmetic Best Arithmetic best is the synonym of target-to-best. 2.3.2.8 Arithmetic Better This scheme differs with arithmetic best by replacing pn,best in (4) with an individual dominating pn,i. A synonym for this scheme is target-to-better. 2.3.2.9 Arithmetic Random Likewise, in this scheme, pn,best in (4) is replaced by an individual randomly chosen from Pn. Its synonym is target-to-random.

30

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2.3.2.10 Trigonometric Trigonometric differential mutation was proposed by Fan and Lampinen in 2003 [54]. Mutant vn+1,i is generated as

∑ ( f (x ) − 3

n , pk

x

n+1, v ,i

=

x

n , p1

+x

n , p2

+x

3

n , p3

+

j =1

(

)

( ))(x

f x

(

n, p j

n, p j

− x n , pk

(

)

)

f x n , p1 + f x n , p2 + f x n , p3

)

k = mod( j ,3) + 1 (11)

where individuals p n , p1 , p n , p2 , and p n , p3 may be chosen from Pn in different ways such as current, best, better, and random as aforementioned. Equivalently, the trigonometric differential mutation can be regarded as a special differential mutation in which the center of p n , p1 , p n , p2 , and p n , p3 acts as differential mutation base and p n , p1 , p n , p2 , and p n , p3 donates equally to vector differences. It can be seen that the intrinsic control parameter F does not show up in trigonometric differential mutation which might be one of its most attractive features. 2.3.2.11 Directed Directed differential mutation was also proposed by Fan and Lampinen in 2003 [55]. Mutant vn+1,i is generated as

( (

)( )

3 ⎡ f x n , p1 ⎤ n , p1 x n +1,v ,i = x n , p1 + ∑ ⎢1 − x − x n , pi n , pi ⎥ x f i =2 ⎣ ⎦

)

(12)

where individual p n , p1 dominates individuals p n , p2 and p n , p3 . It can be seen from (12) that individual p n , p1 serves as both differential mutation base and donor. Similarly, the intrinsic control parameter F does not show up.

2.3.3 Crossover Crossover has been thought unessential for differential evolution [4]. It is even not applied in some differential evolution strategies [3], [56]. However, recent studies hint that its significance in differential evolution might be seriously underestimated [4]. Crossover has been extensively studied in genetic algorithms. Almost all crossover schemes there can be implemented in differential evolution straightforward or after minor adjustment. Besides the exponential crossover mentioned earlier, some of the crossover schemes commonly applied in differential evolution are summarized here. In most evolutionary algorithms, the child cn+1,i is required to be different from its parents. This convention is followed here although it is not absolutely necessary in differential evolution.


31

2.3.3.1 Binary Crossover Binary crossover may be one of the most common crossover schemes in differential evolution. In this scheme, as shown in Fig. 2.4, cn+1,i inherits an optimization parameter from either vn+1,i or pn,i according to the result of a Bernoulli experiment of crossover probability pc, where β is a real random number uniform in [0,1]. do j = 1, N xjn+1,c,i = xjn,i end do L=0 do j = 1, N ȕ = rand(0, 1) if (ȕ pc) then L=L+1 xjn+1,c,i = xjn+1,v,i end if end do if (L = 0) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn+1,v,i else if (L = N) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn,i end if

Fig. 2.4 Fortran-style Pseudo-code of Binomial Crossover

A demonstrative example is shown in Fig. 2.5. cn+1,i inherits parameters x2n+1,v,i, , x7n+1,v,i, and x8n+1,v,i from vn+1,i and x1n,i, x3n,i, x5n,i, x6n,i from pn,i. Therefore, x4 the crossover length is 4. n+1,v,i

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

0

1

0

1

0

0

1

1

28.69

16.22

57.82

68.12

26.99

82.96

99.28

85.86

Bernoulli experiments

xn+1,c,i

Fig. 2.5 Binomial Crossover

32

A. Qing

2.3.3.2 One-Point Crossover One-point crossover randomly selects a single crossover point r (1 0, p(t ) − p(u ) = 0 st ≠ su , μ st su > 0, p(t ) − p(u ) = 1

(2)

otherwise

K is a very large value defined in the configuration files of the network which makes it undesirable to allocate the same or adjacent frequencies to TRXs that are installed in the same sector, st is a sector in which transceiver t is installed, and su is the other sector in which transceiver u is installed, Cc is co-channel interferences and Ca is adjacent-channel interferences, p(t) is the frequency assigned to transceiver t, and p(u) is the frequency assigned to transceiver u. In this paper, we follow the setting in [13], i.e., K=100.000, signaling threshold CSH=6dB, adjacent channel interference rejection CACR =18dB. CSH is the minimum quality signalling threshold, that is, if the carrier-to-interference ratio is lower than this threshold there will be a degradation of the communication quality. On the other hand, CACR is a hardware specific constant that measures the ability of the receiver to receive the wanted signal in the presence of an unwanted signal at an adjacent channel. For more detailed explanation about the mathematical formulation, please refer to [14].

7.2.3 Separation Cost Technical limitations in the construction of sectors and BTSs mean that certain combinations of TRX channel are not permitted. These constraints include (1) Site Channel Separation Any pair of frequencies at a site (BTS) must be separated by a certain fixed amount, typically 2 channels for a large problem. If a BTS uses high power TRXs, its channel separation should be larger. Violation of this separation regulation involves a cost Csite. In our case, this cost is the violated number of site channel separations. (2) Sector Channel Separation This is similar to the previous one, but at sector level. In conclusion, any pair of frequencies at a sector must be separated by a certain fixed amount, typically 3 channels for a large problem. It is important to observe that sector channel separation generally is larger than site channel separation, due to shorter distances

Application of Differential Evolution to a Multi-Objective Real-World FAP

159

between involved TRXs. Similarly, violation of sector channel separation regulation involves a cost Csector. Likewise, this cost is the number of violated sector channel separations. In total, the separation cost is

CS (p ) = Csite (p ) + Csec tor (p )

(3)

7.3 Multi-objective Differential Evolution with Pareto Tournaments Multi-objective differential evolution with Pareto tournaments [15] is based on classic differential evolution [4][5][6][7][8]. It incorporates Pareto tournaments to address multi-objective FAP.

7.3.1 Algorithm Structure The pseudocode for DEPT algorithm is given in Fig. 7.2 where Np is population size. It can be seen that the algorithm structure look almost identical with that of classic differential evolution given in Fig. 2.1 of this book, except that new individual pt+1,i is generated through tournament between ct+1,i and pt,i. t=0 initialize population Pt evaluation population Pt while time limit is not reached do for i = 1, Np create a child ct+1,i evaluate child ct+1,i Pareto tournament between ct+1,i and pt,i to get individual pt+1,i end for t=t+1 end while Fig. 7.2 Pseudocode for DEPT Algorithm

7.3.2 Pareto Tournament Pareto tournament [16] is implemented in DEPT to select individuals that survive to the next generation. An individual has a fitness value that is a scalar value to be minimized. In multi-objective scenario, this fitness value is defined as i i fitness i = N p N isDo min ated + N do min ates

(4)

160

M.S. Maximiano et al.

where NiisDominated is the number of individuals that dominate individual i while Nidominates is the number of individuals that are dominated by individual i. Child ct+1,i joins population Pt to compute NiisDominated and Nidominates, and accordingly fitness, for both pt,i and ct+1,i. A tournament between pt,i and ct+1,i is then implemented based on their fitness value. 1. 2. 3. 4. 5.

The individual with smaller fitness wins. In case of a tie, the winner is determined according to their relevant dominance. If pt,i and ct+1,i continue to tie, the winner will be the one having smaller interference cost. If pt,i and ct+1,i tie again, the winner will be the one having smaller the separation cost. If tie happens again, pt,i survives.

7.3.3 Problem Domain Knowledge Whenever possible, knowledge about the problem domain has to be incorporated to create valid solutions. In our practice, an operation is performed to assure that a frequency is assigned to a TRX installed in a sector only if this frequency does not produce adjacent-channel or co-channel interferences within this same sector. It is triggered when a frequency value is to be replaced by a new value. The set of available frequencies are maintained for every sector, and are continuously updated. At the beginning, valid frequencies are available to all sectors. After that, when a new frequency f is assigned to a TRX, f as well as its respective adjacent frequencies f + 1 and f – 1, is removed from the set of frequencies available to the sector in which the TRX is installed. Meanwhile, the older frequency value reappears in the set.

7.4 Multi-objective Variable Neighborhood Search 7.4.1 Variable Neighborhood Search Variable neighborhood search (VNS) [17]–[18] is a metaheuristic based on trajectories. It starts from an initial solution and performs several changes of neighborhood within a local search. The algorithm increases neighborhood size (k parameter) when the search does not move forward. Following the notations in [17], denote Nk, 1≤k≤Nk, as a finite set of preselected neighborhood structures, and Nk(S) the set of solutions in the kth neighborhood of current solution S. Several neighborhoods Nk can be produced from the solution space of S. VNS gradually applies a mutation operator to generate a new solution S' from S, conducted according to the type of environment used in each iteration Nk(S). If this solution S' improves the current solution S, the next iteration will start again to use the first environment E1. Otherwise, the algorithm will go to the next neighborhood environment.


161

The mutation operator was implemented using specific information from the domain of the problem. It assigns a random frequency value for a TRX from unused frequencies of the sector in which the TRX is installed. The set of available frequencies of the specific sector is updated accordingly.

7.4.2 Multi-objective Variable Neighborhood Search The multi-objective variable neighborhood search (MO-VNS) is developed to solve multi-objective optimization problems by implementing dominance concept in VNS to select better solution. The pseudocode of MO-VNS is given in Fig. 7.3. It starts from a random initial solution. Neighborhood solutions are generated by mutation until no further improvements are possible. It also changes the environment when the obtained solution is worse than the current solution.

ParetoSolutionsĸ0 while time limit is not reached do SĸgenerateSolution(at random) if S dominates any Pareto solution in ParetoSolutions add S into ParetoSolutions remove solutions in ParetoSolutions dominated by S end if kĸ1 while k kmax do SƍĸmutationInEnvironmentk(S) if Sƍ dominates any Pareto solution in ParetoSolutions add Sƍ into ParetoSolutions remove solutions in ParetoSolutions dominated by Sƍ end if if Sƍ dominates S SĸSƍ kĸ1 else kĸ k+1 end if end while end while return paretoSolutions

Fig. 7.3 Pseudocode for Multi-objective Variable Neighborhood Search

162


At the beginning, the set of pareto solutions is empty. It will contain all nondominated solutions that represent the best solutions found so far. The size of the set of pareto solutions is not always the same because it changes along the run. In every iteration, a comparison between existent pareto solutions and the candidate solution is performed to see if it is a better solution.

7.4.3 Greedy Mutation The greedy mutation, as shown in Fig. 7.4, considers all potential neighborhood solutions. At the beginning, the sector in which a TRX is installed and the corresponding unused frequencies are identified. Then, the best solution in terms of dominance is chosen from all possible solutions by testing all frequencies available to that TRX. for TRXid = 0 to allTRXInNetwork do if randomNumber() mutation probability do sectorĸgetSector(TRXid) unusedFreqĸgetUnusedFreq4Sector(sector) populationĸgenerateSolution(unusedFreq) computeDominance(population) bestĸselectBestSolution(population) end if end for return best

Fig. 7.4 Pseudocode for Greedy Mutation

This procedure uses a temporary population that only contains the candidate neighborhood solutions. The individuals inside are eliminated right after the mutation operation.

7.4.4 Multi-objective Skewed Variable Neighborhood Search The multiobjective skewed variable neighbourhood search (MO-SVNS), as shown in Fig. 7.5 where 0≤α≤1.0 is a quality magnitude parameter, implements Pareto dominance in skewed variable neighbourhood search (SVNS) [17]–[18] to solve multi-objective optimization problems. It restarts from another initial solution to explore remote areas when it is lost in a local optimum which may be far from the global optimum. However, unlike MO-VNS, it is more receptive in accepting a new solution, even if it is slightly


163

worse than the previous solution. It also increases the size of neighborhood when the search does not move forward. ParetoSolutionsĸ0 while time limit is not reached do SĸgenerateSolution(at random) if S dominates any Pareto solution in ParetoSolutions add S into ParetoSolutions remove solutions in ParetoSolutions dominated by S end if kĸ1 while k kmax do SƍĸmutationInEnvironmentk(S) if Sƍ dominates any Pareto solution in ParetoSolutions add Sƍ into ParetoSolutions remove solutions in ParetoSolutions dominated by Sƍ end if if Sƍ dominates S+Į*distance(S, Sƍ) SĸSƍ kĸ1 else kĸ k+1 end if end while end while return paretoSolutions

Fig. 7.5 Pseudocode for Multi-objective Skewed Variable Neighborhood Search

The concept behind is to keep the best found solution and quantifys the distance between two distinct solutions S and S′ using a bi-dimensional distance function that represents the two different goals to be minimized. The distance function is the difference between the lower cost value of S and S′ and the highest cost value of all solutions inside the set of Pareto Solutions.

7.5 Experiments and Results 7.5.1 Experimental Setup 7.5.1.1 Used GSM Instances Two different real-world FAP instances currently operating in Seattle and Denver were used. These instances have different sizes among which the Denver instance is larger and therefore most complex to tackle.

164


Fig. 7.6 Topology of the Denver GSM Instance

Fig. 7.7 Topology of the Seattle GSM Instance

The Denver instance has 2612 TRXs, installed in 711 sectors distributed in 334 BTSs. Every TRX has only 18 valid frequencies to be assigned to it. The Seattle instance is smaller. It has 970 TRXs installed in 1180 sectors distributed in 503


165

BTSs. Each TRX has 15 valid frequencies to be assigned to it. The range of valid frequencies may differ from TRXs. Figs. 7.6 and 7.7 show the network topology for each of the instances. In these figures, every triangle represents a sectorized antenna in which the TRXs operate. 7.5.1.2 Encoding A solution is coded as a vector of integers. As an example, Fig. 7.8 shows the representation of a solution for the Denver instance. Each gene corresponds to the frequency value assigned to the TRX. In our experiments scenario, the range of valid frequencies for different TRXs can be different. Therefore, for every TRX it is kept a set of valid frequencies. In terms of memory consumption, store all this information becomes an important overhead.

Fig. 7.8 Solution Encoding

7.5.1.3 Computational Facilities All experiments have been carried out under exactly the same conditions: PC with 2.66Ghz Intel Pentium processor and 3GB RAM running Windows XP operating system. The code was developed under the .NET framework 3.5 with C# language through object-oriented programming (OOP) approach. 7.5.1.4 Termination Conditions and Process Monitoring An execution is performed during 30 minutes. The minimum and average fitness and objective values of all solutions in the Pareto front are saved every 2 minutes. At the end of each run, the final result is also saved. 7.5.1.5 Confidence Building In order to provide results with a statistical confidence, 30 independent runs are implemented for each experiment.

166


7.5.2 Methodology and Metrics Evaluating the performance of multi-objective algorithms is far from being a trivial task since there are multiple solutions instead of a good solution. We have incorporated two complementary measurements besides common statistical comparisons to assess solution quality: hypervolume indicator that defines the volume of the objective space dominated by a Pareto front, and coverage relation [10] to determine best Pareto front. Both are applied to all non-dominated solutions as comparison criterion for assessing Pareto optimal solutions. 7.5.2.1 Hypervolume The hypervolume indicator [9] is truly useful because it rewards convergence towards the Pareto front. It defines the measure of the region simultaneously dominated by every Pareto front point and bounded above by a reference point. In our case, the hypervolume represents the two-dimensional area dominated by the Pareto front. The Pareto front with higher hypervolume value is a better configuration plan for FAP. It is necessary to define a search space by specifying upper bound and lower bound. In our practice, the upper bound is reference point. The upper bound was (300000, 2500) for the Denver instance and (70000, 200) for the Seattle instance. The lower bound points were (130000, 1200) and (4000, 15) respectively. 7.5.2.2 Coverage Relation Coverage relation [9] was used as an additional performance indicator. In terms of dominance, solution x1 covers solution x2, if x1 dominates or ties with x2. Given two sets of non-dominated solutions, we compute the fraction of a set of solutions covered by the other set of solutions. A higher percentage of coverage means a better solution set.

7.5.3 Tuning of the DEPT Parameters DEPT has many variants and involve three intrinsic control parameters. The tuning process aims to find the best variant and its corresponding optimal intrinsic control parameter values. It starts from DE/Rand/2/Bin with CR=0.2 and F=0.1, the best parameters based on a previous work with a mono-objective version of FAP [19][20]. The findings are applied in the final experiments. 10 instead of 30 independent runs are implemented for the tuning process. In addition, we have considered three different time limits, 120, 600, and 1800 seconds. Statistical analysis over minimum, average and standard deviation of the two objective functions and the fitness value has been performed. The tuning is performed in the following order: 1. 2. 3. 4.

population size crossover probability mutation intensity differential evolution strategies


167

7.5.3.1 Population Size Population sizes tested are 10, 25, 50, 75 and 100. Figs. 7.9 and 7.10 show the corresponding Pareto fronts. In general, higher NP value leads to better

Fig. 7.9 Pareto Fronts of Experiments against Population Size Using the Denver Instance

Fig. 7.10 Pareto Fronts of Experiments against Population Size Using the Seattle Instance

168


hypervolume. NP=75 works best for both instances, with 24.8% of hypervolume for the Denver instance and 23.8% for Seattle instance. 7.5.3.2 Crossover Probability The next experiment was focused on crossover probability (CR). The CR parameter normally uses values between [0, 1]. Our first set of values was 0.1, 0.25, 0.5, 0.75, and 0.9. Results are tabulated in Table 7.1 where IC means Interference Cost, SC stands for Separation Cost and HV represents the HyperVolume. We clearly identified that the best solutions were founded using CR of 0.1. A CR=0.1 has higher mean hypervolume and smaller standard deviation value (1.1%).

Table 7.1 Results for Experiments against Crossover Probability (CR)

CR IC Seattle SC HV IC Denver SC HV

0.1

0.25

0.5

0.75

0.9

Best

42306.3

45582.5

47664.6

47516.7

48379.7

Avg

46524.7

48284.3

50018.4

51134.9

51037.7

Std

3344.8

1888.3

1886.2

1975.1

2542.1

Best

64

84

91

97

91

Avg

100.0

108.4

117.0

111.6

116.9

Std

24.2

15.4

17.0

9.4

22.3

(%)

28.3

22.0

18.9

18.0

18.3

Best

229511.2

233629.2

235176.8

236098.1

237040.9

Avg

236070.5

238991.0

240475.4

240831.8

241193.0

Std

6148.3

3854.8

3891.8

3318.2

3535.5

Best

1644

1695

1719

1729

1722

Avg

1750.6

1758.1

1773.7

1784.5

1776.9

Std

78.6

40.0

32.8

37.9

39.2

(%)

26.7

23.9

22.6

22.1

22.0

27.5

22.9

20.8

20.0

20.1

Mean HV (%)

Inspired by the above finding, we further investigate if smaller crossover probability is better. The second set of crossover probability, 0.01, 0.02, 0.03, 0.05, 0.07, and 0.09, is therefore tested. Results are tabulated in Table 7.2. Hypervolume is the highest at 0.01.


169

Table 7.2 Results for Experiments against Crossover Probability (CR) CR Best

0.01 34882.4

0.02 37774.2

0.03 39066.0

0.05 40693.5

0.07 41746.3

0.09 42604.6 46763.2

Avg

37677.0

41320.4

42985.6

45208.6

45989.1

Std

2300.1

2833.5

3112.7

3724.5

3510.9

3589.9

Best

32

36

42

47

55

59

Avg

73.8

76.9

82.3

85.5

91.0

95.1

Std

25.1

25.6

26.7

26.2

25.7

24.6

(%)

46.4

40.9

37.5

33.9

31.1

29.2

Best

226637.1

226734.7

225774.4

226621.7

228018.0

228558.5

Avg

232924.9

233724.5

233053.9

234546.1

236359.9

236690.3

Std

3175.0

4329.7

5646.1

5947.0

6402.0

5624.7

Best

1634

1625

1606

1612

1628

1640

Avg

1713.6

1717.5

1702.3

1712.6

1712.6

1728.2

Std

41.1

56.0

64.3

67.3

58.8

69.6

(%) HV Mean HV (%)

28.3

28.4

29.4

28.8

27.8

27.1

37.3

34.6

33.4

31.4

29.5

28.2

IC Seattle SC HV IC Denver SC

7.5.3.3 Mutation Intensity The tested mutation intensity values are 0.1, 0.25, 0.5, 0.75 and 0.9. Results are shown in Table 7.3. Higher F is more favorable. Table 7.3 Results for Experiments against Mutation Intensity (F) F

IC

Seattle SC

HV

IC

Denver SC

HV

0.1

0.25

0.5

0.75

0.9

Best

34704.0

34621.4

34882.7

34654.1

34311.6

Avg

37737.7

38029.6

38434.7

37772.9

37602.5

Std

2384.5

2603.3

2755.5

2363.1

2452.8

Best

33

28

27

31

29

Avg

72.6

68.2

65.1

73.4

71.7

Std

24.2

25.3

23.7

25.9

26.2

(%)

46.3

47.6

47.5

46.7

47.6

Best

225994.0

226297.3

225867.5

225872.1

225865.6

Avg

232824.8

233180.0

233830.9

232124.7

231808.6

Std

3370.1

3542.1

3768.6

3627.9

3528.9

Best

1628

1609

1605

1611

1615

Avg

1715.4

1697.1

1700.4

1701.4

1699.0

Std

43.0

47.2

47.2

45.9

45.7

(%)

28.6

29.1

29.3

29.3

29.2

37.5

38.3

38.4

38.0

38.4

Mean HV (%)

170


7.5.3.4 DEPT Scheme DEPT schemes differ themselves in terms of base of differential mutation, number of difference vectors, and crossover schemes. Different DEPT scheme may behave differently. The tested DEPT schemes are grouped according to the implemented crossover scheme. Results for DEPT schemes involving binomial crossover are given in Table 7.4 while results for DEPT schemes applying exponential crossover are shown in Table 7.5.

Table 7.4 Results for Experiments against DEPT Schemes Using Binomial Crossover (1: rand/1/bin, 2: rand/2/bin, 3: rand/3/bin, 4: best/1/bin, 5: best/2/bin, 6: best/3/bin, 7: best-torandom/1/bin) DEPT Scheme Seattle

IC

2

3

4

5

6

7

Best

34316.6

34366.2

34929.5

35379.8

34859.8

35135.6

34317.6

Avg

37842.2

37607.5

37933.2

38521.6

38066.1

38215.4

37375.5

Std

2115.5

2447.2

2188.0

2317.5

2360.1

2310.4

2269.5

Best

32

29

33

29

32

31

30

Avg

70.1

71.6

73.6

71.6

73.9

71.5

71.4

Std

24.1

26.1

24.4

25.0

26.0

24.8

26.3

HV

(%)

46.8

47.6

45.9

46.2

46.1

46.2

47.6

IC

Best

227092.5

226454.4

226562.1

226177.4

226618.3

226795.7

226983.0

Avg

233811.1

232203.9

232800.7

232957.6

232110.4

232356.9

233092.7

Std

3791.5

3350.2

3540.5

3826.0

3005.9

3192.4

3471.3

Best

1610

1612

1622

1624

1618

1618

1607

Avg

1704.6

1710.4

1711.7

1702.7

1705.9

1716.0

1705.6

Std

48.1

48.6

47.2

45.9

44.2

48.7

42.2

(%)

28.8

29.0

28.6

28.8

28.8

28.7

28.9

37.8

38.3

37.3

37.5

37.5

37.4

38.2

SC

Denver

1

SC

HV

Mean HV (%)

It is apparent that DEPT schemes using exponential crossover outperform those using binomial crossover. It is further noticed that DE/rand2Best/1/exp is the most profitable for both instances. In terms of differential mutation base, it can be seen from Figs. 7.11-7.14 that random base is more beneficial. These results are much clearer from results by rand/*/bin. However, as shown in Fig. 7.14, in the Seattle instance, DE/RandToBest/1/Exp clearly outperforms all other approaches.


171

Table 7.5 Results for Experiments against DEPT Schemes Using Exponential Crossover (1: rand/1/exp, 2: rand/2/exp, 3: rand/3/exp, 4: best/1/exp, 5: best/2/exp, 6: best/3/exp, 7: best-to-random/1/exp) 1 Seattle

IC

SC

Denver

2

3

4

5

6

7

Best

20034.4

20560.0

19954.5

20400.1

19287.6

20220.7

18123.0

Avg

26083.2

26006.0

26062.3

26383.4

26047.1

26121.7

23897.4

Std

2179.6

2136.4

2350.2

2298.0

2511.8

2371.0

2144.3

Best

25

25

22

27

21

23

27

Avg

71.7

71.2

70.3

73.2

74.0

72.0

70.9

Std

23.6

22.6

23.6

23.3

25.1

24.8

23.7

HV

(%)

67.1

66.9

68.2

65.9

68.7

67.5

69.4

IC

Best

200990.1

202740.2

203580.0

201437.5

201981.7

202988.4

203283.1

Avg

226315.2

228028.3

226350.3

226367.9

226860.6

225976.8

227214.2

Std

7531.0

10216.0

6738.6

6571.7

6359.7

6592.8

6411.9

Best

1427

1455

1438

1433

1448

1443

1444

Avg

1661.2

1586.2

1661.2

1659.3

1654.0

1661.8

1661.4

Std

65.1

73.0

61.3

60.0

62.4

54.9

51.6

(%)

43.9

43.4

42.8

43.6

43.0

42.9

42.7

55.5

55.1

55.5

54.8

55.9

55.2

56.0

SC

HV

Mean HV (%)

DEPT Schemes Using Random Base

DEPT Schemes Using Best Base

Fig. 7.11 Pareto Fronts Obtained Using */*/bin over the Denver Instance

The results obtained when comparing the random schemes against the schemes using the best-so-far approach, do not express a so clear difference between the two approaches. Indeed, in Figs. 7.12 and 7.14 that are showing the Pareto Fronts for the schemes using the exponential crossover it is possible to identify that the differences between the two approaches are smaller, especially for the Denver instance Fig. 7.12.

172




Fig. 7.12 Pareto Fronts Obtained Using DEPT Schemes */*/exp over the Denver Instance



Fig. 7.13 Pareto Fronts Obtained Using DEPT Schemes */*/bin over the Seattle Instance



Fig. 7.14 Pareto Fronts Obtained Using DEPT Schemes */*/exp over the Seattle Instance

7.5.3.5 Findings According to the above tuning, DE/rand-to-best/1/exp is chosen. The corresponding optimal intrinsic control parameter values are: NP=75; CR=0.01; F=0.9.


173

7.5.4 Empirical Results Now, DEPT is compared with MO-VNS [21], MO-SVNS [21], greedy MO-VNS (GMO-VNS), and greedy MO-SVNS (GMO-SVNS) where α=0.4 for MO-SVNS and greedy MO-SVNS. Hypervolumes are tabulated in Table 7.6 while the final Pareto fronts by different algorithms are plotted in Figs. 7.15 and 7.16. DEPT clearly outperforms MO-VNS and MO-SVNS. It is also seen that the greedy mutation in GMO-VNS and GMO-SVNS significantly improves MO-VNS and MO-SVNS. Table 7.6 Hypervolumes Obtained by Different Algorithms

DEPT

MO-VNS

MO-SVNS

GMO-VNS

GMO-SVNS

Denver

72.3

22.4

22.4

75.0

81.1

Seattle

43.4

17.9

18.8

57.5

56.9

Fig. 7.15 Pareto Front for the Denver Instance

Now, let’s have a look at the coverage relation which is shown in Table 7.7. It can be seen that GMO-VNS is 100% better than MO-VNS and GMO-SVNS is 100% better than MO-SVNS. Furthermore, it appears that GMO-SVNS is better than GMO-VNS, which fully agrees with our previous results [21]. Moreover, we see that both GMO-SVNS and GMO-VNS are in general better than DEPT.

174


Fig. 7.16 Pareto Front for the Seattle Instance

Table 7.7 Percentage of Non-dominated Solutions Obtained by Algorithms B Covered by Non-dominated Solutions Obtained by algorithm A

Algorithm A DEPT

MO-VNS

MO-SVNS

GMO-VNS

GMO-SVNS

Algorithm B MO-VNS MO-SVNS GMO-VNS GMO-SVNS DEPT MO-SVNS GMO-VNS GMO-SVNS DEPT MO-VNS GMO-VNS GMO-SVNS DEPT MO-VNS MO-SVNS GMO-SVNS DEPT MO-VNS MO-SVNS GMO-VNS

Denver 100 100 0 0 0 75.0 0 0 0 57.1 0 0 100 100 100 4.2 100 100 100 80.0

Seattle 100 100 0 12.5 0 63.6 0 0 0 55.6 0 0 24.1 100 100 12.5 10.1 100 100 20.0

mean 100 100 0 6.3 0 69.3 0 0 0 56.3 0 0 62.0 100 100 8.3 55.1 100 100 50.0


175

7.6 Conclusions In this paper we have tackled real-world frequency assignment problem for GSM mobile networks through a multi-objective optimization approach. Differential evolution with Pareto tournaments have been proposed to solve the multiobjective optimization problem. Comparative study against MO-VNS, MO-SVNS, GMO-VNS, and GMO-SVNS over two real-world instances has been conducted. Final results show that fine-tuned DEPT outperforms both MO-VNS and MOSVNS while performs worse than both GMO-SVNS and GMO-VNS, among which GMO-SVNS performs best.

Acknowledgement This work was partially funded by the Spanish Ministry of Science and Innovation and FEDER under the contract TIN2008-06491-C04-04 (the M* project). Thanks also to the Polytechnic Institute of Leiria, for the economic support offered to Marisa Maximiano to make this research.

References [1] Eisenblätter, A.: Frequency assignment in GSM networks: Models, heuristics and lower bounds, Ph.D. Thesis, Technische Universität Berlin (June 2001) [2] GSM Association, GSM World, http://www.gsmworld.com/newsroom/market-data/ market_data_summary.htm (last accessed January 2010) [3] Shirazi, S.A.G., Amindavar, H.: Fixed Channel assignment using new dynamic programming approach in cellular radio networks. Computers Electrical Engineering 31(4-5), 303–333 (2005) [4] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (March 1995) [5] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [6] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [7] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [8] Storn, R.: Differential evolution (DE) for continuous function optimization (an algorithm by Kenneth Price and Rainer Storn), http://www.icsi.berkeley.edu/~storn/code.html (last accessed October 23, 2009) [9] Fonseca, C.M., Paquete, L., López-Ibáñez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE Congress Evolutionary Computation, Vancouver, BC, Canada, July 16-21, pp. 1157–1163 (2006)

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[10] Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms - A comparative case study. In: 5th Int. Conf. Parallel Problem Solving Nature, Amsterdam, The Netherlands, September 27-30, pp. 292–304 (1998) [11] Kuurne, A.M.J.: On GSM mobile measurement based interference matrix generation. In: IEEE 55th Vehicular Technology Conf., Birmingham, AL, May 6-9, vol. 4, pp. 1965–1969 (2002) [12] Luna, F., Estébanez, C., León, C., Chaves-González, J.M., Alba, E., Aler, R., Segura, C., Vega-Rodríguez, M.A., Nebro, A.J., Valls, J.M., Miranda, G., Gómez-Pulido, G.A.: Metaheuristics for solving a real-world frequency assignment problem in GSM networks. In: Genetic evolutionary computation Conf., Atlanta, GA, July 12-16, pp. 1579–1586 (2008) [13] Mishra, A.R.: Fundamentals of Cellular Network Planning and Optimisation: 2G/2.5G/3G...Evolution to 4G. John Wiley, New York (2004) [14] Luna, F., Blum, C., Alba, E., Nebro, A.J.: ACO vs EAs for solving a real-world frequency assignment problem in GSM networks. In: Genetic Evolutionary Computation Conf., London, UK, July 7-11, pp. 94–101 (2007) [15] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Parameter analysis for differential evolution with Pareto tournaments in a multiobjective FAP. In: Corchado, E., Yin, H. (eds.) IDEAL 2009. LNCS, vol. 5788, pp. 799–806. Springer, Heidelberg (2009) [16] Weicker, N., Szabo, G., Weicker, K., Widmayer, P.: Evolutionary multiobjective optimization for base station transmitter placement with frequency assignment. IEEE Trans. Evolutionary Computation 7(2), 189–203 (2003) [17] Hansen, P., Mladenovic, N.: Variable neighborhood search: Principles and applications. European J. Operational Research 130(3), 449–467 (2001) [18] Hansen, P., Mladenovic, N., Pérez, J.A.M.: Variable neighbourhood search. Computers Operations Research 24(11), 1097–1100 (1997) [19] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: A hybrid differential evolution algorithm to solve a real-world frequency assignment problem. In: Int. Multiconference Computer Science Information Technology, Polskie Towarzystwo Informatyczne, Wisla, Poland, October 20-22, pp. 201–205 (2008) [20] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Analysis of parameter settings for differential evolution algorithm to solve a realworld frequency assignment problem in GSM networks. In: 2nd Int. Conf. Advanced Engineering Computing Applications Sciences, Valencia, Spain, September 29October 4, pp. 77–82 (2008) [21] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Multiobjective frequency assignment problem using the MO-VNS and MO-SVNS algorithms. In: World Congress Nature Biologically Inspired Computing, Coimbatore, India, December 9-11, pp. 221–226 (2009)

Chapter 8

RNN Based MIMO Channel Prediction Chris Potter

*

Abstract. In this work, differential evolution (DE) is combined with particle swarm optimization (PSO) and another evolutionary algorithm (EA) to create a novel hybrid PSO-EA-DEPSO algorithm. The alteration between PSO, PSO-EA, and DEPSO provides additional diversity to counteract premature convergence. This hybrid algorithm is then shown to outperform PSO, PSO-EA, and DEPSO when applied to wireless MIMO channel prediction.

8.1 Introduction Multiple-input multiple-output (MIMO) wireless communication systems have been shown to provide significant gains in both spectral efficiency and reliability compared to traditional single-input single-output systems [1]-[2]. These results, however, are based on the assumption that the transmitter and/or receiver have perfect knowledge of the channel state information (CSI). One possible alternative is to estimate the channel at the receiver and send the channel state information back to the transmitter. The performance using this approach suffers when the transmitted CSI has become outdated due to channel fluctuation. It may be possible to reduce this effect by sending back a prediction of the CSI. Linear predictors [3] have been used for narrow-band prediction in [4]-[6]. One drawback is their difficulty in estimating the correlation coefficients of the channel in the presence of received data that has undergone non-linear distortions [7]. One solution to this problem are neural networks, which if trained properly, are well suited for non-linearities since they are equipped with arbitrary activation functions. A multi-layer perceptron (MLP) neural network predictor was utilized in [8], while a hybrid network was employed in [9]-[10]. In this work, a recurrent neural network (RNN) is chosen for narrow-band channel prediction. Their advantage over feed-forward neural network predictors Chris Potter Dynetics Inc e-mail: [email protected] *

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 177–206. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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is the potential to learn temporal statistics from previous neural network outputs. In [7], a RNN was trained online by the extended Kalman filter and real-time recurrent learning algorithms for narrow-band channel prediction. A disadvantage of these online training algorithms is the necessity of an accurate measurement of the channel prediction error (which requires full CSI knowledge). In contrast, a RNN trained off-line by PSO-EA was utilized in [11] for time series prediction. All of the prediction techniques mentioned up to this point have been for SISO systems. In this work, the main focus will be on wireless systems with multiple antennas at both the transmitter and receiver. Although particle swarm optimization (PSO) [12]-[13] utilizes the personal and global bests of the swarm when updating the particles, stagnation around local extremum can occur. One solution to this problem is to combine PSO with other evolutionary algorithms (EA) such as differential evolution (DE) [14]-[18]. This in theory will provide additional diversity to the swarm through mutation, crossover, and selection. Examples of these hybrid algorithms include PSO-EA [11] and DEPSO [19]. In this work, a new hybrid PSO-EA-DEPSO training algorithm is proposed. This algorithm is shown to outperform PSO, PSO-EA, and DEPSO as well as the Levinson-Durbin algorithm with respect to mean squared error (MSE) when applied to the problem of multiple-input multiple-output (MIMO) wireless channel prediction. The robustness of this new algorithm is illustrated by varying the temporal and spatial correlation of the MIMO channel after the weights have been trained. To analysis how the MIMO RNN predictor trained by the new PSO-EADEPSO algorithm impacts the performance on the wireless link, new expressions for the received SNR, array gain, and probability of error for a MIMO beamforming system when the transmitter and receiver only have access to a channel prediction are derived. This work abstains from the common assumptions that the channel prediction error is Gaussian and/or independent of the true CSI. An analysis of these expressions reveal that the array gain decreases with SNR and is larger for spatially correlated channels. Also, the diversity gain is shown to match the perfect CSI case up until the channel prediction error saturates the received signal. The reduction of the average probability of error before saturation is strictly due to coding gain.

8.2 Received Signal Model 8.2.1 Received Signal Model A MIMO wireless flat fading baseband communication system with Nt transmitting antennas and Nr receiving antennas is modeled at discrete time k by

y (k ) = H (k ) ⋅ x(k ) + n(k )

(1)

RNN Based MIMO Channel Prediction

179

where y(k) is an Nr-dimensional vector containing received signals, H(k)={hmn(k)} is an Nr× Nt channel matrix describing the complex channel gain between the mth receiving antenna and nth transmitting antenna, x(k) is an Nt-dimensional vector containing transmitted symbol xi(k), 1≤i≤ Nt, of constellation C, and n(k) is an Nrdimensional vector of complex white noise with mean zero and variance variance No, i.e.,

ni ~ CN(0, N o ) 1 ≤ i ≤ N r iid

(2)

Denoting Pt and T respectively as the total transmitted power and symbol period, the transmitted symbols must satisfy

E x(k ) 2 ≤ PtT 2

where E(·) and

(3)

2

⋅ 2 denote the expectation operator and two norm, respectively.

It is assumed the MIMO channel can induce non-linear distortions such as low noise amplification and non-linear scattering on the transmitted signal. These distortions are lumped into the mapping f: C → C where C is the set of complex numbers. For virtually any wireless communications application, the received signal must remain finite and thus it is assumed that f is bounded on C.

8.2.2 Optimization Problem The purpose here is to predict the channel state information (CSI) that minimizes the mean squared error between the predicted and measured channel. This can be cast into an minimization problem by defining the following objective function

C (k ) = E hˆmn (k ) − hmn (k ) where

[

2

1 ≤ m ≤ N r ,1 ≤ n ≤ N t

(4)

]

hˆmn (k ) = F hˆmn (k ),K, hˆmn (k − N p ) is the prediction, hˆmn (k − 1) , …,

hˆmn (k − N p ) are the Np most recent estimates, and F maps the estimates to the

prediction. Since it was only required that f be bounded, the possibility that the derivative of f may not exist requires the use of training algorithms that do not rely on gradient information.

8.3 Hybrid PSO-ES-DEPSO Training Algorithm In this work, a new algorithm is proposed that is a hybrid version of PSO, EA, and DEPSO. The block diagram is displayed in Fig. 8.1.

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Fig. 8.1 Hybrid PSO-EA-DEPSO Training Algorithm

The motivation behind the algorithm is to alternate between PSO, EA, and DEPSO to continually provide diversity for the particles/parents. PSO is implemented for one fourth of the total iterations to converge quickly on the potential solution. PSO-EA and DEPSO then alternate for the remaining iterations to prevent premature convergence, since DEPSO can be more favorable in dense solution spaces where a small change in the personal best could lead the swarm to a lower fitness solution, while PSO-EA may be desirable in sparse solution spaces when the swarm needs a “nudge” that is independent of any personal/global bests to migrate from a local minimum in search of solutions with better fitness.

8.4 MIMO Channel/Beam-Forming Models 8.4.1 Channel Model The MIMO sub-channels are represented by [20]

[

]

I Q (k ) + jg mn (k ) 1 ≤ m ≤ N r ,1 ≤ n ≤ N t g mn (k ) = f g mn

(5)

where f, as mentioned previously, is arbitrary but bounded on C, M

I (k ) = 2 M ∑ cos[φn + 2πf d kTs cosα n ] g mn n =1

(6)


181

is the in-phase component, M

Q (k ) = 2 M ∑ cos[ψ n + 2πf d kTs sin α n ] g mn

(7)

n =1

is the quadrature component, Ts is the sampling period, fd is the maximum doppler frequency, and

αn =

2πn −π +θ 4M

(8)

øn, ψn and θ are U[-π, π], where U[a, b] is a uniform random variable between a and b. The system is assumed to be operating in an urban environment in which the receiver is spatially uncorrelated due to scattering from adjacent objects. The overall channel is thus modeled by

H (k ) = G (k ) ⋅ ΦTX (k )

(9)

where

ΦTX =

∑ E[h( ) (k )h( ) (k )] Nr

1 Nr

r =1

H r;

(10)

r;

h(r;) denotes the rth row of H(k), the superscript H stands for Hermitian operator. These correlation matrices are dependent on many parameters, including the angle of arrival, transmitting and receiving antenna distances, and angular spread [21]. Since the focus of this work will not be emphasized on any particular antenna geometry, spatial generality will be maintained and a Toeplitz structure will be considered

ΦTX

⎡ 1 ⎢ ⎢ γt = ⎢ γ t4 ⎢ ⎢ M ⎢γ ( N t −1)2 ⎣ t

γ t4 L γ t( N −1) M 1 γt O γt 1 O γ t4 O O O γt L γ t4 γ t 1 γt

t

2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(11)

where γt is the transmit spatial correlation factor. It should be noted that the simplicity of this model does not prevent accurate approximations for a variety of antenna configurations [22]. Stacking the columns of H(k) into the Nt Nr-dimensional column vector vec[H(k)], it follows that

vec[H (k )] ~ CN (0, Ψ )

(12)

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where

Ψ = ΦTX ⊗ Φ RX

(13)

⊗ denoting the kronecker product [23]. This form will prove useful when deriving the array and diversity gains in Section 8.

8.4.2 Channel Estimation Model For a meaningful prediction, it is necessary for the RNN to learn the statistics of the fading process. This can be accomplished by supplying the RNN with previously obtained channel estimates. Following the same procedure as those in [24]-[26], the channel is written in terms of its minimum mean squared error

H (k ) by

(MMSE) estimate

H (k ) = H (k ) + W (k ) where

(14)

[

]

H (k ) and W(k) are uncorrelated with hmn (k ) ~ CN 0,σ h2 (k ) and

[

]

wmn (k ) ~ CN 0,σ w2 (k ) . The channel estimation error between the mth receiver and nth transmitter is thus

σ w2 (k ) = σ h2 (k ) − σ h2 (k )

(15)

8.4.3 MIMO Beam-Forming When channel state information is available at both the transmitter and receiver, spatial diversity can be achieved. Suppose that the transmitter sends the symbol x(k) across all antennas. To maintain equal transmit power with respect to the SISO case, it is required that

E x(k ) = PtT = Es 2

where Es is the average symbol energy [27]. x (k ) = Next, the symbol is pre-processed by ~ H is then post-processed by u (k) to yield

(16)

v (k )x(k ) . The received symbol

~ y (k ) = u H (k ) ⋅ y (k ) = u H (k ) ⋅ H(k ) ⋅ ~ x (k ) + u H (k ) ⋅ n(k )

(17)

The average received energy

E~ y (k ) = Es u H (k ) ⋅ H (k ) ⋅ v (k ) + σ n2 2

2

(18)

is now maximized with respect to u(k) and v(k), subject to the constraints

u(k ) 2 = v(k ) 2 = 1 . 2

2


183

Defining the Lagrangian as

[

]

[

]

Λ(u, v ) = u H (k ) ⋅ H(k ) ⋅ v(k ) − λ1 (k ) u(k ) 2 − 1 − λ2 (k ) v(k ) 2 − 1 (19) 2

2

2

where λ1(k), λ2(k) ∈ C, and taking the derivative yields

∇u * ( k )Λ(u, v ) = H (k ) ⋅ v (k ) ⋅ v H (k ) ⋅ H H (k ) ⋅ u(k ) − λ1 (k )u(k )

(20)

∇ λ* (k )Λ(u, v ) = u H (k ) ⋅ u(k ) − 1

(21)

∇ v * (k )Λ(u, v ) = H H (k ) ⋅ u(k ) ⋅ u H (k ) ⋅ H (k ) ⋅ v(k ) − λ2 (k )v(k )

(22)

∇ λ* (k )Λ (u, v ) = v H (k ) ⋅ v(k ) − 1

(23)

1

2

Setting these to zero, it follows immediately that

H(k ) ⋅ v(k ) =

λ1 (k )u(k )

(24)

v (k ) ⋅ H H (k ) ⋅ u(k ) H

u H (k ) ⋅ u(k ) = 1 H H (k ) ⋅ u(k ) =

(25)

λ2 (k )v(k )

(26)

u (k ) ⋅ H (k ) ⋅ v (k ) H

v H (k ) ⋅ v (k ) = 1

(27) H

H

Multiplying the left of Eqs. (24) and (26) by u (k) and v (k), respectively, yields

λ1 (k ) = v H (k ) ⋅ H H (k ) ⋅ u(k ) ⋅ u H (k ) ⋅ H(k ) ⋅ v(k ) = λ2 (k )

(28)

Inserting λ1(k) and λ2(k) into Eqs. (24) and (26) yields

H(k ) ⋅ v(k ) = u H (k ) ⋅ H(k ) ⋅ v(k )u(k ) = σ (k )u(k )

(29)

H H (k ) ⋅ u(k ) = v H (k ) ⋅ H H (k ) ⋅ u(k )v(k ) = σ (k )v(k )

where

σ (k ) = λ1 (k ) = λ2 (k )

This shows that Eq. (10) is maximized when u(k) and v(k) are chosen to be the left and right singular vectors, respectively, that correspond to the maximum singular value. The received symbol when u(k)= u1(k) and v(k)= v1(k), the left and right singular vectors corresponding to the maximum singular value σ H ( k ) , 1

respectively, is

u1H (k ) ⋅ y (k ) = σ H 1 (k )x(k ) + u1H (k ) ⋅ n(k )

(30)

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This signal processing technique is known throughout the literature as MIMO beam-forming or dominant eigenmode transmission. Letting one can easily show that

n~(k ) = u1H (k ) ⋅ n(k ) ,

E n~(k ) = N o 2

(31)

and remains white. The average received energy is

E~ y (k ) = Esσ H2 1 (k ) + σ n2 2

(32)

When the transmitter and receiver only have a prediction of the channel matrix

ˆ (k ) = U ˆ (k ) ⋅ D ˆ (k ) ⋅ V ˆ H (k ) , the received symbol is H

ˆ (k ) ⋅ D ˆ (k ) ⋅ V ˆ H (k ) ⋅ vˆ (k )x(k ) + u H (k ) ⋅ n (k ) (33) uˆ 1H (k ) ⋅ y (k ) = u1H (k ) ⋅ U 1 1 To supply the transmitter with the dominant right singular vector for preprocessing, a feedback link must be established. Let td be defined as the total delay arising from processing and transmission latencies. A constraint on this delay to keep the CSI from becoming stale is [28]

td = 0.423 f d

(34)

where fd is the maximum doppler frequency. This seems discouraging, since for a maximum doppler frequency of 100 Hz, the total delay must be less than 4.23 ms. However, with the aid of accurate doppler estimation [29]-[30], this delay constraint can be relaxed. For example, assuming a worst case approximation error of 0.5 Hz, the maximum delay increases to 846 ms, a tolerable value for most wireless latencies [31].

8.5 Recurrent Neural Network for Channel Prediction Recurrent neural networks are fundamentally different from their feedforward counterparts in that they possess an internal state of what has previously been been processed by the network. This state gives the network the ability to better describe data possessing temporal correlations than feed-forward networks. An analogous statement is that a feed-forward neural network, by the universal approximation theorem, can represent any function defined on a fixed input space; however, a recurrent neural network can utilize its internal state to represent data that is temporally correlated over potentially unbounded input spaces. The recurrent neural network used for prediction is shown in Fig. 8.2.


185

z−1

• •

z−1 d1(k−1)

d1(k)

+ •

dϑ(k−1)

•

• •

+

s(k )

dϑ(k)

•

s(k − N p )

•

A Fig. 8.2 Recurrent Neural Network Channel Predictor

The output of the activation functions are

⎤ ⎡ N p +ϑ d j (k ) = φ j ⎢ ∑ a ji ri (k )⎥ 1 ≤ j ≤ ϑ ⎦⎥ ⎣⎢ i =1 where r(k) =[d1(k-1), …,

(35)

dϑ (k − 1) , s(k), …, s(k-Np)]T is the RNN input and

øj(x)=tanh(x), 1≤j≤ ϑ , are the non-linear activation functions. The RNN predictor is implemented in two stages. During the first stage, d1(k-1), … dϑ (k − 1) =0. The outputs are then fed back to form r(k). After the second stage, the RNN prediction

hˆmn (k + 1) is set equal to d1(k). This process is

repeated for all NrNt channel coefficients. Unless otherwise stated, Np=5 and ϑ =2 is chosen. This yields Nw= ϑ (Np+ ϑ )=14 neural network weights which are used to construct the 2×7 matrix A={aji}.

8.6 Training Procedure The RNN in Fig. 8.2 is trained off-line with data generated from Eq. (10). To be specific, let Ntrain denote the number of training samples. Then for each 0≤k≤Ntrain, the training data can be generated using the aforementioned channel models in following sequence of steps. • •

Choose the number of transmit antennas, Nt, and receive antennas, Nr. Choose the spatial correlation factor γt.

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Use γt and Nt to generate ФTX. Choose fd and Ts to generate at discrete time k, the sub-channels gmnI(k) and gmnQ(k) for 0≤m≤Nr, 0≤n≤Nt , in Eqs. (6) and (7), respectively. Perform the non-linear mapping induced by f to obtain gmn(k) in Eq. (5) and form the matrix G(k)={gmn(k)}. Calculate H(k) using Eq. (9).

For each sub-channel, a RNN whose dynamics are described in the previous section, is trained with s(k-i)=h(k-i), 0≤i≤Np-1 on iteration η with fitness function

C (η ) =

∑ [h (k ) − hˆ (k ,η )]

N train 1 N train

A mn

A mn

2

(36)

k =1

where A={I, Q}. For each iteration, P sets of RNN weights are updated by a training algorithm chosen by the user and are given fitness values determined by C(η). When a specified number of iterations have occurred or the fitness function has reached a desired value, the RNN weights are frozen and brought online. At this point, the RNN does not know the channel coefficients and must instead use the estimates for channel prediction. This is done by setting s(k-i)= h (k-i), 0≤i≤Np-1. This method of training the RNN offline is less restrictive than online training [7], [32]-[33], where it is assumed that the instantaneous value of the error (and hence full channel knowledge) is known at the receiver. In addition, a more robust training method than that in [34] is proposed in this study, where the RNN weights had to be retrained whenever the channel changed. Unless otherwise specified, the parameters used for training are tabulated in Table 8.1. The value of Nw was given in Section 4 and is used to calculate τ. The remaining parameters were found on a trial and error basis and generally speaking are dependent on the optimization problem at hand. Table 8.1 Parameter Values for Training Algorithms

parameter

value

description

Nw

14

Number of RNN weights

Vmax

2

Maximum PSO velocity

Xmax

4

Maximum PSO position

w

0.8

PSO Inertia Weight

c1

1

PSO Cognitive Weight

c2

1.5

PSO Social Weight

P

40

Number of PSO Particles

pc

0.5

Crossover Probability for DEPSO

L

7

DEPSO parameter

τ

0.3265

EA Parameter


187

8.7 Numerical Results At this point, it is illustrative to present a few examples. First, the new PSO-EADEPSO algorithm is compared with PSO, PSO-EA, and DEPSO algorithms. In the second example, the robustness of the training algorithm is investigated by varying fdTs, the normalized doppler frequency. In the third and final example, a RNN is compared to a linear predictor using the PSO-EA-DEPSO algorithm.

8.7.1 Algorithm Comparison To provide a comparison of the training algorithms described in the previous section, a 2×2 (Nr×Nt) spatially uncorrelated MIMO channel with f(x)=x, Ntrain=100, γt=0, σw2(k)=0.001, and normalized doppler frequency fdTs=0.05 is predicted using PSO, DEPSO, PSO-EA, and the new hybrid PSO-EA-DEPSO algorithm. For each iteration and training algorithm, the fitness values corresponding to the global best of h11(k) for 50 random trials are collected. The fitness values are plotted for the I and Q components in Figs. 8.3 and 8.4 respectively.

Fig. 8.3 Mean squared error of in-phase component when fdTs=0.05

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C. Potter

Fig. 8.4 Mean squared error of quadrature component when fdTs=0.05

For the first 50 iterations, the proposed hybrid algorithm is operating in PSO mode and thus the performance resembles that of PSO. For the remaining iterations, the hybrid algorithm uses its diversity obtained by alternating between DEPSO and PSO-EA to outperform the competition.

8.7.2 Robustness of PSO-ES-DEPSO Algorithm Although accurate training of the weights is important, the robustness to different channel conditions is also critical for the RNN predictor. To investigate this, the weights are trained with the new hybrid PSO-EA-DEPSO training algorithm when f(x)=x, Ntrain=100, γt=0, σw2(k)=0.01, and normalized doppler frequency fdTs=0.1 are brought online to predict a channel with the same parameters except a time varying fdTs. The in-phase (I) and quadrature (Q) channel coefficients along with their predictions are illustrated in Figs. 8.5 and 8.6 respectively.


189

Fig. 8.5 In-phase channel coefficients for varying fdTs

Fig. 8.6 Quadrature channel coefficients for varying fdTs

Next, the effect of spatial correlation is investigated by training the weights with f(x)=x, Ntrain=100, γt=0, σw2(k)=0.01, normalized doppler frequency fdTs=0.1 and bringing them online for channel prediction with the same parameters except γt=0.7. The I and Q components are plotted in Figs. 8.7 and 8.8 respectively. The

190

C. Potter

accuracy of these predictions verify the robustness of the RNN equipped with the new PSO-EA-DEPSO training algorithm.

Fig. 8.7 In-phase channel coefficients for γt=0.7

Fig. 8.8 Quadrature channel coefficients for γt=0.7


191

8.7.3 Linear and Nonlinear Predictors with PSO-EA-DEPSO Algorithm The strength of RNNs and neural networks in general is their ability to describe non-linearities. These can be present in the transmitter/receiver hardware due to devices such as low noise amplifiers (LNAs) and automatic gain controllers (AGCs), or in the wireless environment when non-linear scatterers distort the electromagnetic signal. A byproduct of these non-linearities is that the signal can become non-stationary, which greatly complicates the prediction problem. To exemplify this, a RNN using the new hybrid algorithm is compared to a fifth order feed-forward linear predictor using the Levinson-Durbin Algorithm [35] for f(x)=exp(-x), Ntrain=100, γt=0, fdTs=0.5, and σw2(k)=0.01. Observing Figs. 8.9 and 8.10, the RNN outperforms the linear predictor. This example suggests the RNN trained with the PSO-EA-DEPSO algorithm is capable of predicting certain non-linear, non-stationary channels better than the Levinson-Durbin linear predictor.

Fig. 8.9 In-phase channel coefficients for non-linear channel

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C. Potter

Fig. 8.10 Quadrature channel coefficients for non-linear channel

8.7.4 Non-convexity of the Solution Space Now that the performance and robustness of the new hybrid algorithm has been established, the (non-)convexity of the optimization problem is investigated. For ten random trials, the RNN weights obtained during training and the corresponding fitness values are displayed in Table 8.2 when f(x)=x, Ntrain=100, γt=0, σw2(k)=0.01, and fdTs=0.1. Table 8.2 RNN weights trained by the new PSO-EA-DEPSO algorithm and their corresponding fitness values for ten random trials fitness

weights

1

2

3

4

5

6

7

8

9

10

0.0199

0.0249

0.0062

0.0744

0.0246

0.0529

0.0051

0.0103

0.0433

0.0057

-0.1852

-0.2938

0.0988

-2.3740

-1.3550

-0.7193

-0.0915

-2.9857

-3.0606

-0.9186

-0.0481

-0.0695

0.5724

-0.5708

-0.8356

-0.3019

0.2574

2.5382

-0.2169

1.6185

-0.2177

-0.6067

-0.2857

-0.9908

1.9431

1.3150

-0.9999

1.9331

-0.8632

0.9503

1.5102

2.4664

1.6808

-0.2108

-3.0000

-2.2981

3.0000

-3.0000

2.1963

-0.2512

-2.1005

-3.0403

-2.9904

2.8505

-0.7334

0.6472

-3.0000

-1.0616

-2.9998

-3.0000

-0.3749

0.1797

0.7507

-2.2392

2.2325

-0.7102

-0.5636

3.0000

3.0002

2.5535

2.5595

2.3267

2.3462

-0.3983

1.0659

2.3951

3.0000

0.5107

-1.8604

0.9262

5.8845

3.0412

-3.0000

-1.4910

2.2044

3.0129

-1.7209

-3.0000

-2.2399

-1.5449

2.1256

-2.8989

-3.0000

2.6533

3.0000

3.0463

-3.0000

3.0000

3.0015

-3.0000

2.4032

-2.2344

-1.6517

3.0411

-0.4324

-0.7710

-0.6671

1.1447

1.2529

-1.0727

-6.0000

2.7577

3.0000

-4.0465

-0.5025

-0.3501

1.0829

-1.6065

-0.2251

3.0000

0.4291

-1.7064

0.6771

0.9490

2.1018

1.8101

2.0453

-0.7452

-0.2497

-2.6953

3.0298

3.0116

-3.0000

1.7369

-0.0355

0.1754

-3.0000

0.9632

-3.0001

0.0360

3.0480

-3.0000

-0.1994

-1.1739

-2.9869

-3.0000

-0.7356

1.7151

2.1385

1.2633


193

A quick inspection of the RNN weights indicate for each run, the swarm has found a distinct local minimum, which suggests the optimization problem is nonconvex. The fact that each run yields a different local minimum also demonstrates the difficulty of this optimization problem. The proposed hybrid algorithm in this scenario seeks to find the “best” local minimum by exploring its diversity through alternation of the PSO-EA and DEPSO algorithms.

8.8 Performance Measures of RNN Predictors Now that good training performance has been established, the impact of prediction error at the receiver is investigated. Up to this point it has been a common assumption in the literature to assume that the prediction error is Gaussian and/or independent of the true CSI. A contribution of this work is to derive new performance measures that do not rely on these assumptions. The approach taken will start with a new approximation for the array gain followed by a new upper bound on the probability of error. This bound will show the effect of prediction error on the diversity and coding gains before saturation. These two expressions also relate known parameters (e.g. number of antennas, spatial correlation, SNR, etc.) with performance and hence aid in the analysis. This is followed by new tight approximations for the array gain and probability of error that are based on parameter estimation. All of these measures are dependent on the received SNR, which will be the starting point of the derivation. With the aid of Eqs. (16) and (30), it follows that the instantaneous received SNR for the MIMO RNN beam-former when the receiver has perfect CSI is

ρ bf (k ) = ρσ H2 (k )

(37)

1

where

ρ=Es/No

is

the

SNR

of

a

SISO

AWGN

channel.

Writing

vˆ 1 (k ) = v1 (k ) + Δv1 (k ) and uˆ 1 (k ) = u1 (k ) + Δu1 (k ) the received SNR for

the MIMO RNN predictor is

ρbf (k ) = σ H2 (k ) [β (k ) + ρ −1 ]

(38)

1

where

β (k ) = σ H (k )v1H (k ) ⋅ Δv1 (k ) + σ H (k )Δu1H (k ) ⋅ u1 (k ) + Δu1H (k ) ⋅ H (k ) ⋅ Δv1 (k ) (39) 2

1

1

represents the effective noise due to channel prediction error.

Remark 8.1. When perfect CSI is available at the transmitter and receiver, β=0 and Eq. (38) agrees with Eq. (5.48) in [36]. The correlation between

σ H2

1

(k ) and β(k) requires their joint probability density

function (pdf) to calculate exact expressions for the array gain and probability of error. The approach taken in this work will be the derivation of accurate approximations that conserve this dependence. Before proceeding to these results, however, it is first noted in Eq. (39) that for small prediction errors

σ H2

1

(k ) and

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C. Potter

β(k) are approximately uncorrelated. This observation leads to a new closed form approximation for the array gain in the low prediction error regime.

Theorem 8.2. The array gain for a MIMO RNN beam-former with low channel prediction error is approximated by.

[

η = E[ρbf (k )] = Nt N r α +

(

)

Nt + N r 2 3 N t N r +1

](1 − α ) (

μ β ( k ) + ρ −1

(μ β (

)

2

+ 2σ β2 ( k ) ( k )

k) +ρ

)

−1 3

(40)

where Nt

α=

∑ λΨ j − N t j =1

(41)

N t ( N r −1)

Proof. Representing Eq. (38) by its Taylor series expansion about the means

[

η = μσ

2 H1 ( k )

]

, μβ (k ) it follows that [15]

[

]

E ρbf (k ) = E{12 [γ 1 (k ) + γ 2 (k ) + γ 3 (k )]} where

[

γ 1 (k ) = σ H2

[

γ 2 (k ) = 2 σ H2

1 (k )

1 (k )

− μσ 2

H1 ( k )

]

2 ∂ 2 ρ bf (k )

- μσ 2

H1 ( k )

][β (k ) − μ

[

∂ σ H2 1 ( k )

(43)

]

2

]

∂ 2 ρ bf ( k ) β ( k ) ∂σ 2 H1 ( k ) ∂β ( k )

γ 3 (k ) = [β (k ) − μ β (k ) ]2 ∂ [β (k )]

(42)

∂ 2 ρ bf ( k )

(44) (45)

2

Computing these derivatives yields

γ 1 (k ) = 0 γ 2 (k ) = −

(46)

⎤ ⎡ 2 ⎢σ H2 1 ( k ) − μ 2 ⎥ β ( k )− μ β ( k ) σ H1 ( k ) ⎦ ⎥ ⎣⎢

[

] (47)

[μ β ( ) + ρ ]

−1 2

k

γ 3 (k ) = [β (k ) − μ β ( k ) ]2

2 σ H2 1 ( k )

[μ β (

k)

+ ρ −1

]

(48)

3

Inserting Eqs. (46)-(48) into Eq. (42) and taking the expectation yields

[

]

E ρ bf (k ) =

μ

σ2

H1 ( k )

μ β (k ) + ρ

−1

−

[

cov σ H2 1 ( k ) , β ( k )

[μ β ( ) + ρ ] k

−1 2

]

+

σ β2 ( k ) μ

σ2

H1 ( k ) −1 3

[μ β ( ) + ρ ] k

(49)


195

σ H2

Assuming the prediction error is low enough that

1

( k ) and β(k) are

approximately uncorrelated, it immediately follows that

[

]

E ρbf (k ) = μσ 2 The pdf of

σ H2 (k ) 1

H1 ( k )

[μ β ( ) + ρ ] + 2σ β ( ) [μ β ( ) + ρ ] −1 2

k

2

k

−1 3

k

(50)

was extended from the diversity equation of correlated

branches in a SIMO system [38] to MIMO systems in [39] with mean

μσ

2 H1 ( k )

[

= Nt N r α +

(

)

Nt + Nr 2 3 N t N r +1

](1 - α )

(51)

where Nt

α=

∑ λΨ j − N t j =1

(52)

N t ( N r −1)

which was to be proven.

Remark 8.3. As the accuracy of the prediction improves, which η→ μσ 2

H1 ( k )

μ β ( k ) , σ β2 (k ) →0, from

, which is in agreement with [36].

The probability of error was approximated for SISO systems in [27] by

⎡ Pe (k ) = LQ ⎢ ⎣

ρ h ( k ) d m2 2 2

⎤ ⎥ ⎦

(53)

where Q(·) is the Gaussian Q function [40] while L and dm are respectively the number of the nearest neighbors and minimum Euclidean distance in the normalized constellation. Inserting Eq. (38) into Eq. (53), the average probability of error for a MIMO beam-forming system is

⎧ ⎡ E[Pe (k )] = LE ⎨Q ⎢ ⎩ ⎣

0.5σ H2 1 ( k ) d m2

β ( k )+ ρ

−1

⎤⎫ ⎥⎦ ⎬ ⎭

(54)

Observing this expression, the average probability will behave similar to the perfect CSI case until the noise floor becomes saturated by β(k). This is mathematically justified with the following result.

Theorem 8.4. The average probability of error before saturation due to channel prediction error for a MIMO RNN predictor has diversity gain and coding gain with probability p described by

Gd = N t r [Φ RX (k )]

(55)

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C. Potter

Gc = where

d m2

{

δ L det [Φ RX ( k )]N r

(56)

}

1 Gd

δ = 4[ρβ p min (N t , N r ) + 1]

(57)

and βp(k) represents the 100pth percentile of β(k).

Proof. Noting first that

H (k ) F = 2

r [H i (k )]

∑σ i =0

2 H i (k )

≤ σ H2 1 (k )r [H (k )] ≤ σ H2 1 (k ) min( Nt , N r )

(58)

The beam-forming SNR is lower-bounded with probability p by

ρbf (k ) ≥

ρ H (k ) F ρ min ( N t , N r )β p +1 2

(59)

Substituting Eq. (59) into Eq. (53) and applying the Chernoff bound [40], the probability of error is bounded by

Pe (k ) ≤ L exp⎛⎜ − ⎝

ρ H (k ) δ

Thus

d m2

⎞ ⎟ ⎠

(60)

( )

E[Pe (k )] ≤ LM H (k ) 2 − F

2 F

ρd m2 δ

(61)

where Mx(s)=E(esx) is the moment generating function (MGF). This has been computed in [36] as r (Ψ )

M x (s ) = ∏ 1+ s1λΨ

(62)

i

i =1

Inserting this into Eq. (54) yields r (Ψ )

E[Pe (k )] ≤ L∏

i =1 1+

1 2 ρd m δ

(63)

λ Ψi

Applying the approximation (1+x)-1≈x-1 for large x, the average probability of error can be written as

E[Pe (k )] ≤ L =L

[( )] ρd m2 δ

[( )] ρd m2 δ

− r (Φ TX ) N r

− r (Ψ ) r (Ψ )

∏λ i =1

−1 Ψi

=L

det[ΦTX (k )]

−Nr

[( )] ρd m2 δ

− r (Ψ )

det[Φ RX (k ) ⊗ ΦTX (k )]

⎧⎪ ρd m2 =⎨ ⎪⎩ δ L det (ΦTX )− N r

[

−1

⎫⎪ 1 [r (Φ TX ) N r ] ⎬ ⎪⎭

]

− r (Φ TX ) N r

(64)


197

which implies that for small prediction errors (i.e. before the noise floor due to channel prediction error saturates the signal)

Gc =

{

δ [L det (Φ σX

d m2

)]− N r }

1 [ N r r ( ΦσX

Gd = N t r (Φ RX )

)]

(65) (66)



the desired result.

Remark 8.5. If only μβ(k) and σ β(k) are available, the Bienaymé-Chebyshev inequality can be employed to Theorem 8.4 to obtain with probability p=1-N-2

β (k ) ≤ Nσ β (k ) + μ β (k )

(67)

Theorem 8.4 suggests that until saturation, the probability of error will maintain the same diversity order as the perfect CSI case. The penalty instilled by the prediction error is a loss in coding gain. Before illustrating this, new tight approximations for the average probability of error and array gain are derived. Noting through Figs. 8.11 and 8.12 that the cumulative density function (c.d.f.) of ρbf0.5(k) is accurately fitted by a gamma distribution with scale parameter θ and shape parameter κ, the probability of error is accurately approximated by the following result.

Fig. 8.11 Comparison of simulated and fitted c.d.f.'s for a MIMO spatially uncorrelated channel

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C. Potter

Fig. 8.12 c.d.f.'s for a MIMO spatially correlated ( γ t

= 0.7 ) channel

Theorem 8.6. The average probability of error for a MIMO RNN predictor before saturation is tightly approximated by

E[Pe (k )] ≈ 1.135 where

Τ1 =1 F1

(

L 4π θ κ Γ (k )d m

( ) [ d m2 4

) (

; ; ( d 1θ )2 −1 F1

k −1 1 2 2

(

m

−κ 2

dm 2

Γ ( k 2−1 )Τ1 + Γ( k2 )Τ2

(

; ; 12 0.99 +

k −1 1 2 2

(

Τ2 =(0.99d m + θ1 )1 F1 κ2 , 32 , 12 0.99 + dm1θ

) )− 2

(,

1 κ θ 1 1 2

F

))

1 2 d mθ 3 2

]

(68)

(69)

, (d 1θ )2 m

)

(70)

Proof. Approximating the Gaussian Q function by [41]

[1 − e Q(x ) ≈

− ( Ax )

2

]e

−x2 2

(71)

B 2π x

with A=1.98 and B=1.135 while recalling the pdf of a Gamma distribution, the probability of error is

E[Pe (k )] ≈

1

B 4π θ Γ (k )d m κ

∫

∞

0

[1 − e

− ( Ad m x )

2

]e

− ( d m x )2 4 κ − 2 − x θ

x

e

dx

(72)


199

Making the substitution y=x2 and utilizing [42], the result follows after several manipulations.

Remark 8.7. The array gain for a MIMO RNN predictor is accurately approximated by

η ≈ θ 2κ (κ + 1)

(73)

Proof. This result follows immediately after performing the transformation y=x2 and calculating the mean. These two results provide closed form expressions that depend only on the scale and shape parameters of the Gamma distribution which can be found via maximum likelihood estimation. Figs. 8.13 and 8.14 illustrate the parameter values used for this work.

Fig. 8.13 Parameters of fitted Gamma distribution for a MIMO spatially uncorrelated channel

The array gain is investigated along with its approximations from Theorem 8.2 and Remark 8.7 for a 2×2 MIMO RNN predictor trained with f(x)=x, fdTs=0.1, Ntrain=200, γt=0. The weights are brought online for the prediction of a channel using the same parameter values in Fig. 8.15 for various channel estimation errors. The same parameters values are used with the exception of γt=0.7 to consider the effects of spatial correlation in Fig. 8.16. For both scenarios, the approximation due to Remark 8.7 is superior. The array gain is slightly larger for the spatially correlated case which is clear from Eq. (51).

200

C. Potter

Fig. 8.14 Parameters of fitted Gamma distribution for a MIMO spatially correlated (γt=0.7) channel

Fig. 8.15 Array gain for a 2×2 MIMO beam-forming fast fading uncorrelated channel


201

Fig. 8.16 Array gain for a 2×2 MIMO beam-forming fast fading spatially correlated (γt=0.7) channel

Fig. 8.17 Average Probability of error for a 2×2 MIMO beam-forming fast fading spatially uncorrelated channel

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Fig. 8.18 Average Probability of error for a 2×2 MIMO beam-forming fast fading spatially correlated (γt=0.7) channel

Fig. 8.19 Average Probability of error comparison for a 2×2 MIMO beam-forming fast fading spatially uncorrelated channel


203

Fig. 8.20 Average Probability of error for a 2×2 MIMO beam-forming fast fading spatially correlated (γt=0.7) channel

The average probability of error was simulated for a 2×2 MIMO system when f(x)=x, fdTs=0.1, γt=0 using 8×105 BPSK symbols with symbol period T=Ts and is plotted versus the theoretical average probability of error given by Eq. (54) for various channel estimation errors in Fig. 8.17 and with γt=0.7 in Fig. 8.18. For both cases, the RNN was trained with f(x)=x, γt=0, fdTs=0.1, with σw2=0.001. As proposed in Theorem 8.4, an increase in the prediction error leads to a decrease in coding gain but maintains full diversity gain until saturation. Next, the approximation due to Theorem 8.6 is plotted for comparison in Figs. 8.19 and 8.20 and is seen to be in good agreement with both the simulated probability of error and (54) until saturation.

8.9 Conclusions A recurrent neural network trained off-line by a novel PSO-EA-DEPSO was used to predict a MIMO channel. This training algorithm was shown to be superior to PSO, PSO-EA, and DEPSO for different fast fading scenarios. The RNN predictor was then shown to outperform a linear predictor trained by the Levinson-Durbin algorithm. New expressions for the received SNR, array gain, average probability of error, and diversity order for the MIMO RNN predictor were then derived. These expressions differed from numerous works in that the prediction error was not assumed to be independent of the actual CSI and/or Gaussian. The array gain

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for spatially correlated systems was shown to be slightly higher. It was verified through simulation that increasing the prediction error caused a loss in coding gain but still were able to achieve full diversity gain up until saturation.

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[37] Bain, L., Engelhardt, M.: Introduction to Probability and Mathematical Statistics. Brooks/Cole, Pacific Grove (1992) [38] Lee, W.: Mobile communication engineering. McGraw Hill, New York (1982) [39] Fu, J., Taniguchi, T., Karasawa, Y.: The largest eigenvalue characteristics for mimo channel with spatial correlation. J. Global Optimization 6(2), 109–133 (1995) [40] Goldsmith, A.: Wireless Communications. Cambridge University Press, New York (2005) [41] Karagiannidis, G., Lioumpas, A.: An improved approximation for the gaussian q function. IEEE Communications Letters 11(8), 644–646 (2007) [42] Gradshteyn, S., Ryzhik, I., Jeffrey, A.: Table of Integrals, Series, and Products, 5th edn. Academic, San Diego (2007)

Index

A Absorber 54, 64 absorption band 55 academic misconducts 1 active pattern 48, 50 adaptive simulated annealing 20 adjacent channel 157, 158, 160 adjacent channel interference rejection 158 advantage 37, 49, 116, 135, 177 algorithm 19, 20, 73, 112, 134, 151, 177 anechoic chamber 60, 145 anisotropic electromagnetic material 57 annealed Nelder & Mead strategy 20 antenna 48, 49, 134, 157, 165, 178, 181 antenna array 48, 49, 50, 61 application problem 9, 47 arithmetic crossover 32, 33, 34, 35 arithmetic multi-point crossover 33 arithmetic one-point crossover 33, 34 arithmetic operation 19 arithmetic recombination 26 arithmetic two-point crossover 34 array element 48, 49, 55, 61 array factor 48 array gain 178, 193, 194, 197, 199, 203 artifacts 11 Ask 4 available frequency 155 average symbol energy 182 B base transceiver stations 156 base vector 22 basis functions 137 benchmark application problems 47 Bernoulli experiments 26, 27, 35 best solution 162, 168 bibliography 2, 4

binary crossover 31 bioelectromagnetics 61 binomial crossover 34, 170, broadband planar microwave absorber 55 BTS 156 C calibration 51 candidate solution 162 case study 6, 7 cell 57, 112, 156, 157 channel 58, 59, 157-160, 177-206 channel coefficient 59, 185, 186, 188 channel interference 157, 158, 160 channel state information 177, 179, 182 Chebyshev polynomial fitting problem 19 Chinese 3, 4 Chinese Electronic Periodical Services 3 citeSeerX 4, 5 classic differential evolution 28, 36, 156, 159 classification 9, 10, 37, 43, 45, 74, 79, 98, 101 clustering procedure 111, 112 co-channel 157, 158, 160 co-evolution 9 co-evolution differential evolution 37 comparative evaluation 9 comparison 9, 20, 38, 58, 77, 84, 89, 112, 162, 187 complex permittivity 108 computational electromagnetics 45, 59,61, 151 Computer Science Bibliographies 4 constraint 9, 23, 49, contrast 109, 178 contrast source 129 control parameters 9, 20. 27, 58, 81, 166 conventional antenna array 48

208

Index

convergence 19, 22, 59, 77, 83, 166 cost function 76, 94, 109 coverage 4, 43, 58, 156 coverage relation 156, 166 crossover 8, 9, 81, 141, 166, 186 crossover intensity 33 crossover length 26 crossover probability 22, 81, 166, 186 cultural differential evolution 4 cut and try 52 D data insufficiency 76 database 3, 55 DE 20, 22, 125, 128, 177, 178 DE particle swarm optimization Decoding 9, 54 degradation of network quality 157 Denver 163. 166 DEPT 156, 166, 170 DEPT scheme 170 Defect 22 deterministic optimization algorithms 60 diagnostic applications 107 diagonal matrix 79, 110 dielectric materials 107 differential evolution 1, 19, 43, 73, 107, 135 differential evolution equation 4 differential evolution strategy 8, 28 differential evolution with Pareto tournament 159 differential genetic algorithm 10 differential mutation 8, 9, 19, 170 differential mutation base 22, 29, 170 digital resources 2 digital platform 2 directed differential mutation 30 disadvantage 9, 38, 178 distance function 163 diversity 22, 177 dominance 2, 19, 160 donor 24, 30 Doppler double square loop array 56 dynamic differential evolution 36, 81 dynamic range ratios 49 E economic differential evolution 4 effective anisotropy 56

effective wave numbers 56 efficiency 37, 51, 177 electromagnetic compatibility 57, 60, 134 electromagnetic composite materials 56, 57 electromagnetic fields 45, 73, 136 electromagnetic formation flight 60 electromagnetic inverse problem 44, 45, 47, 55 electromagnetic materials 56, 57 electromagnetic energy 59, 60 electromagnetic spectrum 43 electromagnetic structure 53 electromagnetic theory 53, 61 electromagnetic waves 43, 54, 73 electronic resources 4 Elsevier 4 emission 60, 133, 142, 144 encoding 9, 54, 165 Engineering Village 2 3 English 4, 43 epistatic problems 38 equivalent circuit method 55 error figures 112, 116, 125 estimation 182 evaluation 9, 20, 73, 107 evolution mechanism 9, 10 evolution strategy 8, 28 evolutionary algorithm 10, 19, 47, 77, 156, 177 evolutionary crimes 58, 61 evolutionary operation 8, 10, 24, 141 evolutionary programming 59 excitation amplitude 49, 50 experiments 26, 163, 166 exponential crossover 26, 35, 170, 171 external medium 108 F FAP 155, 156, 159, 161 FAP solution 155, 158 feed forward 177, 184, 191 filters 55 finite difference time domain method 55 finite element method 55 fitness 159, 160, 165, 180 Foldy-Lax model 78 frequency 55, 187 frequency assignment 57, 155 frequency assignment problem 155 frequency planning 57

Index

209

frequency selective surfaces 54, 55 frequency spectrum 155 frequency value 160, 165 functional 53, 108, 110, 156 G generation 24, 58, 101, 159 genetic algorithm 10, 30, 54, 55, 77, 134 genetic annealing algorithm 19 genetic differential evolution 4 geographical differential evolution 4 geological differential evolution 4 geometrical center 29 geometrical properties 107 geophysical prospecting 107 global optimum 37, 38, 162 global search 19, 37, 53, 140 GMO-SVNS 173, 175 GMO-VNS 173, 175 Google 4 Google Scholar 4 graphic 4, 54 Greedy mutation 162, 173 Greedy variants 156 Green’s function 79, 108 grey-level representation 111 GSM 57, 155 GSM networks 57, 155, 156 H Hankel function 75, 109 helical antenna 52 history 8, 19, 22, 50 horn antenna 52, 53 hybrid differential evolution 37, 57, 59 hybridization 9 hypervolume 156, 166 I IEEE Explore 3 ill-posedness 76, 107 imaging 107, 109 impedance-matching tuner 54 inception 1, 19 incident field 108 infinite cylinders 108 initial guess 73, 77 initial population 22, 24, 25 initial solution 37, 160, 161, 162

initialization 9, 28 insight 2, 7, 20 instability 77 Institute of Scientific and Technical Information of China 3 Integral Fredholm operators 108 interference cost 156, 157, 158 interference matrix 157 interferences 57, 133, 138, 157, 160 International Contests on Evolutionary Optimization 20 intrinsic control parameters 8, 20, 58, 166 inverse scattering 10, 73, 107 inverse scattering problems 10, 73, 77, 101, 107 ISI Web of Science 3 isotropic electromagnetic material 57 isotropic point sources 48 iterative multiscaling approach 107, 108, 110, 112, 129 K Kronecker product 182 L language 4, 43, 165 large problem 158 laser diode 52 least Squares 74, 78, 134, 140 lens antenna 52 Levinson-Durbin algorithm 178, 191, 203 limit of number of generations 27 limited radio spectrum 157 linear antenna array 48, 49, 50 literature survey 1, 2, 43 local minimum 144, 180, 193 local optimizer 22 local optimum 162 local shape functions 74, 76 logic dominance function 29 lower bound 166, 196 M mathematical formulation 78, 156, 158 mathematical model 7, 8, 156 Maxwell’s equations 59 mean 29 mean squared error 178, 179, 182 measurement 53, 81, 144

210

Index

measurement domain 108 measurement points 112, 135, 138, 140 medical imaging 170 memetic differential evolution 59 metaheuristics 133, 135 method of moment 76, 112 Microsoft Bing 4 microstrip antenna 53 microwave circuit 61 microwave & RF 50, 51, 52 microwave tomography 130 milestone 22 MIMO 58, 177 minimization process 108 misconception 10, 61 mixed optimization parameters 54, 55 MO-SVNS 156 MO-VNS 156, 161, 175 mobile communications 155 mobile stations 156 model approximation 110 moment generating function 196 mother-child competition 19 moving phase center antenna array 48, 50 multi-layer perceptron 177 multi-objective 37, 61, 155, 159, 166 multi-objective differential evolution 37, 159 multi-objective FAP 156, 159 multi-objective optimization 161, 162, 174 multi-objective skewed variable neighborhood search 162 multi-point crossover (or M-point crossover) 32, 33 multilayered medium 53 multiple-input multiple-output 177, 178 multiple signal classification 74, 78, 79, 81, 101 multiresolution strategy 111 multistatic response matrix 76, 79 mutant 22, 26, 30 mutation 9, 19, 58, 160, 169, 178 mutation intensity 21, 22, 28, 81, 166 mutual coupling 48, 49, 61 N National Knowledge Infrastructure 3 natural real code 19, natural selection 27 network 51, 59, 134, 155 network quality 155, 157

Newton’s method 59 Noise filtering 111 noisy problems 38 non-intrinsic control parameters 8 non-uniqueness 58, 77 nondestructive testing 107 nonlinear operator 109 nonlinearity 52, 107 normalized vector difference 28 notation 22, 23 numerical algorithms 59 O objective function 22, 23, 52, 140, 166, 179 objective function evaluation 28 objective space 166 one-point crossover 32, 33, 141 operator 78, 108, 141, 155, 157, 179, 181 optimal control in minimal time 54 optimization 31, 52, 59, 77, 177, 193 optimization algorithm 9, 20, optimization parameters 23, 54, 76, 101, 140 optimization problem 45, 50, 52, 76, 107, 140, 156, 179, 193 optimization vector 76, 78 originality 1 P paper platform 2 parallel processing 110 parameterization 110 penetrable bodies 108 parasitic effect 51 Pareto 156, 159, 165, 173, 175 Pareto front 165, 166, 167, 171, 173 Pareto solution 162, 163 Pareto tournament 156, 159, 175 particle swarm optimization 59, 177 pattern nulling 49 perfect electric conductor 108 performance 9, 20, 36, 48, 77, 166, 177 performance indicator 166 periodic arrays 53, 54, 55, 56 periodic moment method 55 permeability 56, 57, 136 permeability tensor 57 permittivity 57, 74, 80, 108, 136 permittivity tensor 56, 57

Index

211

personal library 10 phased antenna array 50 phase center 48, 50 phaseless near field antenna measurement 53 pixel-wise approach 55 plain electromagnetic structure 53, 54 planar antenna array 49 Planck length 43 plane wave 112 platform 2, 49, 149 polarization current 108 population 19, 21, 25, 77, 81, 143, 162 population size 22, 28, 77, 81, 159, 166, 167 population-based incremental learning 58 practical advice 21 practical usage advice 7 prediction 59, 60, 135, 150, 177-206 principle of pattern multiplication 48 probability distribution function 28 probability of error 178, 193, 195, 196, 203 problem feature 37 propagation medium 109 pseudocode 159, 161 PSO evolutionary algorithm 177, 188 PSO-EA-DEPSO algorithm 177, 187, 191 Q Q function 195, 198 QoS 155 Qualitative methods 74, 78 quality of service 155 R radar 45, 49, 59 radio network design 45, 58 random frequency 161 random search 58 real material database 55 real-world 57, 155, 156, 163 real-world instances 156, 175 recombination coefficient 26, 28 recurrent neural networks 184 relative permittivity 57, 74 reliability 8, 37, 59, 177 reputation 9, 20 resolution 55, 110 retrieval procedure 111

RF circuits 50, 51 RF low noise amplifier 44, 50 ripple 49 robustness 8, 38, 178 S sampling period 181 scanned beam antennas 53 scatterer 45, 46, 73, 77, 78, 101, 111, 126, 191 scattered electric field 75, 81, 107, 112 scattering equation 108 scattering potential 109, 110, 112, 113 ScienceDirect 3 Scopus 3, 4 search engines 4 search space 107, 125, 143, 166 Seattle 163, 166, 168, 172 sector 157, 160, , 162, 164 sector channel 158, 159 sector channel separation 158, 159 selection 9, 19, 22, 27 semi-anechoic chambers 60, 133, 137, 151 separation constraints 155 separation cost 156, 157, 158, 159, 160, 168 Shannon limit 58 shape 48, 56, 59, 74, 76, 108, 150, 197 side lobe 48 signal-to-noise ratio 58 signaling threshold 158 simplicity 28, 37, 49, 75, 181 simulated annealing 20, 54, 58 single-input single-output 177 singular vector 80, 183, 184 singular value 79, 80, 183 site channel separation 158 social differential evolution 4 spectral-domain method 55 SpringerLink 3 staggered dipole array 56 standard deviation 158 state of the art 1, 2, 36 statistical confidence 165 stochastic algorithms 60 stochastic optimization method 110 survival of the fittest 27 susceptibility 60 SVNS 156, 162, 175 symbol period 177, 203 synthesis 48, 49, 51, 55

212

Index

system level evaluation 38 system level parametric study 1, 38 T T-matrix method 55, 56 tabu search 59 tapering 49 TCQ model 56, 57 technical limitations 158 telecommunications 57, 155 termination 9, 24, 28, 36, 81, 84, 111, 165 termination conditions 9, 24, 27, 36, 165 termination procedure 111 test bed 9, 20 theory of differential evolution 7 thresholding operation 111 time modulation 49, 50 time-modulated antenna array 48, 49 time sequence 49 topical review 1, 36, 37 topology 165 total field 109 toy function 20 traffic demand 157 transceiver 156, 157, 158 transmit power 182 transverse magnetic 75, 112 trial and error 20, 54, 55, 186 trial parameter vector 19 trigonometric differential mutation 30

TRX 157, 158, 160, 161, 165 two norm 179 U undesired interferences 157 upper bound 166, 193 usage 7, 9, 22 user friendliness 19 V variable interferences variable neighborhood search 59, 160-163 vector difference 22, 24, 26, 28, 30 vector spectral-domain method 55 VNS 156, 160, 161 W wave number 56, 57, 75 website 4, 21 weighted difference vector 19 weighting function 109, 110 Wiley Interscience 3 wireless 50, 52, 57, 177-179, 184, 191 wireless communication 50, 52, 57, 177, 179 Y Yahoo 4

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