CuuDuongThanCong.com Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms CuuDuongThanCong.com Adaptation, Learning, and Optimization, Volume Series Editor-in-Chief Meng-Hiot Lim Nanyang Technological University, Singapore E-mail: emhlim@ntu.edu.sg Yew-Soon Ong Nanyang Technological University, Singapore E-mail: asysong@ntu.edu.sg Further volumes of this series can be found on our homepage: springer.com Vol Jingqiao Zhang and Arthur C Sanderson Adaptive Differential Evolution, 2009 ISBN 978-3-642-01526-7 Vol Yoel Tenne and Chi-Keong Goh (Eds.) Computational Intelligence in Expensive Optimization Problems, 2010 ISBN 978-3-642-10700-9 Vol Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms, 2010 ISBN 978-3-642-12833-2 CuuDuongThanCong.com Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms 123 CuuDuongThanCong.com Ying-ping Chen Natural Computing Laboratory Department of Computer Science National Chiao Tung University 1001 Ta Hsueh Road HsinChu City 300 Taiwan E-mail: ypchen@cs.nctu.edu.tw ISBN 978-3-642-12833-2 e-ISBN 978-3-642-12834-9 DOI 10.1007/978-3-642-12834-9 Adaptation, Learning, and Optimization ISSN 1867-4534 Library of Congress Control Number: 2010926027 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 CuuDuongThanCong.com Preface As genetic and evolutionary algorithms (GEAs) have been employed to handle complex optimization problems in recent years, the demand for improving the performance and applicability of GEAs has become a crucial and urgent issue The exploitation of linkage is one of many mechanisms that have been integrated into GEAs This concept draws an analogy between the genetic linkage in biological systems and the variable relationships in optimization problems Most of the papers on the subjects of detecting, understanding, and exploiting linkage in GEAs are scattered throughout various journals and conference proceedings This edited volume serves as an archive of the theoretical viewpoints, insightful studies, and the state-of-art development of linkage in GEAs This book consists of papers written by leading researchers who have investigated linkage in GEAs from different points of view The 11 chapters in this volume can be divided into parts: (I) Linkage & Problem Structures; (II) Model Building & Exploiting; and (III) Applications Part I consists of chapters that deal primarily with the nature and properties of linkage and problem structures Thorough understanding of linkage, which composes the target problem, on the fundamental level is a must to devise GEAs better than what are available today The next chapters in Part II discuss issues regarding depicting linkage structures by establishing probabilistic models or presenting insights into relationship networks These chapters develop adequate techniques for processing linkage, facilitating the analysis of problem structures and optimization tasks Part III consists of chapters that present applications that incorporate intermediate analysis solutions, allowing linkage to be exploited by, and incorporated into, practical problem-solving More work on applying linkage to real-world problems should be encouraged, and this edited volume represents a significant step in that direction I hope that this book will serve as a useful reference for researchers working in the areas of detecting, understanding, and exploiting linkage in GEAs This compilation is also suitable as a reference textbook for a graduate level course focusing on linkage issues The collection of chapters can quickly CuuDuongThanCong.com VI Preface expose practitioners to most of the important issues pertaining to linkage For example, practitioners looking for advanced tools and frameworks will find the chapters on applications a useful guide I am very fortunate and honored to have a group of distinguished contributors who are willing to share their findings, insights, and expertise in this edited volume For this, I am truly grateful Hsinchu City, Taiwan December 2009 CuuDuongThanCong.com Ying-ping Chen Contents Part I: Linkage and Problem Structures Linkage Structure and Genetic Evolutionary Algorithms Susan Khor Fragment as a Small Evidence of the Building Blocks Existence Chalermsub Sangkavichitr, Prabhas Chongstitvatana 25 Structure Learning and Optimisation in a Markov Network Based Estimation of Distribution Algorithm Alexander E.I Brownlee, John A.W McCall, Siddhartha K Shakya, Qingfu Zhang DEUM – A Fully Multivariate EDA Based on Markov Networks Siddhartha Shakya, Alexander Brownlee, John McCall, Fran¸cois Fournier, Gilbert Owusu 45 71 Part II: Model Building and Exploiting Pairwise Interactions Induced Probabilistic Model Building David Icl˘ anzan, D Dumitrescu, B´eat Hirsbrunner 97 ClusterMI: Building Probabilistic Models Using Hierarchical Clustering and Mutual Information 123 Thyago S.P.C Duque, David E Goldberg CuuDuongThanCong.com VIII Contents Estimation of Distribution Algorithm Based on Copula Theory 139 Li-Fang Wang, Jian-Chao Zeng Analyzing the k Most Probable Solutions in EDAs Based on Bayesian Networks 163 Carlos Echegoyen, Alexander Mendiburu, Roberto Santana, Jose A Lozano Part III: Applications Protein Structure Prediction Based on HP Model Using an Improved Hybrid EDA 193 Benhui Chen, Jinglu Hu Sensible Initialization of a Computational Evolution System Using Expert Knowledge for Epistasis Analysis in Human Genetics 215 Joshua L Payne, Casey S Greene, Douglas P Hill, Jason H Moore Estimating Optimal Stopping Rules in the Multiple Best Choice Problem with Minimal Summarized Rank via the Cross-Entropy Method 227 T.V Polushina Author Index 243 Index 245 CuuDuongThanCong.com Part I Linkage and Problem Structures CuuDuongThanCong.com 232 T.V Polushina where α is called the smoothing parameter, with 0.7 < α ≤ Clearly, for α = we have our original updating rule To complete specification of the algorithm, one must supply values for N2 and ρ , initial parameters u0 , and a stopping criterion We use stopping criterion from [12] To identify T , we consider the following moving average-process Bt (K) = t ∑ γs ,t = s, s = 1, , s ≥ K, K s=t−K+1 where K is fixed; Ct (K) = Next define t K−1 {∑s=t−K+1 (γs − Bt (K)) } Bt (K)2 Ct− (K, R) = Ct+ j (K) j=1, ,R and Ct+ (K, R) = max Ct+ j (K), j=1, ,R respectively, where R is fixed We define stopping criterion as T = min{t : Ct+ (K, R) − Ct− (K, R) ≤ ε }, Ct− (K, R) (5) where K and R are fixed and ε is a small number, say ε ≤ 0.01 The Cross-Entropy Method for the Problem We solve maximization problem max ES(x, R), x∈X (6) where X = {x = (x(1) , , x(k) ) : conditions (1) are hold}, R = (R1 , , RN ) is a random permutation of numbers 1, 2, , N, S(x) is an unbiased estimator of ES(x, R) S(x) = N1 ∑ (Rnτ1 + + Rnτk ) N1 n=1 As in [14] we consider a 3-dimensional matrix of parameters u = {ui jl } (i) ui jl = P{X j = l}, i = 1, , k; j = k − i + 1, , N − i + 1; l = 0, , N − CuuDuongThanCong.com (7) Estimating Optimal Stopping Rules in the Multiple Best Choice Problem 233 It follows easily that (i) f (x j ; u) = N−1 ∑ ui jl I{x(i)j =l} l=0 We can see that (t−1) N (t) ui jl = (t−1) Wni j I{S(Xn )≥γt }Wni j ∑n=1 I{Xni j =l} (t−1) N2 I{S(Xn )≥γt }Wni j ∑n=1 , (0) = ui jXni j (t−1) ui jXni j , (8) (i) where Xn = {Xni j }, Xni j is a random variable from f (x j ; ut−1 ) Formula (4) becomes (8) Numeric Results In this section we present numerical results for N = 10 and k = For different set of (δ (1) , δ (2) ) we calculate v, which is decreased from 11 to (figure 1) A Monte Carlo technique (N1 = 50000) is applied for finding the gain v for 5000 different sets (δ (1) , δ (2) ) Then we show that the CE algorithm allows to find v = 5.8638, and the optimal sets (δ (1) , δ (2) ) While the minimal v by Monte Carlo technique is about We use the CE method with simulation parameters ρ = 0.1, α = 0.7, N2 = 200, N1 = 100, Nlast = 5000, K = 6, R = 3, ε = 0.01 We run the algorithm 100 times Figure shows the histogram of solutions that were obtained by the CE method Note that the alogithm finds the optimal sets (δ (1) , δ (2) ) and neighbouring optimal sets Figures 3, show how the CE method works Initially we have uniform distribu(0) tion, that is, ui jl = 0.1 for i = 1, 2; j = − i, , 11 − i; l = 0, , For example, (t) u120 is considered It is situated in the first subdiagram in position (figure 3) En(t) (0) larged diagram u120 is showed on figure At first u120 = 0.1 Then with iterations (t) (1) t u120 increases and amounts to This implies that P{X2 = 0} = Thus in set δ (1) in the first position is situated Similarly, zeros are the second and the third elements of set δ (1) Simulation parameters are ρ = 0.1, α = 0.7, N2 = 200, N1 = 100, Nlast = 500, K = 6, R = 3, ε = 0.01, repeat times By simulation we obtained δ (1) = (0, 0, 0, 1, 1, 2, 3, 4, 10), δ (2) = (0, 0, 1, 1, 2, 2, 3, 5, 10) Nikolaev M.L [10] shows that for N = 10 theoretical optimal are δ (2) = (0, 0, 1, 2, 2, 3, 4, 5, 10), δ (1) = (0, 0, 1, 1, 2, 2, 3, 5, 10) We also compare v with theoretical (δ (1) , δ (2) ) and v with modelling (δ (1) , δ (2) ) (table 2) For this table N1 = 50000 We can see that v with theoretical (δ (1) , δ (2) ) CuuDuongThanCong.com 234 T.V Polushina 12 11 10 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Fig Gain v for different (δ (1) , δ (2) ) 10 20 30 40 50 60 70 80 90 100 Fig Bar graph for N = 10 is differ from v with modelling (δ (1) , δ (2) ) slightly Difference connects that N1 and Nlast are not big Then we use the CE method for different N, using the following parameters N1 = 100, N2 = 200, Nlast = 500, ρ = 0.1, α = 0.7, ε = 0.1, K = 6, R = This method CuuDuongThanCong.com Estimating Optimal Stopping Rules in the Multiple Best Choice Problem 235 0.5 1 9 9 9 9 (t) u1 jl for N = 10 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Fig Values Table v with theoretical (δ (1) , δ (2) ) and with modelling (δ (1) , δ (2) ) δ (1) δ (2) v theoretical (0,0,1,1,2,2,3,5,10) (0,0,1,2,2,3,4,5,10) 5.9124 modelling (0,0,0,1,1,2,3,4,10) (0,0,1,1,2,2,3,5,10) 5.8638 Table Value mean, maximum and minimum v for different N N mean v max v v standard error CPU time 10 15 20 4.5908 5.8698 6.6448 7.5668 4.6060 5.7880 6.7620 7.8260 4.5800 5.6760 6.5320 7.3580 0.0074 0.0458 0.0927 0.2793 10.7470 15.6188 26.2562 35.1936 has been implemented in MatLab on a PC (AMD Athlon 64 2.01 GHz) Using this algorithm parameters after repetitions, we obtain the results summarized in table Figure shows values v with standard error and for different N, N = 3, , 20 Modelling parameters are ρ = 0.1, α = 0.7, N2 = 100, N1 = 100, Nlast = 500 ε = 0.1, K = 6, R = Method was repeated times for each N Nikolaev M.L shows CuuDuongThanCong.com 236 T.V Polushina 0.5 1 9 9 9 9 (t) u2 jl for N = 10 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Fig Values 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 Fig Values 15 (t) u120 20 25 for N = 10 that asymptotic v is 7.739 [10] We can see that v is bigger than the expected value for big N It arises from N! >> N1 , N2 , Nlast , which are used for simulation Notwithstanding N1 , N2 , Nlast are much smaller than N! the cross-entropy method finds optimal stopping rules that are nearly optimal CuuDuongThanCong.com Estimating Optimal Stopping Rules in the Multiple Best Choice Problem 237 7.5 6.5 5.5 4.5 3.5 10 12 14 16 18 20 Fig Gain v with standard error for N = 3, , 20 Genetic Algorithm In this section, the Genetic Algorithm (GA) is considered The GA is a stochastic search technique Choosing an appropriate genotype or representation to encode the important features of the problem to be solved is the first step in applying the GA to any problem The fitness function is defined over the genetic representation and measures the quality of the represented solution The GA can develop a population of potential good solution by applying the genetic operators, and finally find a good solution of the problem The individuals of the population are called ”chromosomes” In genetic operators, crossover and mutation are important operators which influence the behavior of GA The purpose of crossover consists in the combination of useful string segments from different chromosomes to form new, hopefully better performing offspring Crossover is the process in which the chromosomes are mixed and matched in a random fashion to produce a pair of new chromosomes (offspring) The purpose of mutation is give population diversity to the offspring to selecting a gene at random with a mutation rate and perturbing its value Mutation operator is the process used to rearrange the structure of the chromosomes to produce a new one Following Wang, Okazaki [16], we propose an improved GA by modifying the crossover and mutation behavior Let N p is the population size and Nc is the current number of children generated First Nc is set to zero Then (N p − Nc )/2 pairs of parent chromosomes are randomly selected, and the difference-degree for every pair of parent chromosomes are calculated The difference-degree di of i parent pair is defined as follow: di = NNdg , where Ng is the size of chromosome, and Nd is the number of different genes between the two parent chromosomes A new parameter called setting difference-degree Ds is introduced If di is larger than the Ds , then CuuDuongThanCong.com 238 T.V Polushina the crossover is applied with 100% certainty on the parent pair to generate two children After crossover, the total number of children generated is calculated If the total number Nc is found to be smaller than the population size N p , than mutation is performed with 100% certainty on parent pairs chromosomes with di less than the Ds The above procedure is performed in a loop until the total number of children is equal to the population size The genetic search process is described as follows: GA stars with an initial set of random solutions for the problem under consideration This set of solutions is known as the population Evaluate the fitness of each individual in the population Repeat until termination: a Select best-ranking individuals to reproduce b Breed new generation through crossover and mutation and give birth to offspring c Evaluate the individual fitnesses of the offspring d Replace worst ranked part of population with offspring [8] The technique of GAs requires a string representation scheme (chromosomes) In this chapter we put that each element of sets (δ (1) , , δ (k) ) correspond to a locus of chromosome Than we solve a combinatorial optimization problem by the GA So we consider the problem (2) The fitness function is calculated from (7) For population creation we use vector of parameters u = {ui jl }, same as in the CE method Than for initial population GA is applied, and we find the first approximation to problem (2) The same as the CE method instead of updating the parameter vector we use the following smoothed version ut = α ut + (1 − α )ut−1, i = 1, , n The selection operator is applied to select parent chromosomes from the population A Monte Carlo selection technique is applied A parent selection procedure functions as: Calculate the fitness of all population members using (7) Return the first population member whose fitness is among the best f it · N1 % members of population Repeat step for the second population member and check that the new selected member is not the same as the first member; and so on Fitness value f it is a given arbitrary constant The selected chromosomes to crossover will be crossed to produce two offspring chromosomes by using crossover operator Crossover operator is described as follows Let a pair of parent chromosomes (P1 , P2 ) Select two random number to be aligned to the parents The genes are exchanged so that portion of genetic codes from P1 is transferred to P2 , and conversely If the chromosome has large size genes, the cutting section is differing from small to large, which reflects the flexibility of the approach CuuDuongThanCong.com Estimating Optimal Stopping Rules in the Multiple Best Choice Problem 239 The mutation operator is used to rearrange the structure of a chromosome The swap mutation is used, which is simply selecting two genes at random and swapping their contents Mutation helps to increase the searching power In order to explain the need of mutation Consider the case where reproduction or crossover may not produce a good solution to a problem During the creation of a generation it is possible that the entire population of strings is missing a vital gene of information that is important for determining the correct or the most nearly optimum solution In that case mutation becomes very important This generational process is repeated until a termination condition has been reached We use stopping criterion (5) Numeric Results of GA Process We also use the GA for different N and parameters for simulation are N2 = 100, N1 = 500, f it = 0.25, α = 0.95, ε = 0.01 K = 6, R = Using this algorithm parameters after repetitions, we obtain the results summarized in Table Table Minimum, maximum, mean value v and standard error for different N by GA method N mean v max v v standard error CPU time 10 15 20 4.5023 5.7726 6.6572 7.8912 4.8470 5.8910 6.9820 8.0160 4.3900 5.5130 6.1140 7.1370 0.0702 0.1096 0.2084 0.7264 1.0318 8.2614 12.6725 17.9326 8.5 7.5 6.5 5.5 4.5 3.5 10 12 14 16 18 20 Fig Gain v with standard error for N = 3, , 20 by GA method CuuDuongThanCong.com 240 T.V Polushina Figure shows value v with standard error for different N, N = 3, , 20 Modelling parameters are α = 0.95, f it = 0.25, N2 = 100, N1 = 500, ε = 0.1, K = 6, R = Method was repeated times for each N The simulation results show that the genetic algorithm can generate solutions with bigger dispersion compared with the cross-entropy method Solutions by crossentropy method are more closely optimum but this method need a lot more computational resource Besides, if k, N and simulation parameters are sufficiently great, the genetic algorithm will generate better solutions Conclusions In this chapter we have proposed how CE method can be used to solve the multiple best choice problem The CE method is better than the GA algorithm for solution this problem But for N tends to infinity, the CE method gives heavy error, and it should be modified If N1 , N2 , Nlast increase, the cross-entropy method finds optimal stopping rules that are nearly optimal The methodology can also be extended to more general models Acknowledgements T.V Polushina would like to thank G Yu Sofronov for his invaluable advices and helpful discussion References Bruss, F.t., Ferguson, T.S.: Minimizing the expected rank with full information J.Appl Prob 30, 616–626 (1993) Chow, Y.S., Robbins, H., Siegmund, D.: Great Expectations: The Theory of Optimal Stopping Houghton Mifflin, Boston (1971) Chow, Y.S., Moriguti, S., Robbins, H., Samuels, S.M.: Optimal selection based on relative rank Israel J.Math 2, 81–90 (1964) Dynkin, E.B., Yushkevich, A.A.: Theorems and Problems on Markov Processes Plenum, New York (1969) Gabor, C.P., Halliday, T.R.: Sequential mate choice by multiply mating smooth newts: females become more choosy Behavioral Ecology 8, 162–166 (1997) Gilbert, J.P., Mosteller, F.: Recognizing the maximum of sequence J Am Stat Assoc 61, 35–73 (1966) Cianini-Pettitt, J.: Optimal selection based on relative ranks with a random number of individuals Adv Appl Prob 11, 720–736 (1979) Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning Kluwer Academic Publishers, Boston (1989) Nikolaev, M.L.: On a generalization of the best choice problem Theory Prob Appl 22, 187–190 (1977) 10 Nikolaev, M.L.: Optimal multi-stopping rules Obozr Prikl Prom Mat 5, 309–348 (1998) 11 Pitcher, T.E., Neff, B.D., Rodd, F.H., Rowe, L.: Multiple mating and sequential mate choice in guppies: females trade up Proceedings of the Royal Society B: Biological Sciences 270, 1623–1629 (2003) CuuDuongThanCong.com Estimating Optimal Stopping Rules in the Multiple Best Choice Problem 241 12 Rubinstein, R.Y., Kroese, D.P.: The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization In: Monte-Carlo Simulation and Machine Learning Springer, New York (2004) 13 Shiryaev, A.N.: Optimal Stopping Rules Springer, New York (1978) 14 Sofronov, G.Y., Kroese, D.P., Keith, J.M., Nikolaev, M.L.: Simulation of thresholds in the multiple best choice problem Obozr Prikl Prom Math 13, 975–983 (2006) 15 Tamaki, M.: Minimal expected ranks for the secretary problems with uncertain selection Lecture Notes-Monograph Series, vol 35, pp 127–139 Institute of Mathematical Statistics (2000) 16 Wang, R., Okazaki, K.: Solving facility layout problem using an improved genetic algorithm IEICE Trans Fundam E88–A N2, 606–610 (2005) CuuDuongThanCong.com CuuDuongThanCong.com Author Index Brownlee, Alexander 71 Brownlee, Alexander E.I 45 Khor, Susan Lozano, Jose A Chen, Benhui 193 Chongstitvatana, Prabhas Dumitrescu, D 97 Duque, Thyago S.P.C Echegoyen, Carlos 163 Fournier, Fran¸cois 71 123 Goldberg, David E 123 Greene, Casey S 215 Hill, Douglas P 215 Hirsbrunner, B´eat 97 Hu, Jinglu 193 Icl˘ anzan, David CuuDuongThanCong.com 97 25 163 McCall, John 71 McCall, John A.W 45 Mendiburu, Alexander 163 Moore, Jason H 215 Owusu, Gilbert 71 Payne, Joshua L 215 Polushina, T.V 227 Sangkavichitr, Chalermsub 25 Santana, Roberto 163 Shakya, Siddhartha K 45, 71 Wang, Li-Fang 139 Zeng, Jian-Chao 139 Zhang, Qingfu 45 CuuDuongThanCong.com Index $k$-order Markov probabilistic model 193 Abductive inference 163 Additively decompasable functions (ADF) 33, 40-41, 168 Ali-Mikhail-Haq copula 152 Archimedean copula 143 Artificial evolution 215, 216, 219 Artificial intelligence (AI) 97, 163 Assortativity Bayesian network 48, 72, 73, 112, 124, 163, 165, 169 Bayesian optimization algorithm (BOA) 32, 199 Best choice problem 227, 228 Bioinformatics 216 Boltzmann distribution 49, 50 Building block (BB) 124 Building block hypothesis 20, 25, 27, 46 Building block processing 25, 46, 107, 124, 216 Centrality 4, 9, 10, 11 Chi-square matrix 44 Chi-square structure learning 58 Classification 215, 218, 225 Clayton copula 143 Clustering 4, 10, 29, 123, 127, 141 Common fragment 31, 34, 36, 40 Composite fitness function 193, 195, 201, 208 Computational biology 193, 196 Computational evolution 213, 217 Contiguous subsequence 28, 29, 30, 34 Copula-EDA 139, 141, 145, 146, 151, 153, 156 Correlation coefficient 99, 101, 146, 168, 184 Cross-entropy method 227, 230, 323 Cumulative distribution function (cdf) 144 CuuDuongThanCong.com Deceptive problem 26, 33, 36, 41, 107, 199 Degree distribution 4, 7, 9, 10, 11, 17, 20 Degree mixing 4, 10, 11 Dependency structure matrix genetic algorithm (DSMGA) 124 Disease 215, 216, 217, 218, 220, 221, 224 Disruption of building block 25, 32, 41 Distribution estimation using Markov networks (DEUM) 47, 72, 81 Diversity 8, 29, 34, 41, 62, 163, 175, 220, 237 Efficiency enhancement 107, 123, 135 Empirical copula 153, 154, 156 Entropy 81, 99, 100, 125, 129, 169, 227, 230, 232 Epidemiology 225 Epistasis 21, 45, 215 Estimation of Bayesian networks algorithm (EBNA) 164-166 Estimation of Distribution Algorithm (EDA) 72, 193, 194 Estimation 21, 27, 45, 53, 71, 76, 81, 97, 123, 139, 146, 163, 166, 193, 197, 231 Evolutionary algorithm 3, 46, 97, 123, 194, 200, 203 Evolutionary computation 45, 139, 163, 197 Extended compact genetic algorithm (ECGA) 99, 124, 197 F-measure 45, 51, 56 Fitness evaluation 34, 57, 60, 86, 157 Fitness modeling 45, 50, 56, 59, 64, 72, 77 Fitness prediction correlation 54, 55, 56 Floodfill search strategy 206 Fragment composition 25, 31 246 Fragment crossover 32, 41 Fragment identification 25, 29 Gain 32, 99, 100, 228, 233 Gaussian copula 152, 159 Genetic algorithm (GA) 46, 194, 237 Genetic programming (GP) 216-217, 225 Genome-wide association study 215, 224 Graph coloring 10 Gumbel copula 143 Hammersley-Clifford Theorem 49, 50, 75 Heritability 221, 224 Hierarchical Bayesian optimization algorithm (hBOA) 32, 55 Hierarchical decomposable functions 33 Hierarchical function 32, 109, 123, 134, 217 HP model 193, 195, 196, 200, 208, 213 Hub 4, 7, 9, 10, 11 Hybrid EDA 195, 199, 203, 208, 211, 213 Improved backtracking-based repairing method 193, 195, 204, 210, 211 Incremental model building 124, 135, 136 Inter-level conflict 3, 13, 14, 18, 20, 21 Interaction network 3, 7, 9, 11, 21, 225 Internal coordinates with relative direction 200 Invalid solution 195, 208 Ising model 79, 168, 174 Ising problem 48, 55, 59, 89, 168, 178 Joint distribution function 145, 151 Joint probability distribution (JPD) 47, 48 Kikuchi approximation 50, 55, 76, 85 Knowledge sharing 27 Linkage detection algorithm (LDA) 51-60 Linkage learning genetic algorithm (LLGA) 27 Linkage learning 21, 45, 105, 128, 164 Linkage structure 3, 7, 11, 21 Linkage 4, 29, 46, 110, 164, 224 Local search with guided operators 193, 195, 203, 213 Machine learning 65, 123, 164, 220 Marginal distribution function/margin 151 Marginal product model (MPM) 106 Markov network estimation of distribution algorithm (MN-EDA) 55, 76 CuuDuongThanCong.com Index Markov network factorized distribution algorithm (MN-FDA) 55 Markov network 45, 53, 66, 71, 77 Markov random fields 73 Markovianity property 49 Max-SAT problem 53, 165, 178, 181 Messy genetic algorithm 43 Model building 27, 66, 90, 112, 127 Modularity 7, 11, 21 Most probable configuration 167 Most probable solution 163, 165, 167, 184 Mutual information 30, 81, 99, 120, 141 Optimal stopping rule 227, 240 Optimization 13, 32, 129, 146, 174 Pairwise interaction 97, 100, 120 Path length 4, 9, 10, 11 Precision 51, 56, 65, 148 Probabilistic graphical model 47, 73, 112, 198 Probabilistic model building genetic algorithm (PMBGA) 103 Probabilistic model building 103 Probabilistic model 27, 48, 78, 120, 184, 200 Probability density function (pdf) 144 Protein structure prediction (PSP) 193 Recall 51, 56, 61, 65 Recombination 8, 26, 46, 219 RMHC2 14 Royal road function 26, 35, 38, 40 Sampling 27, 48, 80, 83, 135, 167, 204 Scale-free 7, Schema theorem 25, 40, 46 Single nucleotide polymorphism 216, 224 Sklar's theorem 143, 151, 153 Specificity 3, 19, 21 Structural properties 3, 21 Structure learning 45, 50, 65, 73, 89 Symbolic discriminant analysis 217 t-copula 143 Trap function 33, 109, 130, 132 Uncommon fragment 34, 38, 40 Undirected Graphical Models 50, 70 upGA 9, 10, 11 ... Problems, 2010 ISBN 978-3-642-10700-9 Vol Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms, 2010 ISBN 978-3-642-12833-2 CuuDuongThanCong.com Ying-ping Chen (Ed.) Exploitation. ..Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms CuuDuongThanCong.com Adaptation, Learning, and Optimization, Volume Series Editor -in- Chief Meng-Hiot... slc.khor@gmail.com Y.-p Chen (Ed.): Exploitation of Linkage Learning, ALO 3, pp 3–23 © Springer-Verlag Berlin Heidelberg 2010 springerlink.com CuuDuongThanCong.com S Khor manner within the model of three basic