Modeling epigenetic evolutionary algorithms: An approach based on the epigenetic regulation process
MModeling Epigenetic Evolutionary Algorithms: Anapproach based on the Epigenetic Regulation process
Lifeth ´Alvarez Camacho
Software Engineer
Code: 1047410852
Universidad Nacional de ColombiaEngineering SchoolComputer Systems EngineeringBogot´a, D.C.December 14, 2020 a r X i v : . [ c s . N E ] F e b odeling Epigenetic Evolutionary Algorithms: Anapproach based on the Epigenetic Regulation process Lifeth ´Alvarez Camacho
Software Engineer
Code: 1047410852
Dissertation to apply for the title of
Master in Computer Systems Engineering
AdvisorJonatan G´omez Perdomo, Ph.D.
Computer scientist
Research area
Evolutionary Computing, Evolutionary Algorithms, Epigenetics
Research line
Artificial Life, Optimization
Research group
Research Group in Artificial Life, ALife
Universidad Nacional de ColombiaEngineering SchoolComputer Systems EngineeringBogot´a, D.C.December 14, 2020 itle in English
Modeling Epigenetic Evolutionary Algorithms: An approach based on the EpigeneticRegulation process.
Abstract:
Many biological processes have been the source of inspiration for heuristicmethods that generate high-quality solutions to solve optimization and search problems.This thesis presents an epigenetic technique for Evolutionary Algorithms, inspired by theepigenetic regulation process, a mechanism to better understand the ability of individualsto adapt and learn from the environment. Epigenetic regulation comprises biologicalmechanisms by which small molecules, also known as epigenetic tags, are attached to orremoved from a particular gene, affecting the phenotype. Five fundamental elements formthe basis of the designed technique: first, a metaphorical representation of
EpigeneticTags as binary strings; second, a layer on chromosome top structure used to bind thetags (the
Epigenotype layer ); third, a
Marking Function to add, remove, and modifytags; fourth, an
Epigenetic Growing Function that acts like an interpreter, or decoderof the tags located over the alleles, in such a way that the phenotypic variations can bereflected when evaluating the individuals; and fifth, a tags inheritance mechanism. A setof experiments are performed for determining the applicability of the proposed approach.
Keywords: evolutionary algorithms, evolution, epigenetics, gene regulation. cceptation Note
Thesis WorkApproved
JuryGerman Hernandez P´erezJuryJorge Ortiz Trivi˜noAdvisorJonatan G´omez PerdomoBogot´a, D.C., December 14, 2020 edicatory
To Abba... cknowledgments
I want to thank my advisor, Professor Jonatan Gomez, for his guidance and academicsupport during this research. I value his insights, remarks, motivation, and expertise thatnotably contributed to this thesis.I want to thank Professor Luis Eugenio Andrade, Biologist, for his support at thebeginning of this work, his help with Epigenetics understanding, and the comments andsuggestions during academic meetings.I express my gratitude to Aimer Alonso Gutierrez, Biologist and passionate aboutEpigenetics, for all the comments and suggestions. His knowledge reinforces and refinesdifferent aspects of this thesis.My recognition goes out to the Artificial Life (ALife) Research Group, for all thecomments and suggestions during formal and informal meetings.Finally, I would like to acknowledge my family for their support and constant encour-agement to finish this research. ontents
Contents IList of Tables IVList of Figures VII1. Introduction 1
2. State of the Art 7
ONTENTS II
3. Evolutionary Algorithms with Regulated Genes: ReGen EAs 26
4. ReGen GA: Binary and Real Codification 41
ONTENTS
III4.3.1.2 Rosenbrock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1.3 Schwefel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1.4 Griewank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5. ReGen HAEA: Binary and Real Codification 82
6. Concluding Remarks 96
A. Examples of Individuals with Tags 101B. Standard and ReGen EAs Samples for Statistical Analysis 103Bibliography 113 ist of Tables
IST OF TABLES
V4.17 Results of the experiments for Generational and Steady replacements: Ras-trigin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.18 Results of the experiments for Generational and Steady replacements:Rosenbrock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.19 Results of the experiments for Generational and Steady replacements:Schwefel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.20 Results of the experiments for Generational and Steady replacements:Griewank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.21 Solutions found by different EAs on real functions . . . . . . . . . . . . . . . . . . 694.22 Anova Single Factor: SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.23 Anova Single Factor: ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.24 RAS Student T-tests pairwise comparisons with pooled standard deviation.Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . . . . . 754.25 RAS Student T-tests pairwise comparisons with pooled standard deviation.Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . . . . . 754.26 ROSE Student T-tests pairwise comparisons with pooled standard devia-tion. Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . 764.27 ROSE Student T-tests pairwise comparisons with pooled standard devia-tion. Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . 764.28 SCHW Student T-tests pairwise comparisons with pooled standard devia-tion. Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . 774.29 SCHW Student T-tests pairwise comparisons with pooled standard devia-tion. Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . 774.30 GRIE Student T-tests pairwise comparisons with pooled standard devia-tion. Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . 784.31 GRIE Student T-tests pairwise comparisons with pooled standard devia-tion. Benjamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . 785.1 Results of the experiments for Generational and Steady state replacements . 855.2 Real functions tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Results of the experiments for Generational and Steady state replacements . 875.4 Anova Single Factor: SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.5 Anova Single Factor: ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.6 Student T-tests pairwise comparisons with pooled standard deviation. Ben-jamini Hochberg (BH) as p-value adjustment method. . . . . . . . . . . . . . . . . 93A.1 Individual representation for Binary functions, D = 20. . . . . . . . . . . . . . . . 101A.2 Individual representation for Real functions, D = 2. . . . . . . . . . . . . . . . . . 102 IST OF TABLES
VIB.1 Deceptive Order Three Fitness Sampling: Ten Classic GAs and Ten ReGenGAs with different crossover rates and 30 runs. Best fitness value per run. . 104B.2 Deceptive Order Four Fitness Sampling: Ten Classic GAs and Ten ReGenGAs with different crossover rates and 30 runs. Best fitness value per run. . 105B.3 Royal Road Fitness Sampling: Ten Classic GAs and Ten ReGen GAs withdifferent crossover rates and 30 runs. Best fitness value per run. . . . . . . . . 106B.4 Max Ones Fitness Sampling: Ten Classic GAs and Ten ReGen GAs withdifferent crossover rates and 30 runs. Best fitness value per run. . . . . . . . . 107B.5 Rastrigin Fitness Sampling: Ten Classic GAs and Ten ReGen GAs withdifferent crossover rates and 30 runs. Best fitness value per run. . . . . . . . . 108B.6 Rosenbrock Fitness Sampling: Ten Classic GAs and Ten ReGen GAs withdifferent crossover rates and 30 runs. Best fitness value per run. . . . . . . . . 109B.7 Schwefel Fitness Sampling: Ten Classic GAs and Ten ReGen GAs withdifferent crossover rates and 30 runs. Best fitness value per run. . . . . . . . . 110B.8 Griewank Fitness Sampling: Ten Classic GAs and Ten ReGen GAs withdifferent crossover rates and 30 runs. Best fitness value per run. . . . . . . . . 111B.9 Fitness Sampling: Two standard
HaEa and Two ReGen
HaEa implemen-tations with 30 runs. Best fitness value per run. . . . . . . . . . . . . . . . . . . . . 112 ist of Figures
Marking function: a) shows a chromosomewith no tags on it; b) depicts the addition of four tags to a chromosome; c)illustrates tags’ bit modification in red; and d) presents a chromosome withtwo removed tags. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 General representation of an individual with its epigenotype. The bottomsection shows the tag’s interpretation process to generate a bit string usedto build the individual’s phenotype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Illustrative example of genetic and epigenetic recombination: Simple PointCrossover operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Deceptive Order 3. Generational replacement (GGA) and Steady Statereplacement (SSGA). From top to bottom, crossover rates from 0 . .
0. . 464.2 Deceptive Order 4. Generational replacement (GGA) and Steady Statereplacement (SSGA). From top to bottom, crossover rates from 0 . .
0. . 474.3 Royal Road. Generational replacement (GGA) and Steady State replace-ment (SSGA). From top to bottom, crossover rates from 0 . .
0. . . . . . . 484.4 Max Ones. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . .
0. . . . . . . . . . . 494.5 From top to bottom: Deceptive Order Three and Deceptive Order FourTrap Functions. On the left, EAs with Generational replacement (GGA)and Steady State replacement (SSGA) with Crossover rates from 0 . . . .
0. On thebottom, EAs grouped by Generational replacement (GGA) and SteadyState replacement (SSGA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Rastrigin. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . .
0. . . . . . . . . . . 64VII
IST OF FIGURES
VIII4.8 Rosenbrock. Generational replacement (GGA) and Steady State replace-ment (SSGA). From top to bottom, crossover rates from 0 . .
0. . . . . . . 654.9 Schwefel. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . .
0. . . . . . . . . . . 664.10 Griewank. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . .
0. . . . . . . . . . . 674.11 From top to bottom: Rastrigin and Rosenbrock Functions. On the left,EAs with Generational replacement (GGA) and Steady State replacement(SSGA) with Crossover rates from 0 . .
0. On the right, EAs groupedby Generational replacement (GGA) and Steady State replacement (SSGA). 724.12 From top to bottom: Schwefel and Griewank Functions. On the left,EAs with Generational replacement (GGA) and Steady State replacement(SSGA) with Crossover rates from 0 . .
0. On the right, EAs groupedby Generational replacement (GGA) and Steady State replacement (SSGA). 735.1 Deceptive Order 3. Generational replacement (G
HaEa ) and Steady statereplacement (SS
HaEa ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Deceptive Order 4. Generational replacement (G
HaEa ) and Steady statereplacement (SS
HaEa ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 Rastrigin. Generational replacement (G
HaEa ) and Steady state replace-ment (SS
HaEa ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.4 Schwefel. Generational replacement (G
HaEa ) and Steady state replace-ment (SS
HaEa ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.5 Griewank. Generational replacement (G
HaEa ) and Steady state replace-ment (SS
HaEa ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 EAs with Generational replacement (G
HaEa ) and Steady State replace-ment (SS
HaEa ). On top: Deceptive Order Three and Deceptive OrderFour Trap Functions. On the bottom: Rastrigin and Schwefel functions. . . . 915.7 EAs with Generational replacement (G
HaEa ) and Steady State replace-ment (SS
HaEa ). Griewank function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
HAPTER Introduction
Optimization is a common task in people’s lives. Investors use passive investment strate-gies that avoid excessive risk while obtaining great benefits. A conventional applicationof calculus is calculating a function minimum or maximum value. Manufacturers strivefor the maximum efficiency of their production procedures. Companies lessen productioncosts or maximize revenue, for example, by reducing the amount of material used to packa product with a particular size without detriment of quality. Software Engineers designapplications to improve the management of companies’ processes. The school bus routethat picks a group of students up from their homes to the school and vice-versa, everyschool day, must take into account distances between homes and time. Optimization isan essential process, is present in many activities, contributes to decision science, and isrelevant to the analysis of physical systems.Making use of the optimization process requires identifying some objective, a quantita-tive measure of the system’s performance under consideration. The objective can be profit,time, energy, or any resource numerically represented; the objective depends on problemcharacteristics, named as variables or unknowns. The purpose is to obtain variables valuesthat optimize the objective; variables may present constraints or restrictions, for example,quantities such as the distance between two points and the interest rate on loan must bepositive. The process of identifying objectives, variables, and constraints for a problem isknown as modeling. The first step in the optimization process is to build an appropriatemodel. Once the model is formulated, an optimizer (a problem-solving strategy for solvingoptimization problems, such as equations, analytic solvers, algorithms, among others) canbe implemented to find a satisfactory solution. There is no unique optimization solverbut a set of optimizers, each of which is related to a particular optimization problemtype. Picking a suitable optimizer for a specific problem is fundamental, it may determinewhether the problem is tractable or not and whether it finds the solution [12, 40, 53].1
HAPTER 1. INTRODUCTION sensitivity analysis technique that analyzes the influence of oneparameter on the cost function at a time and exposes the solution susceptibility to changesin the model and data. Interpretation in terms of the applicability of obtained solutionsmay recommend ways in which the model can be refined or corrected. The optimizationprocess is repeated if changes are introduced to the model [12, 40, 53].An optimization algorithm is a method that iteratively executes and compares severalsolutions until it finds an optimum or satisfactory solution. Two optimization algorithmstypes widely used today are deterministic and stochastic. Deterministic algorithms donot involve randomness; these algorithms require well-defined rules and steps to find asolution. In contrast, stochastic algorithms comprise in their nature probabilistic trans-lation rules [12]. The use of randomness might enable the method to escape from localoptima and subsequently reach a global optimum. Indeed, this principle of randomizationis an effective way to design algorithms that perform consistently well across multiple datasets, for many problem types [12, 40]. Evolutionary algorithms are a kind of stochasticoptimization methods.Evolutionary Algorithms (EAs) are a subset of population-based, metaheuristic opti-mization algorithms of the Evolutionary Computation field, which describes mechanismsinspired by natural evolution, the process that drives biological evolution. There aremany types of evolutionary algorithms, the most widely known: Genetic Algorithm (GA),Genetic Programming (GP), Evolutionary Strategies (ES), and Hybrid Evolutionary Al-gorithms (HEAs). The common underlying idea behind all EAs is the same, an initial pop-ulation of individuals, a parent selection process that considerate the aptitude of each in-dividual, and a transformation process that allows the creation of new individuals throughcrossing and mutation. Candidate solutions act like the population’s individuals for anoptimization problem, and the fitness function determines the quality of solutions. Thepopulation’s evolution then occurs after the repeated application of the above mechanisms[12, 21, 33, 40].Many computer scientists have been interested in understanding the phenomenon ofadaptation as it occurs in nature and developing ways in which natural adaptation mecha-nisms might be brought into computational methods. Current evolutionary algorithms aresuitable for some of the most important computational problems in many areas, for exam-ple, linear programming problems (manufacturing and transportation), convex problems(communications and networks), complementarity problems (economics and engineering),and combinatorial problems (mathematics) such as the traveling salesman problem (TSP),the minimum spanning tree problem (MST), and the knapsack problem [21, 46]. However,some computational problems involve searching through a large number of solution pos-
HAPTER 1. INTRODUCTION
HAPTER 1. INTRODUCTION P generation caused by the environment might also be carried over into the P generation or beyond, leading to a kind of long term memory [73].Epigenetics drives modern technological advances, also challenges and reconsiders con-ventional paradigms of evolution and biology. Due to recent epigenetic discoveries, earlyfindings on genetics are being explored in different directions. Both genetics and epige-netics help to better understand functions and relations that DNA, RNA, proteins, andthe environment produce regarding heritage and health conditions. Epigenetics will notonly help to understand complex processes related to embryology, imprinting, cellulardifferentiation, aging, gene regulation, and diseases but also study therapeutic methods.The incorporation of epigenetic elements in EAs allows robustness, a beneficial feature inchanging environments where learning and adaptation are required along the evolutionaryprocess [61, 26, 52, 73]. Adaptation is a crucial evolutionary process that adjusts thefitness of traits and species to become better suited for survival in specific environments[37].Epigenetic mechanisms like DNA Methylation and Histone Modification are vital mem-ory process regulators. Their ability to dynamically control gene transcription in responseto environmental factors promotes prolonged memory formation. The consistent and self-propagating nature of these mechanisms, especially DNA methylation, implies a molecularmechanism for memory preservation. Learning and memory are seen as permanent changesof behavior generated in response to a temporary environmental input [76]. Organisms’ability to permanently adapt their behavior in response to environmental stimulus relieson functional phenotypic plasticity [19]. Epigenetic mechanisms intervene in biologicalprocesses such as phenotype plasticity, memory formation between generations, and epi-genetic modification of behavior influenced by the environment. The previous fact leadsresearchers to improve evolutionary algorithms’ performance in solving hard mathematicalproblems or real-world problems with continuous environmental changes by contemplatingthe usage of epigenetic mechanisms [61]. This approach is inspired by the epigenetic regulation process, a biological mechanismwidely studied by the Epigenetic field. This thesis aims to present a technique that modelsthe adaptive and learning principles of epigenetics. The dynamics of DNA Methylationand Histone Modification has been summarized into five fundamental elements: first,a metaphorical representation of
Epigenetic tags ; second, a structural layer above thechromosome structure used to bind tags (
Epigenotype ); third, a marker (
Marking Function )that comprises three actions: add, modify, and remove tags located on alleles; fourth, atags interpreter or decoder (
Epigenetic Growing function ); and fifth, an inheritance process
HAPTER 1. INTRODUCTION Crossover Operator ) to pass such tags on to the offspring. So that, the technique mayreflect the adaptability of individuals during evolution, the ability of individuals to learnfrom the environment, the possibility of reducing time in the search process, and thediscovery of better solutions in a given time.The epigenetic mechanisms DNA Methylation and Histone Modification have beencharacterized to abstract principles and behaviors (from the epigenotype, tags, marking,reading, and inheritance biological elements) for the metaphorical designing of epigeneticcomponents. In this thesis, the
Epigenotype structure represents individuals’ second layer,where tags are attached. The designed technique takes advantage of such a layer to influ-ence the direction of the search process. The
Epigenotype is made up of tags,
EpigeneticTags are represented with a binary string sequence of 0’s and 1’s, a tag implies a ruleto interpret segments (alleles) of an individual’s genome. The tags contain two sections;the first section denotes an operation, that is, a binary operation that operates on anindividual’s chromosome; the second section of the tags contains the gene size. The genesize indicates the number of alleles on which operates a binary operation. Tags determineindividuals’ gene expression; in other words, how alleles will be expressed, whether 1 or 0,depending on the operation applied to them.The
Marking Function involves writing, modifying, and erasing tags based on ametaphorical representation from writers, erasers, and maintenance enzymes. These ac-tions act with a distributed probability of being applied to a single allele of a chromosome.Also, these epigenetic changes are framed into marking periods; such periods represent theenvironment, an abstract element that has been a point of reference to assess results ofthis technique. This mechanism allows performing the marking process in defined periodsduring the evolution process. The
Epigenetic Growing Function represents the behavior ofreader enzymes to interpret the epigenetic code (tags) on genotypes and then build phe-notypes. The
Epigenetic Growing Function plays the role of tags decoder or interpreter.After scanning the original individual’s genome with its equivalent epigenotype (tags) andapplying the operations to a copy of the chromosome, the
Epigenetic Growing Function generates a resulting bit string to build phenotypes that are evaluated and scored. The
Crossover Operator keeps its essence, but now, it includes tags as part of the exchangeof genetic and epigenetic material between two chromosomes to create progeny. Theseepigenetic components are part of the proposed technique for the evolutionary algorithms’framework.The epigenetic technique is implemented in classic Genetic Algorithms (GAs) andstandard versions of
HaEa (Hybrid Adaptive Evolutionary Algorithm, designed to adaptgenetic operators rates while solving a problem [30, 31]). The epigenetic components de-scribed previously are embedded in the algorithms’ logic. The epigenetic implementationsare named ReGen GA (GA with regulated genes) and ReGen
HaEa ( HaEa with reg-ulated genes). Finally, the validation of the technique is made by comparing GA and
HaEa classic versions versus epigenetic implementations of GA and
HaEa , through a setof experiments/benchmarks to determine the proposed approach applicability. The com-
HAPTER 1. INTRODUCTION
Chapter 2 presents state of the art. An overview of optimization, evolutionary algorithmswith an emphasis on genetic algorithms, and hybrid evolutionary algorithms. The chapteralso describes the biological basis of this research and a brief review of related work in theliterature.Chapter 3 explains the proposed approach in detail. The chapter includes
Tags en-coding, selected operations, gene sizes,
Epigenotype representation, the
Marking Function ,the
Epigenetic Growing Function , the
Crossover Operator , and a generic evolutionaryalgorithm pseudocode with the epigenetic components of the proposed technique.Chapter 4 introduces the implementation of the epigenetic technique on Genetic Algo-rithms. This chapter reports results on selected experimental functions with binary andreal encoding for determining the model’s applicability. Additionally, the chapter presentsthe analysis and discussion of results.Chapter 5 presents the implementation of the epigenetic technique on
HaEa . The
HaEa variations use two genetic operators: single point Crossover (enhanced to includetags) and single bit Mutation. Experimental functions with binary and real encoding arerepresented along with the analysis and discussion of results.Chapter 6 brings this book to a close with a short recapitalization and future researchdirections of this thesis.Appendix A exhibits an example of individuals representation. The appendix includesindividuals with a marked genotype in binary representation and its phenotypic inter-pretation. In Appendix B standard and ReGen EAs Samples for statistical analysis areincluded.
HAPTER State of the Art
Optimization is a significant paradigm with extensive applications. In many engineeringand industrial tasks, some processes require optimization to minimize cost and energyconsumption or maximize profit, production, performance, and process efficiency. Theeffective use of available resources requires a transformation in scientific thinking. Thefact that most real-world tasks have much more complicated circumstances and variablesto change systems’ behavior may help in switching current reasoning. Optimization ismuch more meaningful in practice since resources and time are always limited. Threecomponents of an optimization process are modeling, the use of specific problem-solvingstrategies (optimizer), and a simulator [12, 40, 53].A problem can be represented by using mathematical equations that can be trans-formed into a numerical model and be numerically solved. This phase must ensure thatthe right numerical schemes for discrete or continuous optimization are used. Anotherfundamental step is to implement the right algorithm or optimizer to find an optimalcombination of design variables. A vital optimization capability is to produce or searchfor new solutions from previous solutions, which leads to the search process convergence.The final aim of a search process is to discover solutions that converge at the global opti-mum, even though this can be hard to achieve. In terms of computing time and cost, themost crucial step is using an efficient evaluator or simulator. In most cases, an optimiza-tion process often involves evaluating an objective function, which will verify if a proposedsolution is feasible [12, 40, 53].Optimization includes problem-solving strategies in which randomness is present inthe search procedure (Stochastic) or mechanical rules without any random nature (De-terministic). Deterministic algorithms work in a mechanically and predetermined mannerwithout any arbitrary context. For such an algorithm, it reaches the same final solution ifit starts with the same state. Oppositely, if there is some randomness in the algorithm, itusually reaches a different output every time the algorithm runs, even though it starts with7
HAPTER 2. STATE OF THE ART soft,hard, and wet ) to represent such systems [4, 59]. A characteristic of computing inspired bynature is the metaphorical use of concepts, principles, and biological mechanisms. ALifeconcentrates on complex systems that involve life, adaptation, and learning. By creat-ing new types of life-like phenomena, artificial life continually challenges researchers toreview and think over what it is to be alive, adaptive, intelligent, creative, and whetherit is possible to represent such phenomena. Besides, ALife aims to capture the simpleessence of vital processes, abstracting away as many details of living systems or biologicalmechanisms as possible [4].An example of this is the evolution process by natural selection, a central idea inbiology. Biological evolution is the change in acquired traits over succeeding generations oforganisms. The alteration of traits arises when variations are introduced into a populationby gene mutation, genetic recombination, or erased by natural selection or genetic drifts.Adaptation is a crucial evolutionary process in which traits and species’ fitness adjust forbeing better suited for survival in specific environments. The environment acts to promoteevolutionary change through shifts in development [37]. The evolution of artificial systemsis an essential element of ALife, facilitating valuable modeling tools and automated designmethods [48]. Evolutionary Algorithms are used as tools to solve real problems and asscientific models of the evolutionary process. They have been applied to a large variety ofoptimization tasks, including transportation problems, manufacturing, networks, as wellas numerical optimization [21, 48]. However, the search for optimal solutions to someproblem is not the only application of EAs; their nature as flexible and adaptive methods
HAPTER 2. STATE OF THE ART
Evolutionary Algorithms (EAs) are a subset of population-based, metaheuristic optimiza-tion algorithms of the Evolutionary Computation field, which uses mechanisms inspiredby natural evolution, the process that drives biological evolution. There are many sortsof evolutionary algorithms, the most widely known: Genetic Algorithm (GA), GeneticProgramming (GP), Evolutionary Strategies (ES), and Hybrid Evolutionary Algorithms(HEA). The general fundamental idea behind all these EAs is the same; given a populationof individuals within some environment with limited resources, only the fittest survive. Todefine a particular EA, there are some components, or operators that need to be specified.The most important are: the representation of individuals, an evaluation (fitness) func-tion, an initial population of individuals, a parent selection process that considerate theaptitude of each individual, a transformation process that allows the creation of new indi-viduals through crossing and mutation, and a survivor selection mechanism (replacement)[12, 21, 33, 40].
GAs are adaptive heuristic search computational methods based on genetics and the pro-cess that drives biological evolution, which is natural selection [21]. Holland [33] pre-sented the GA as the biological evolution process abstraction and formulated a theoryabout adaptation. Holland intended to understand adaptation and discover alternativesin which natural adaptation mechanisms might be brought into computer methods. Themost used EA to solve constrained and unconstrained optimization problems is the tradi-tional GA [12, 33, 40], also today, the most prominent and widely evolution models usedin artificial life systems. They have been implemented as tools for solving scientific modelsof evolutionary processes and real problems [48].A Genetic Algorithm explores through a space of chromosomes, and each chromosomedenotes a candidate solution to a particular problem. Bit strings usually represent chro-mosomes in a GA population; each bit position (locus) in the chromosome has one outof two possible values (alleles), 0 and 1. These concepts are analogically brought frombiology, but GAs use a simpler abstraction of those biological elements [46, 47]. The mostimportant elements in defining a GA are the encoding scheme (hugely depends on theproblem), an initial population, a parent selection mechanism, variation operators suchas recombination, mutation, and a replacement mechanism [21, 47]. The GA often re-quires a fitness objective function that assigns a score to each chromosome in the currentpopulation [46, 47]. Once an optimization problem has been set up, the search processtakes place by evaluating the population of individuals during several iterations. In the
HAPTER 2. STATE OF THE ART { } ). The geneticalgorithms obey to a population evolution model where the fittest survive [33]. The binary encoding uses the binary digit, or a bit, as the fundamental unit of information,there are only two possibilities 0 or 1. The genotype simply consists of a string or vectorof ones and zeroes. For a particular problem, it is important to decide the string’s lengthand how it will be interpreted to produce a phenotype. When deciding the genotype tophenotype mapping for a problem, it is essential to ensure the encoding allows all possiblebit strings to express a valid solution to a given problem [21].
Real numbers represent any quantity along a number line. Because reals lie on a numberline, their size is comparable. One real can be greater or less than another and used onarithmetic operations. Real numbers ( R ) include the rational numbers ( Q ), which includethe integers ( Z ), which include the natural numbers ( N ). Examples: 3 . , − . , , / , − k genes is a vector ( x , ..., x k ) with x i ∈ R [21]. Hybridization of evolutionary algorithms is growing in the EA community due to theircapabilities in handling several real-world problems, including complexity, changing en-vironments, imprecision, uncertainty, and ambiguity. For diverse problems, a standardevolutionary algorithm might be efficient in finding solutions. As stated in the literature,standard evolutionary algorithms may fail to obtain optimal solutions for many types ofproblems. The above exposes the need for creating hybrid EAs, mixed with other heuris-tics. Some of the possible motives for hybridization include performance improvement ofevolutionary algorithms (example: speed of convergence), quality enhancement of solu-tions obtained by evolutionary algorithms, and to include evolutionary algorithms as partof a larger system [21, 32].There are many ways of mixing techniques or strategies from population initializationto offspring generation. Populations may be initialized by consolidating previous solutions,using heuristics, or local search, among others. Local search methods may be includedwithin initial population members or among the offspring. EAs Hybridation may involve
HAPTER 2. STATE OF THE ART
How living beings (particularly humans) respond to their environment is determined byinheritance, and the different experiences lived during development. Inheritance is tra-ditionally viewed as the transfer of variations in DNA (Deoxyribonucleic Acid) sequencefrom parent to child. However, another possibility to consider in the gene-environmentinteraction is the trans-generational response. This response requires a mechanism totransmit environmental exposure information that alters the gene expression of the nextgenerations [55].Two examples of trans-generational effects were found in ¨Overkalix, a remote townin northern Sweden, and the Netherlands. The study conducted in ¨Overkalix with threegenerations born in 1890, 1905, and 1920 revealed that the high or low availability offood for paternal grandfathers and fathers (during childhood or their slow growth period)influenced the risk of cardiovascular disease and diabetes mellitus mortality in their malechildren and grandchildren [38, 39, 55]. On the other hand, the study carried out on agroup of people in gestation and childhood during the period of famine experienced be-tween the winter of 1944 and 1945 in the Netherlands, evidenced that people with lowbirth weight, developed with higher probability, health problems such as diabetes, hyper-tension, obesity or cardiovascular disease during their adult life. The research concludesthat famine during gestation and childhood has life-long effects on health. Such effectsvary depending on the timing of exposure and the evolution of the recovery period [41].In this sense, gene expression can be affected in such a way that it reflects habits thatshape an individual’s lifestyle, even the “experiences” of a generation might be passeddown to progeny that have not necessarily lived in similar conditions to their parents.It is in this context that epigenetics, area that studies the modifications that affect gene
HAPTER 2. STATE OF THE ART
HAPTER 2. STATE OF THE ART
Humans have 23 pairs of chromosomes in each body cell; each pair has one chromosomefrom the mother and another from the father. A chromosome is composed of DNA andproteins. The DNA consists of two long chains made up of nucleotides, on which thousandsof genes are encoded. The complete set of genes in an organism is known as its genome.The DNA is spooled around proteins called histones. Both the DNA and the Histonesare marked with chemical tags, also known as epigenetic tags. The histones and theepigenetic tags form a second structural layer that is called the epigenome. The epigenome(epigenotype) comprises all chemical tags adhered to the entire DNA and Histones as away to regulate the genes’ activity (gene expression) within the genome. The biologicalmechanisms that involve attaching epigenetic tags to or removing them from the genomeare known as epigenetic changes or epigenetic mechanisms [26, 52, 73].
DNA methylation mechanism conducts the addition or elimination of methyl groups(
CH3 ), predominantly where cytosine bases consecutively occur [75]. In other words,chemical markers called methyl groups are bound to cytosines at
CpG sites in DNA.Methyl groups silence genes by disrupting the interactions between DNA and the proteinsthat regulate it [50]. Genome regions that have a high density of
CpGs are known as
CpGislands , and DNA methylation of these islands leads to transcriptional repression [28].Methylation is sparsely found but globally spread in indefinite
CpG sequences through-out the entire genome, except for
CpG islands , or specific stretches (approximately onekilobase in length) where high
CpG contents are found. These methylated sequences candrive to improper gene silencing, such as tumor suppressor genes’ silencing in cancer cells.Studies confirm that methylation close to gene promoters differs considerably among celltypes, with methylated promoters associated with low or no transcription [58].
HAPTER 2. STATE OF THE ART Figure 2.1.
Epigenetic Mechanisms.
HAPTER 2. STATE OF THE ART
Dnmt3a and
Dnmt3b methyltrans-ferases. These enzymes catalyze a methyl group’s attachment to the cytosine DNA baseon the fifth carbon ( C5 ). DNA methylation maintenance is preserved when cells divide,and it is carried out by Dnmt1 enzyme. Together, the mentioned enzymes guarantee thatDNA methylation markers are fixed and passed onto succeeding cellular generations. Inthis way, DNA methylation is a cellular memory mechanism that transmits essential geneexpression programming data along with it [9].
Histone modification is a covalent posttranslational change to histone proteins, whichincludes methylation, acetylation, phosphorylation, ubiquitylation, and sumoylation. Allthese changes influence the DNA transcription process. Histone Acetyltransferases, forexmple, are responsible for Histone Acetylation; these enzymes attach acetyl groups tolysine residues on Histone tails. In contrast, Histone Deacetylases (
HDACs ) remove acetylgroups from acetylated lysines. Usually, the presence of acetylated lysine on Histonetails leads to an accessible chromatin state that promotes transcriptional activation ofselected genes; oppositely, lysine residues deacetylation conducts to restricted chromatinand transcriptional inactivation [27].The DNA is indirectly affected, DNA in cells is wounded around proteins called Hi-stones, which form reel-like structures, allowing DNA molecules to stay ordered in theform of chromosomes within the cell’s nucleus as depicted in Fig 2.1. When Histones havechemical labels, other proteins in cells detect these markers and determine if the DNAregion is accessible or ignored in a particular cell [50].
Epigenetic regulation comprises the mechanisms by which epigenetic changes such asmethylation, acetylation, and others can impact phenotype. Regulatory proteins con-duct the epigenetic regulation process. These proteins have two main functions; the firstinvolves switching specific genes on or off (gene activation); the second is related to therecruitment of enzymes that add, read or remove epigenetic tags from genes [73].
Gene regulatory proteins recruit enzymes to add, read, and remove epigenetic tags; theseprocesses are performed on the DNA, the Histones, or both, as explained previously [73].These enzymes are seen as epigenetic tools, a family of epigenetic proteins known as read-ers, writers, and erasers [7]. Epigenetics involves a highly complex and dynamically re-versible set of structural modifications to DNA and histone proteins at a molecular level;
HAPTER 2. STATE OF THE ART
Writ-ers add to DNA or Histones chemical units ranging from a single methyl group to ubiquitinproteins. For example, DNA methyltransferases (
DNMTs ) are responsible for introduc-ing the
C5-methylation on CpG dinucleotide sequences. Such molecular structures notonly influence the relation between DNA and histone proteins but also recruit non-codingRNAs ( ncRNAs ) and chromatin remodellers. On the other hand, the specialized domain-containing proteins that recognize and interpret those modifications are
Readers ; the bind-ing interactions recognize through so-called reader modules specific modification codes ormarks within the chemically modified nucleic acids and proteins and then perform con-formational changes in chromatins and provide signals to control chromatin dynamics.Finally,
Erasers , a dedicated type of enzyme expert in removing chemical markers, guar-antee a reversible process. In order to achieve that, a group of eraser enzymes catalyzesthe removal of the written information, ensuring a balanced and dynamic process [7, 9, 73].
As explained above, epigenetics means “upon”, “above” or “over” genetics. Epigeneticsdescribes a type of chemical reaction resulting from epigenetic modifications that alterDNA’s physical structure without altering its nucleotide sequence. These epigenetic mod-ifications cause genes to be more or less accessible during the transcription process. Ingeneral, environmental conditions influence the interactions and chemical reactions of theepigenotype, which can mark genes with specific chemical labels that direct actions suchas gene silencing, activation or repression of a gene (activity), which translates into amodification in its function [26, 52, 73].Epigenetic mechanisms, in particular, Histone and DNA modifications, go beyond theidea of switching genes off and on. Gene silencing refers to a mechanism where large re-gions of a chromosome become transcriptionally inactive due to the compact wrapping ofhistone proteins that restricts the activity of DNA polymerases, which situate nucleotideunits into a new chain of nucleic acid [62]. DNA regions that are highly packed are knownto be part of the heterochromatin structure. In contrast, DNA relatively broadened formwhat is known as euchromatin. For a long time, it was assumed that heterochromatinis transcriptionally deedless compared to euchromatin. Nevertheless, many recent studieshave questioned this conception of transcriptionally silent heterochromatin [63]. Thosestudies indicate that the concept of equivalence between open chromatin with active tran-scription and compact chromatin with inactive transcription is not always applicable to
HAPTER 2. STATE OF THE ART
HATs ) allows the H3 and H4 histone tails acetylation. This mechanism promotes interactions between DNA and his-tones. The result is a relaxed structure surrounding the core promoter that is available tothe general transcription process. Activator proteins interact with the general transcrip-tion factors to intensify DNA binding and initiation of transcription. The earlier meansthat activator proteins’ recruitment helps raise the transcription rate, leading to geneactivation [16]. Methylation is related to both gene activation and repression; and eachmechanism depends on the degree of methylation. Inactive genes or silent chromosomeregions are highly methylated in their CpG islands compared with the same gene on theactive chromosome [23, 63].There are other important considerations around the expression of genes. Gene regula-tion mechanisms (silencing, repression, activation) depend mostly on the cell’s epigeneticcondition, which controls the gene expression timing and degree at a specific time [63].Silencing gene expression is not just about switching chromatin areas entirely off, or generepression fully suppressing a gene function. The dynamics of these mechanisms alsoinvolve decreasing the level of transcription by gradually reducing gene expression, de-pending on tags bind location or regions, and how many tag groups are attached. So, it ispossible to evidence sections of the chromosome where the gene expression is not totallyinactivated but strongly reduced. In the same way, active genes and regions with expres-sion levels moderated. The binding of proteins to particular DNA elements or regulatoryregions to control transcription and mechanisms that modulate translation of mRNA mayalso be moderated [13, 16, 23, 63].
Today epigenetic modifications such as DNA methylation and histone tail modificationsare known as essential regulators in the consolidation and propagation of memory. These
HAPTER 2. STATE OF THE ART
Nowadays, epigenetics is known not only because of its relevance for medicine, farming, andspecies preservation, but also because studies have revealed the importance of epigeneticmechanisms in inheritance and evolution. Particularly, evidencing epigenetic inheritancein systems where non-DNA alterations are transmitted in cells. Also, the organism di-versity broadens the heredity theory and defy the current gene-centered neo-Darwinianversion of Darwinism [36]. Epigenetics as science does not intend to oppose early ideasof evolutionary theory. In fact, some authors suggest considering modern epigeneticsas neo-Lamarckian [56] or close to the original argument proposed by Baldwin (knownas Baldwin effect) [69]. Early authors were undergoing studies that are expanding theknowledge about inheritance and evolution. Currently, the epigenetics research commu-
HAPTER 2. STATE OF THE ART
HAPTER 2. STATE OF THE ART There is a predominant focus in the literature on the in-depth study of epigenetic mecha-nisms, especially those that may be associated with the diagnosis, prevention, and treat-ment of diseases with apparently less emphasis on what mechanisms do at the phenotypiclevel of an individual, particularly between generations. Models/Strategies focused onepigenetic changes occurring during one generation’s life span or transmitted throughgenerations, or at an individual level have been the target to identify the most recentachievements around this topic in the evolutionary algorithms community. Those modelshave been developed with different approaches. Several authors have worked on hybridstrategies to improve the solution capacity of population-based global search methods, sothe adaptive behavior of populations can be rapidly manifested under selective pressure.Such strategies aim to address a wider variety of computational problems by mimicking bi-ological mechanisms or social changes. Below, some approaches are briefly described; theyentail adaptation and learning behaviors, two characteristics that this thesis is studying.Dipankar Dasgupta et al. (1993) [17] introduce the structured Genetic Algorithm(sGA). Though this strategy does not mention epigenetic mechanisms, it involves geneactivation, an essential mechanism in gene regulation to control genes states: repression(different from silencing) and expression. This genetic model includes redundant geneticmaterial and a gene activation mechanism that utilizes a multi-layered structure (hierar-chical) for the chromosome. Each gene in higher levels acts as a switchable pointer thatactivates or deactivates sets of lower-level genes. At the evaluation stage, only the activegenes of an individual are translated into phenotypic functionality. It also includes a long-term distributed memory within the population enabling adaptation in non-stationaryenvironments. Its main disadvantage is the use of a multi-level representation with op-tional search spaces that could be activated at the same time, leading to express a bitstring that may be too long for the problem solution.Tanev and Yuta (2008) [71] describe the first model mentioning Epigenetics in the EAcommunity. In this model, they focus on an improved predator-prey pursuit problem.They present individuals with double cell, comprising somatic cell and germ cell, bothwith their respective chromatin granules. In the simulation, they use the Modificationof Histones to evidence the role this mechanism plays in regulating gene expression andmemory (epigenetic learning, EL). The Genetic Programming Algorithm defines a set ofstimulus-response rules to model the reactive behavior of predator agents. The beneficialeffect of EL on GP’s performance characteristics is verified on the evolution of predatoragents’ social behavior. They report that EL helps to double improve the computationalperformance of the implemented GP. Additionally, the simulation evidences the pheno-typic variety of genotypically similar individuals and their preservation from the negativeeffects of genetic recombination. Phenotypic preservation is achieved by silencing partic-ular genotypic regions and turning them on when the probability of expressing beneficialphenotypic traits is high.
HAPTER 2. STATE OF THE ART
HAPTER 2. STATE OF THE ART
HAPTER 2. STATE OF THE ART
HAPTER 2. STATE OF THE ART
The Selfish Gene book. A ‘meme’ denotes the ideaof a unit of imitation in cultural evolution, which in some aspects, is analogous to thegene in GAs. Examples of memes are tunes, ideas, catch-phrases, clothes fashions, ways ofmaking pots, food, music, or ways of building arches [18]. The MAs extend the notion ofmemes to cover conceptual entities of knowledge-enhanced procedures or representations.The MA combines the population-based global search and the local search heuristic madeby each individual, capable of performing local refinements without genetic representationconstraints. The earlier may represent a high computational cost due to the separatedindividual learning process or local improvement for a problem search. Moscato coinedthe name ‘Memetic Algorithm’ to cover a wide range of techniques where the evolutionarysearch is extended by adding one or more phases of local search, or the use of problem-specific information.
HAPTER 2. STATE OF THE ART
HAPTER Evolutionary Algorithms with Regulated Genes:ReGen EAs
The previous chapter described optimization and evolutionary processes as inspiration todesign problem solvers; also, an epigenetics overview, the relation between epigeneticsand evolution, memory and adaptation from epigenetics point of view, and the differentapproaches that implemented Epigenetics into Evolutionary Algorithms.State of the art shows that epigenetic mechanisms play a fundamental role in biologicalprocesses. Some of such processes are phenotype plasticity, memory consolidation withingenerations, and environmentally influenced epigenetic modifications. The earlier leadsresearchers to think about applying epigenetic mechanisms to enhance evolutionary algo-rithms performance in solving hard mathematical problems or real-world problems withcontinuous environmental changes [61].This approach is not supported on the main idea of switching genes off and on (geneactivation mechanism), or silencing chromosome sections like most of the approaches pre-viously described. Epigenetic mechanisms, in particular, Histone and DNA modifications,go beyond the idea of activating and deactivating genes. As mentioned in state of the art,these mechanisms also involve decreasing or promoting the level of transcription by gradu-ally reducing or increasing expression, depending on tags bind location or regions, and howmany tag groups are attached [13, 16, 23, 63]. Methylation, for example, is sparsely found,but globally spread in indefinite
CpG sequences throughout the entire genome, except for
CpG islands , or specific stretches where high
CpG contents are found [58].Based on the preceding, this thesis assumes individuals’ chromosomes to be entirelyactive; that is to say, epigenetic states on/off do not restrict gene/allele expression. Indi-viduals’ genotype is regulated by designed epigenetic tags that encode different meaningsfrom on and off states. Tags encode rules with specific operations to be applied to the26
HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS
Epigenetic tags that are not off/on states, instead they represent read-ing rules to interpret sections (alleles) of an individual’s genome. Second, a structurallayer above the chromosome structure used to bind tags (
Epigenotype ). Third, a marker(
Marking Function ) to add, remove and modify tags, this process is performed betweendefined marking periods, simulating periods where individuals’ genetic codes are affectedby external factors (as seen in study cases of ¨Overkalix and Dutch famine). Fourth, a tagsinterpreter or decoder (
Epigenetic Growing Function ) to generate individuals’ phenotypesfrom their epigenotype-genotype structures. The marker and decoder are based on differ-
HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS
Crossover Operator ) to passsuch tags onto the offspring (transgenerational non-genetic inheritance over subsequentgenerations).The purpose of this chapter is to introduce the proposed technique to design epigeneticevolutionary algorithms with binary encoding. The epigenetic components of the modelare described as follows: section 3.1 shows a detailed description of
Tags’ representation inthe model; section 3.2 briefly characterizes the
Epigenotype ; section 3.3 explains the
Mark-ing process ; section 3.4 describes the
Epigenetic Growing function ; section 3.5 illustratesadjustments on a
Crossover operator to inherit tags in succeeding generations; and at theend, the Pseudocode of the epigenetic EA and a summary of this chapter are presented insections 3.6 and 3.7 respectively.
Epigenetic tags in the ReGen EA, are represented with a binary string sequence of 0’sand 1’s, and are located on alleles. Each set of tags is built with -bits, the first threebits represent a bit operation (Circular shift, Transpose, Set to, Do nothing, Right shift byone, Add one, Left shift by one, and Subtract one) and the last -bits represent the genesize. Note that, the decimal representation of the -bits from to is used withno changes, but for the decimal value is thirty two. The first 3-bits string sequenceuses a one-to-one mapping to a rule that performs a simple bit operation on chromosomes.The gene size says the number of alleles that are involved in the bit operation. Fig. 3.2shows the tags’ representation in the ReGen EA. Eight operations have been defined according to the -bits binary strings depicted inFig. 3.2. Each combination maps to a simple bit operation to be applied on a copy ofthe chromosome. The operations only impact the way alleles are read when evaluatingthe entire chromosome. An operation can be applied in a way that can affect a specificnumber of alleles/bits in later positions based on the -bits binary string that denotes thegene size l . The Bit Operations are described as follows: ( ). Circularly shifts a specified number of bits to the right: starting at the markedbit up to l bits ahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bitmarked with the shift tag and l the gene size encoded by the tag. If the mark is read bythe decoder, the decoded bit string y will have y k + i = x k + i − for all i = 1 , ..., l − y k = x k + l − . HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS ( ). Transposes a specified number of bits: starting at the marked bit up to l bitsahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bit marked withthe transposition tag and l the gene size encoded by the tag. If the mark is read by thedecoder, the decoded bit string y will have y k + i = x k + l − − i for all i = 0 , , ..., l − ( ). Sets a specified number of bits to a given value, the value of the marked bit: startingat the marked bit up to l bits ahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bit marked with the set-to tag and l the gene size encoded by the tag. If the markis read by the decoder, the decoded bit string y will have y k + i = x k for all i = 0 , , ..., l − ( ). Does not apply any operation to a specified number of bits: starting at the markedbit up to l bits ahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bitmarked with the do-nothing tag and l the gene size encoded by the tag. If the mark isread by the decoder, the decoded bit string y will have y k + i = x k + i for all i = 0 , , ..., l − ( ). A right arithmetic shift of one position moves each bit to the right by one. Thisoperation discards the least significant bit and fills the most significant bit with the pre-vious bit value (now placed one position to the right). This operation shifts a specifiednumber of bits: starting at the marked bit up to l bits ahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bit marked with the right shift by one tag and l thegene size encoded by the tag. If the mark is read by the decoder, the decoded bit string y will have y k = x k and y k + i = x k + i − for all i = 1 , ..., l − ( ). Adds one to a specified number of bits: starting at the marked bit up to l bitsahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bit marked with theadd one tag and l the gene size encoded by the tag. If the mark is read by the decoder,the decoded bit string y will have y k + l − − i = x k + l − − i + 1 + carry for all i = 0 , , ..., l − HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS ( ). A left arithmetic shift of one position moves each bit to the left by one. Thisoperation fills the least significant bit with zero and discards the most significant bit. Thisoperation shifts a specified number of bits: starting at the marked bit up to l bits ahead.Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bit marked with the left shiftby one tag and l the gene size encoded by the tag. If the mark is read by the decoder, thedecoded bit string y will have y k + i = x k + i +1 for all i = 0 , , ..., l − y k + l − = 0. ( ). Subtracts one to a specified number of bits: starting at the marked bit up to l bits ahead. Let x be a binary string x = ( x , x , x , ..., x n ), and x k be a bit marked withthe subtract one tag and l the gene size encoded by the tag. If the mark is read by thedecoder, the decoded bit string y will have y k + l − − i = x k + l − − i + borrow − borrowed − i = 0 , , ..., l −
1. When 1 is subtracted from 0, the borrow method is applied. Theborrowing digit (zero) essentially obtains ten from borrowing ( borrow = 10), and the digitthat is borrowed from is reduced by one ( borrowed = 1).These operations have been selected due to their simplicity and capacity to generate,discover, and combine many possible building blocks.
Set to operation, for example, itcan be dominant depending on the optimization problem when maximizing or minimizinga function. The current operations combine short, high-fitness schemas resulting in high-quality building blocks of solutions after the epigenetic growing function is applied. If anyallele has tags bound to it, regions of the chromosome will be read as the
Operation states.In section 3.4, the epigenetic growing function and the application of these
Bit Operations are explained in more detail.
As mentioned above, the last -bits of a tag represent the gene size. The gene size de-termines the number of alleles involved in the bit operation during the decoding process,Table 3.1 briefly shows some binary strings and their respective values. These genes sizeshave been proposed based on the order- i schemas of the functions selected to performexperiments and the transformation of binary strings of 32 bits to real values. Fig. 3.2depicts the complete structure of a tag. Table 3.1.
Gene Sizes
String Value String Value String Value String Value HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS The
Epigenotype is a structural layer on the chromosome top structure used to attachtags. This second layer represents individuals’ epigenome, and it is a structure withthe same size as an individual’s chromosome. This epigenetic component holds a set ofepigenetic changes that influence the direction of the search process. It is coded as amultidimensional vector ( m · n ), where m is the tag length, and n is the length of theindividual’s chromosome. The Marking function adds tags to or removes tags from alleles of any chromosome in thesolution space. Additionally, it can modify the binary string tags. The markingprocess works with a marking rate, that is, the probability of applying the function onevery bit of a chromosome. When the probability is positive, the function generates aprobability of adding a tag to one single allele, removing a tag from one single allele,or modifying a tag on any allele. These actions cannot happen simultaneously and aremutually exclusive. Tags are randomly added or removed from any allele. Also, the modify action randomly changes any of the eight positions in the binary string. The distributionof these actions is given as follows in Equation 3.1. P Marking = No Marking 0 . . . .
006 (3.1)The probability of marking a single bit of a chromosome is defined by taking intoaccount three factors. First, biologically epigenetics marks of the sort of Methylation,for example, are dispersed in indeterminate
CpG sequences on the genome, except for
CpG islands , or specific areas where high
CpG contents are present. Despite that, theycan affect gene expression. They are powerful because of what they encode, not for thequantity, this means for better or worse, a few tags can have the influence or potentialto make individuals bits being interpreted in such a way that good or poor results couldbe obtained. The second factor aims to avoid chromosomes over-marking, if each bit ismarked, it could cause over-processing during the decoding process. The third factor isrelated to the definition of a marking probability that allows the
Marking function tokeep the marking process balanced; a probability value that ensures tags diversity and aconsiderable number of marked positions.
HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS Figure 3.1.
General representation of the
Marking function: a) shows a chromosome with notags on it; b) depicts the addition of four tags to a chromosome; c) illustrates tags’bit modification in red; and d) presents a chromosome with two removed tags.
Based on previous considerations, experiments to define a marking probability, involvetuning the marking process with different rate values from to . The higher therate, the less effective is the marking function. When the rate is reduced, the markingfunction reveals an equilibrium between the applied actions and the obtained solutions,after running experiments with lower rates from to , the rate of has shown tobe enough to influence the search process and help ReGen EAs in finding solutions closer tothe optimum. Consequently, the probability of marking a single bit has been set to .Following the definition of such a probability rate, the probability distribution for adding,removing, and modifying is set based on the significance of having a considerable numberof tags and a variety of them. If tags are added, they should be eventually removed, orchromosomes will be over marked; for this, the approach gets rid of tags with the sameprobability as the add tag action. Then, the modify tag action uses a lower probabilityto recombine the of a tag and generate different decoding frames. The influence orimpact of the designed actions is mostly that they altogether:1. Let individuals have a reasonable quantity of tags,2. Allow bit combination for tags, and3. Ensure discovering building blocks during tags interpretation that could generatesolutions that are not neighbor to current solutions to escape from a local optimum. This action writes tags on any chromosome.
Add is a metaphorical representation of writer enzymes. Fig. 3.1 presents a chromosome (image a ) with no tags. In image b , four tagsare added at positions , , , and , based on the defined add tag probability of . HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS This action modifies tags on any chromosome.
Modify is a metaphorical representationof maintenance enzymes. In Fig. 3.1, image c illustrates modified tags at positions and , bits in red changed. This action is applied under the defined modify tag probability of and then randomly changes any of the eight positions in the binary string with arate of 1 . /l , where l is the tag’s length. This action erases tags from any chromosome.
Remove is a metaphorical representationof eraser enzymes. Fig. 3.1, depicts a chromosome with removed tags. In Image d , twotags at positions and are not longer bound to the chromosome. The tag removal isperformed with the defined remove tag probability of . HA P T E R . E V O L U T I O NA R YA L G O R I T H M S W I T H R E G U L A T E D G E N E S : R E G E N E A S Figure 3.2.
General representation of an individual with its epigenotype. The bottom section shows the tag’s interpretation process to generate a bitstring used to build the individual’s phenotype.
HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS This function is a metaphorical representation of reader enzymes. The epigenetic growingfunction generates bit strings from individuals’ genotypes-epigenotypes for eventual pheno-types creation. Tags allow this function to build different individuals before the quality orfitness of each solution is evaluated. The growth happens in the binary search space (codedsolutions), but is reflected in the solution space (actual solutions). From a mathematicalpoint of view, the search space is transformed and reduced when the tags’ interpretationis performed; this ensures both exploration and exploitation. The first to reach differentpromising regions in a smaller search space and the second to search for optimal solutionswithin the given region. The bigger are gene size values, the less variety of building blockswill result during decoding. Tags may lead individuals to be represented as closer feasi-ble solutions to some extreme (minimum or maximum) in the search space. When thisfunction is applied, chromosomes grow in the direction of minimum or maximum points,depending on the problem. This process differs from mutations or hyper-mutations whichmodify chromosomes and maintain genetic diversity from one generation of a populationof chromosomes to another on a broader search space.The Epigenetic Growing function acts like an interpreter or decoder of tags locatedover a particular allele. This function scans each allele of a chromosome and the tagsthat directly affect it, so that, the phenotypic variations can be reflected when evaluatingindividuals. During the decoding process, alleles are not changed; the chromosome keepsits binary encoding fixed. This means the individual’s genotype is not altered. Notethat the scope of the
Operations to be applied depends on the gene size indicator. If an
Operation has been already applied, and there is another one to be applied, the epigeneticgrowing function considers the interpretation of the previous bits in order to continue itsdecoding process. An example of the prior process to phenotype generation is illustratedin Fig. 3.2. The example shows the decoding process for each bit with or without tags.On the top of Fig. 3.2, a chromosome with a size of and eight tags is depicted.Alleles in positions
1, 9, 14, 18, 23, 26, 29 and are marked with colored tags. Thedecoding starts from left to right. The first position is scanned, as it has a tag bound to it,the function initiates a tag identification. The tag in red is , the first three bits indicate an operation that is Set to . It means to set a specified number of bits to thesame value of the allele, which is . The specified number of bits to be Set to , is indicatedby the last five bits of the tag, which are , this refers to a gene size ( l ) of -bits.Then, the resulting interpretation is to set all bits to , starting at the marked bit up tothe gene size minus one ( l − l + 1) and keeps the previous result. This processis repeated until the entire chromosome is scanned; each result is concatenated to generatea final bit string; the length of the chromosome and the resulting string keep fixed. At thebottom of Fig. 3.2, a final interpretation is shown, the concatenated string is the sourceto build the phenotype, which is evaluated and gives the score for the individual. HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS The epigenetic tags added to the chromosome are inherited by the offspring duringCrossover. Transgenerational epigenetic inheritance transmits epigenetic markers fromone individual to another (parent-child transmission) that affects the offspring’s traitswithout altering the sequence of nucleotides of DNA. The idea here is to perform therecombination process based on the selected operator, as usual, the only difference is thattags located on alleles will be copied along with the genetic information. This modelpresents the Single Point Crossover as an illustrative example of genetic and epigeneticrecombination. So, a calculated cross point x will be applied to the chromosome at x posi-tion. By doing this process, the offspring will inherit alleles with their tags. Fig. 3.3 showsthe exchange of genetic code and epigenetic tags. A Simple Point Crossover operation isperformed over given parents at cross point . Offspring 1 inherited from
Parent 1 , partof the genetic code plus its tags in positions
1, 9 and . From Parent 2 , it also inheritedpart of the genetic code plus some tags in positions
11, 15 and . Offspring 2 got partof the genetic code plus its tags in positions and from Parent 1 . From
Parent 2 , itgot part of the genetic code plus its tags in positions and . HA P T E R . E V O L U T I O NA R YA L G O R I T H M S W I T H R E G U L A T E D G E N E S : R E G E N E A S Figure 3.3.
Illustrative example of genetic and epigenetic recombination: Simple Point Crossover operation.
HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS The sequence of steps for the proposed ReGen EA is defined in Algorithm 1 and Algo-rithm 2. Note that the pseudo-code includes the same elements of a generic evolutionaryalgorithm. The epigenetic components are embedded, as defined in Algorithm 1. TheReGen EA’s behavior is similar to EA’s standard versions until defined marking periodsand tags decoding processes take place. Note that the reading process is different whenchromosomes are marked, phenotypes are built based on tags interpretation. This processfirstly identifies the operation to be applied on a specific section of a chromosome andsecondly the gene size to define the scope of the bit operation, as depicted in Fig. 3.2.The epigenetic EA incorporates a pressure function to perform the marking processduring a specific period. A range of iterations determines a period. Marking periodsrepresent the environment, an abstract element that has been a point of reference to assessthe results of this model. At line 7, the function markingPeriodON validates the beginningof defined periods, this indicates that the marking process can be performed starting fromiteration a to iteration b . Any number of marking periods can be defined. Periods could bebetween different ranges of iterations. Additionally, the epiGrowingFunction is embeddedat line 10. The epiGrowingFunction interprets tags and generates a bit string used tobuild the phenotype before initiating the fitness evaluation of individuals. The standardEA uses the individual’s genotype to be evaluated; in contrast, the epigenetic techniqueuses the resulting phenotype from the tags decoding process. This technique is calledReGen EA, which means Evolutionary Algorithm with Regulated Genes. Algorithm 1
Pseudo code of a ReGen EA initialize population with random candidate solutions evaluate each candidate repeat select parents recombine pairs of parents mutate the resulting offspring if markingPeriodON (iteration) then applyMarking (offspring) end if phenotypes ← decode ( epiGrowingFunction (offspring)) evaluate phenotypes of the new candidates select individuals for the next generation until Termination condition is satisfied
HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS Algorithm 2
Pseudo code of a ReGen EA function markingPeriodON ( it ) start ← startV alue end ← endV alue if start ≥ it ≤ end then return true end if return f alse end function function applyMarking (offspring) mark ← P Marking // probability of 0.02 notM odif y ← P Adding + P Removing // same probability of 0.35 to add and remove for each allele ∈ offspring i chromosome do if mark then if notM odif y then if add then if notMarked then add tag end if else if isMarked then remove tag end if end if else if isMarked then modify any tag’ bit with a rate of 1.0/tag length end if end if end if end for end function function epiGrowingFunction (offspring) bitStrings ← offspring if offspring isMarked then bitStrings ← read offspring marks end if return bitStrings end function HAPTER 3. EVOLUTIONARY ALGORITHMS WITH REGULATED GENES: REGEN EAS This chapter describes the proposed epigenetic technique under the scope of this thesis.Five fundamental elements form the basis of the designed technique (ReGen EA): first,a metaphorical representation of
Epigenetic Tags as binary strings; second, a layer onchromosome top structure used to bind tags (
Epigenotype ); third, a
Marking Function to add, remove, and modify tags; fourth, a
Epigenetic Growing Function that acts likean interpreter, or decoder of tags located on the
Epigenotype ; and fifth, tags inheritanceby the offspring during
Crossover . The abstraction presented in this chapter describes away to address a large number of computational problems with binary and real encoding.This technique may find approximately optimal solutions to hard problems that are notefficiently solved with other techniques.
HAPTER ReGen GA: Binary and Real Codification
Genetic Algorithm with Regulated Genes (ReGen GA) is the implementation of the pro-posed epigenetic model on a classic GA. The general terminology of a GA includes popu-lation, chromosomes, genes, genetic operators, among others. The ReGen GA has a layerto attach tags and involves two functions named
Marking and
Epigenetic Growing . Thefirst function simulates periods in which individuals’ genetic codes are affected by externalfactors, represented by the designed tags. The second function generates bit strings fromgenotypes and their respective epigenotypes for phenotypes formation (see Algorithm 3).Also, the ReGen GA uses Simple Point
Crossover operator to perform recombination andtransmission of epigenetic markers from one individual to its descendants.This chapter aims to present the application of the proposed epigenetic model. Thisimplementation intends to address real and binary encoding problems. Experimentalfunctions with binary and real encoding have been selected to determine the model appli-cability. The experiments will evidence the effect of the tags on population behavior. Insection 4.1, experimental setups and parameters configuration used for the selected func-tions are described. In section 4.2, a set of binary experiments is presented, implementingDeceptive (orders three and four), Royal Road, and Max Ones functions. Additionally,some experimental results and their analysis are exhibited in subsections 4.2.2 and 4.2.3.In section 4.3, a set of real experiments is presented, implementing Rastrigin, Rosenbrock,Schwefel, and Griewank functions. Also, some experimental results and their analysis arereported in subsections 4.3.2 and 4.3.3. At the end of this chapter, a summary is given insection 4.4. 41
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Algorithm 3
Pseudo code of the ReGen GA initialize population with random candidate solutions evaluate each candidate repeat select parents recombine pairs of parents mutate the resulting offspring if markingPeriodON (iteration) then applyMarking (offspring) end if phenotypes ← decode ( epiGrowingFunction (offspring)) evaluate phenotypes of the new candidates select individuals for the next generation until Termination condition is satisfied
The following configuration applies to all experiments presented in this chapter. It iswell known that an algorithm can be tweaked (e.g., the operators in a GA) to improveperformance on specific problems, even though, this thesis intends to avoid giving too manyadvantages to the performed GA implementations in terms of parametrization. The classicGA and the ReGen GA parameters are tuned with some variations on only two standardoperators, Single Bit Mutation and Simple Point Crossover. Also, for all experiments,three marking periods have been defined, note that the defined number of periods are justfor testing purposes. Marking Periods can be appreciated in figures of reported resultsdelineated with vertical lines. Vertical lines in blue depict the starting point of markingperiods and gray lines, the end of them.The set up for classic GAs includes: 30 runs; 1000 iterations; population size of 100individuals; a tournament of size 4 for parents selection; generational (GGA) and steadystate (SSGA, in which replacement policy is elitism) replacements to choose the fittestindividuals for the new population; each bit in the chromosome has a mutation rate of1 . /l , where l is the chromosome length, while the single point crossover rates are set from0 . . . /l , where l is the chromosome length, while the single point crossover rates are set from0 . .
0; a marking probability of 0 .
02 (the probability to add a tag is 0 .
35, to remove a
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION .
35, and to modify a tag 0 .
3) and three marking periods have been defined. Suchperiods start at iterations 200, 500, and 800, with a duration of 150 iterations each.It is worth mentioning that the crossover rate of 0 .
7, along with the mutation rateof 1 . /l , are considered good parameters to solve binary encoding problems [3, 47], eventhough, five crossover rates are used to evaluate tags inheritance impact. Table 4.1 showsa summary of the general setup for the experiments. Table 4.1.
General configuration with 5 different Crossover rates
Factor Name
Classic GA ReGen GA
Mutation Operator Rate 1 . /l . /l Crossover Operator Rate 0.6 - 1.0 0.6 - 1.0Marking Rate none none
Tournament Tournament
Chromosomes encoding is one of the challenges when trying to solve problems with GAs;encoding definition depends on the given problem. Binary encoding is the most traditionaland simple, essentially because earlier GA implementations used this encoding type. Thissection reports experiments with four different binary functions.
Performing experiments use binary encoding for determining the proposed technique ap-plicability. In binary encoding, a vector with binary values encodes the problem’s solution.Table 4.2 shows a simple example of functions with a single fixed bit string length. Thesefunctions have been chosen to work on the first approximation to test this technique. Beaware that this does not mean a different bit string length is not allowed. Any lengthvalue can be set. The selected functions and fixed string length values are just for thepurpose of making the experiments simpler and easier to understand.
Table 4.2.
Experimental Functions
Function Genome Length Global Optimum
Deceptive 3
360 3600
Deceptive 4
360 450
Royal Road
360 360
Max Ones
360 360
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION The deceptive functions proposed by Goldberg in 1989 are challenging problems for conven-tional genetic algorithms (GAs), which mislead the search to some local optima (deceptiveattractors) rather than the global optimum [29]. An individual’s fitness is defined as indi-cated in Table 4.3 and Table 4.4 for Deceptive order three and Deceptive order four trap,respectively.
Table 4.3.
Order Three Function
String Value String Value Table 4.4.
Order Four Trap Function
String Value String Value The Max Ones’ problem (or BitCounting) is a simple problem that consists of maximizingthe number of 1’s in a chain. The fitness of an individual is defined as the numberof bits that are 1. Formally, this problem can be described as finding a string x =( x , x , x , ..., x n ), where x i ∈ { , } , which maximizes the following Equation 4.1: f ( x ) = n (cid:88) i =1 x i (4.1)The Royal Road function developed by Forrest and Mitchell in 1993 [25], consistsof a list of partially specified bit strings (schemas) with a sequence of 0’s and 1’s. Aschema performs well when all bits are set to 1. For the experiments, order-8 schemas areconfigured. HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION
45A simple Royal Road function, R is defined by Equation 4.2. R consists of a list ofpartially specified bit strings (schemas) s i in which (‘ ∗ ’) denotes a wild card (i.e., allowedto be either 0 or 1). A bit string x is said to be an instance of a schema s, x ∈ s , if x matches s in the defined (i.e., non-‘ ∗ ’) positions. The fitness R ( x ) of a bit string x isdefined as follows: R ( x ) = (cid:88) i =1 δ i ( x ) o ( s i ) , where δ i ( x ) = x ∈ s i Based on the defined configuration, both classic GA and ReGen GA are compared toidentify the behavior of tags during individuals’ evolution. Results are tabulated fromTable 4.5 to Table 4.8, these tables present the binary functions: Deceptive order three(D3), Deceptive order four trap (D4), Royal Road (RR), and Max ones (MO). Both EAimplementations with generational (GGA), steady state (SSGA) replacements, and fivecrossover rates per technique. For each rate, the best fitness based on the maximummedian performance is reported, following the standard deviation of the observed value,and the iteration where the reported fitness is found. The iteration is enclosed in squarebrackets.Graphs from Fig. 4.1 to Fig. 4.4 illustrate the fitness of best individuals of populationsin the experiments, reported fitnesses are based on the maximum median performance.Each graph shows the tendency of best individuals per technique. For ReGen GA andClassic GA, two methods are applied: steady state and generational population replace-ments. The fitness evolution of individuals can be appreciated by tracking green and redlines that depict the best individual’s fitness for classic GAs. Blue and black lines tracethe best individual’s fitness for ReGen GAs. From top to bottom, each figure displays in-dividuals’ behavior with crossover rates from 0 . .
0. Figures on the right corner showdefined marking periods. Vertical lines in blue depict the starting of a marking period,lines in gray delimit the end of such periods.
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.5.
Results of the experiments for Generational and Steady replacements: Deceptive Or-der 3
Rate Deceptive Order 3Classic GGA Classic SSGA ReGen GGA ReGen SSGA ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . Iterations F i t ne ss Classic GGA Best
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Figure 4.1.
Deceptive Order 3. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.6.
Results of the experiments for Generational and Steady replacements: Deceptive Or-der 4
Rate Deceptive Order 4Classic GGA Classic SSGA ReGen GGA ReGen SSGA . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Iterations F i t ne ss Classic GGA Best
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Figure 4.2.
Deceptive Order 4. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.7.
Results of the experiments for Generational and Steady replacements: Royal Road
Rate Royal RoadClassic GGA Classic SSGA ReGen GGA ReGen SSGA ± . . ± . ± . ± . ± . . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . Iterations F i t ne ss Classic GGA Best
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Figure 4.3.
Royal Road. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.8.
Results of the experiments for Generational and Steady replacements: Max Ones
Rate Max OnesClassic GGA Classic SSGA ReGen GGA ReGen SSGA ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . Iterations F i t ne ss Classic GGA Best
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Figure 4.4.
Max Ones. Generational replacement (GGA) and Steady State replacement (SSGA).From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Three different tests are performed, One-Way ANOVA test, Pairwise Student’s t-test, andPaired Samples Wilcoxon Test (also known as Wilcoxon signed-rank test). The data setReGen EAs Samples in Appendix B is used, the samples contain twenty EA implemen-tations for each of the following functions: Deceptive Order Three, Deceptive Order FourTrap, Royal Road, and Max Ones. The samples refer to the best fitness of a solutionfound in each run, the number of executions per algorithm is 30. Different implementa-tions involve classic GAs and ReGen GAs with Generational (G) and Steady State (SS)population replacements, and crossover rates from 0 . . .
05 (alpha value), then variances differ, such that there is a statisticallysignificant difference between algorithms. When the null hypothesis is false, it bringsup the alternative hypothesis, which proposes that there is a difference. When significantdifferences between groups (EAs) are found, Student’s T-test is used to interpret the resultof one-way ANOVA tests. Multiple pairwise-comparison T-test helps to determine whichpairs of EAs are different. The T-test concludes if the mean difference between specificpairs of EAs is statistically significant. In order to identify any significant difference inthe median fitness, between two experimental conditions (classic GAs and ReGen GAs),Wilcoxon signed-rank test is performed. For the Wilcoxon test, crossover rates are ignored,and EAs are classified into four groups: GGAs vs. ReGen GGAs and SSGAs vs. ReGenSSGAs.Based on the ReGen EAs Samples in Appendix B, the analysis of variance is computedto know the difference between evolutionary algorithms with different implementations.Variations include classic GAs and ReGen GAs, replacement strategies (Generational andSteady State), and crossover rates from 0 . .
0, algorithms are twenty in total. Table 4.9shows a summary for each algorithm and function. The summary presents the number ofsamples per algorithm (30), the sum of the fitness, the average fitness, and their variances.Results of the ANOVA single factor are tabulated in Table 4.10. HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.9.
Anova Single Factor: SUMMARY
Deceptive Order Three Deceptive Order Four Trap Royal Road Max Ones
Groups Count Sum Average Variance Sum Average Variance Sum Average Variance Sum Average VarianceGGAX06 30 103036 3434.533333 119.4298851 11695 389.8333333 20.55747126 6040 201.3333333 852.2298851 10800 360 0GGAX07 30 102898 3429.933333 88.96091954 11662 388.7333333 20.54712644 6616 220.5333333 294.3264368 10800 360 0GGAX08 30 103016 3433.866667 102.3264368 11701 390.0333333 16.3091954 7384 246.1333333 210.4643678 10800 360 0GGAX09 30 103164 3438.8 113.2689655 11713 390.4333333 10.87471264 7880 262.6666667 490.2988506 10800 360 0GGAX10 30 103080 3436 83.31034483 11757 391.9 24.50689655 8504 283.4666667 325.2229885 10800 360 0SSGAX06 30 102898 3429.933333 107.1678161 11626 387.5333333 23.42988506 2976 99.2 364.5793103 10800 360 0SSGAX07 30 103046 3434.866667 78.53333333 11611 387.0333333 11.68850575 2928 97.6 288.662069 10800 360 0SSGAX08 30 103038 3434.6 129.9724138 11686 389.5333333 17.42988506 3304 110.1333333 395.8436782 10800 360 0SSGAX09 30 103020 3434 100.9655172 11649 388.3 25.38965517 3392 113.0666667 563.7885057 10800 360 0SSGAX10 30 103084 3436.133333 148.9471264 11739 391.3 16.56206897 3696 123.2 364.5793103 10800 360 0ReGenGGAX06 30 107326 3577.533333 124.6022989 13340 444.6666667 10.43678161 10720 357.3333333 19.12643678 10800 360 0ReGenGGAX07 30 107364 3578.8 146.3724138 13383 446.1 2.644827586 10744 358.1333333 20.67126437 10800 360 0ReGenGGAX08 30 107328 3577.6 200.3862069 13324 444.1333333 12.32643678 10784 359.4666667 4.11954023 10800 360 0ReGenGGAX09 30 107372 3579.066667 175.6505747 13378 445.9333333 4.616091954 10760 358.6666667 9.195402299 10800 360 0ReGenGGAX10 30 107412 3580.4 163.6965517 13373 445.7666667 7.840229885 10776 359.2 5.95862069 10800 360 0ReGenSSGAX06 30 107234 3574.466667 161.291954 13310 443.6666667 7.609195402 10480 349.3333333 67.67816092 10800 360 0ReGenSSGAX07 30 107202 3573.4 211.0758621 13367 445.5666667 7.21954023 10496 349.8666667 101.2229885 10800 360 0ReGenSSGAX08 30 107394 3579.8 176.3724138 13337 444.5666667 7.564367816 10504 350.1333333 135.4298851 10800 360 0ReGenSSGAX09 30 107486 3582.866667 119.3609195 13375 445.8333333 4.281609195 10648 354.9333333 41.85747126 10800 360 0ReGenSSGAX10 30 107544 3584.8 115.2 13351 445.0333333 7.688505747 10656 355.2 29.13103448 10800 360 0HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.10.
Anova Single Factor: ANOVA
Deceptive Order Three
Source of Variation SS df MS F P-value F critBetween Groups 3141837.193 19 165359.8523 1240.0941 0 1.60449Within Groups 77339.86667 580 133.3445977Total 3219177.06 599
Deceptive Order Four Trap
Source of Variation SS df MS F P-value F critBetween Groups 465618.6183 19 24506.24307 1888.5604 0 1.60449Within Groups 7526.166667 580 12.97614943Total 473144.785 599
Royal Road
Source of Variation SS df MS F P-value F critBetween Groups 6329162.56 19 333113.8189 1453.2537 0 1.60449Within Groups 132947.2 580 229.2193103Total 6462109.76 599
Max Ones
Source of Variation SS df MS F P-value F critBetween Groups 5.53E-25 19 2.91E-26 1 0.459 1.60449Within Groups 1.69E-23 580 2.91E-26Total 1.74E-23 599As P-values for Deceptive Order Three, Deceptive Order Four Trap, and Royal Roadfunctions are less than the significance level 0 .
05, the results allow concluding that there aresignificant differences between groups, as shown in Table 4.10 (
P-value columns). In one-way ANOVA tests, significant P-values indicate that some group means are different, butit is not evident which pairs of groups are different. In order to interpret one-way ANOVAtest’ results, multiple pairwise-comparison with Student’s t-test is performed to determineif the mean difference between specific pairs of the group is statistically significant. Also,paired-sample Wilcoxon tests are computed.ANOVA test for Max Ones’ samples shows that the P-value is higher than the sig-nificance level 0.05, this result means that there are no significant differences betweenalgorithms (EAs) listed above in the model summary Table 4.9. Therefore, no multiplepairwise-comparison Student’s t-tests between means of groups are performed; neither,paired-sample Wilcoxon test is computed. HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10
EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA
EAs F i t ne ss GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10
EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA
EAs F i t ne ss Figure 4.5.
From top to bottom: Deceptive Order Three and Deceptive Order Four Trap Functions. On the left, EAs with Generational replacement(GGA) and Steady State replacement (SSGA) with Crossover rates from 0 . .
0. On the right, EAs grouped by Generational replacement(GGA) and Steady State replacement (SSGA).
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10 EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA EAs F i t ne ss Figure 4.6.
Royal Road Function. On top, EAs with Generational (GGA) and Steady State(SSGA) replacements with Crossover rates from 0 . .
0. On the bottom, EAsgrouped by Generational replacement (GGA) and Steady State replacement (SSGA).
Box plots in Fig. 4.5 and Fig. 4.6 depict the median fitness of EAs’ best solutions(ReGen EAs Samples in Appendix B). On the left, twenty EAs’ variations with differentcrossover rates: Gray (0 . . . . . HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.11.
D3 Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 0.17764919 - - - - - - - - -GGAX08 0.87373395 0.24754713 - - - - - - - -GGAX09 0.21528009 0.00554289 0.14859587 - - - - - - -GGAX10 0.71303353 0.0705529 0.57069862 0.43796354 - - - - - -ReGenGGAX06 1.48E-203 1.40E-209 1.94E-204 1.24E-197 1.54E-201 - - - - -ReGenGGAX07 3.23E-205 3.71E-211 4.47E-206 2.13E-199 2.72E-203 0.75006919 - - - -ReGenGGAX08 1.23E-203 1.22E-209 1.62E-204 1.01E-197 1.26E-201 0.98736535 0.76386956 - - -ReGenGGAX09 1.48E-205 2.02E-211 2.04E-206 8.99E-200 1.23E-203 0.70352874 0.95386568 0.71303353 - -ReGenGGAX10 2.81E-207 5.69E-213 3.77E-208 1.26E-201 2.19E-205 0.42937099 0.69400018 0.43796354 0.74066234 -ReGenSSGAX06 2.62E-199 1.48E-205 3.09E-200 3.08E-193 2.93E-197 0.39042554 0.20794839 0.3822627 0.17764919 0.07707801ReGenSSGAX07 8.27E-198 3.48E-204 9.41E-199 1.09E-191 9.80E-196 0.22880804 0.11278114 0.22116359 0.09315297 0.03350474ReGenSSGAX08 1.73E-206 2.31E-212 2.34E-207 8.46E-201 1.33E-204 0.54845698 0.80065744 0.56134173 0.86012623 0.88238029ReGenSSGAX09 1.73E-210 5.11E-216 2.89E-211 5.83E-205 1.39E-208 0.11646875 0.23327459 0.12124417 0.26599333 0.50716527ReGenSSGAX10 9.47E-213 2.57E-218 2.10E-213 1.95E-207 4.35E-211 0.02681405 0.07375646 0.02823607 0.08927929 0.20079383SSGAX06 0.17764919 1 0.24754713 0.00554289 0.0705529 1.40E-209 3.71E-211 1.22E-209 2.02E-211 5.69E-213SSGAX07 0.94586954 0.14859587 0.80065744 0.24754713 0.77767313 4.02E-203 8.80E-205 3.33E-203 3.86E-205 7.54E-207SSGAX08 0.98736535 0.17392479 0.86012623 0.22116359 0.72683516 1.79E-203 3.86E-205 1.48E-203 1.80E-205 3.30E-207SSGAX09 0.89581456 0.23327459 0.9798171 0.1615203 0.60061513 2.90E-204 6.71E-206 2.37E-204 3.05E-206 5.59E-208SSGAX10 0.69400018 0.06448843 0.54845698 0.4643557 0.9798171 2.32E-201 4.02E-203 1.89E-201 1.79E-203 3.23E-205
Table 4.12.
D3 Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.79147218 - - - - - - - -ReGenSSGAX08 0.11646875 0.05569495 - - - - - - -ReGenSSGAX09 0.00897573 0.00290977 0.39042554 - - - - - -ReGenSSGAX10 0.00105773 0.00027428 0.14533483 0.61389238 - - - - -SSGAX06 1.48E-205 3.48E-204 2.31E-212 5.11E-216 2.57E-218 - - - -SSGAX07 7.65E-199 2.38E-197 4.47E-206 4.57E-210 2.21E-212 0.14859587 - - -SSGAX08 3.22E-199 1.01E-197 2.04E-206 2.05E-210 1.05E-212 0.17392479 0.95386568 - -SSGAX09 4.72E-200 1.44E-198 3.30E-207 3.71E-211 2.66E-213 0.23327459 0.83276418 0.88238029 -SSGAX10 4.51E-197 1.51E-195 1.94E-204 2.06E-208 6.31E-211 0.06448843 0.75006919 0.70352874 0.57069862 HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.13.
D4 Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 0.29677823 - - - - - - - - -GGAX08 0.85224562 0.21774467 - - - - - - - -GGAX09 0.59061606 0.09801927 0.71632051 - - - - - - -GGAX10 0.04153318 0.00125756 0.06818451 0.15999358 - - - - - -ReGenGGAX06 1.17E-246 8.61E-251 6.80E-246 2.34E-244 1.43E-238 - - - - -ReGenGGAX07 5.36E-252 6.26E-256 2.87E-251 8.11E-250 3.13E-244 0.16928761 - - - -ReGenGGAX08 1.31E-244 8.13E-249 7.53E-244 2.76E-242 2.03E-236 0.63324079 0.05347637 - - -ReGenGGAX09 2.19E-251 2.45E-255 1.13E-250 3.43E-249 1.37E-243 0.22613672 0.87161088 0.07995101 - -ReGenGGAX10 8.61E-251 8.57E-255 4.55E-250 1.43E-248 6.17E-243 0.29677823 0.76019487 0.11371468 0.87161088 -ReGenSSGAX06 8.29E-243 4.82E-247 4.98E-242 1.94E-240 1.63E-234 0.34659749 0.01507201 0.67268757 0.02432668 0.03819933ReGenSSGAX07 4.55E-250 4.82E-254 2.64E-249 8.27E-248 3.71E-242 0.3961808 0.63324079 0.16928761 0.73618084 0.85224562ReGenSSGAX08 2.82E-246 1.97E-250 1.60E-245 5.62E-244 3.60E-238 0.91925507 0.14042521 0.69248564 0.19171249 0.25184377ReGenSSGAX09 4.97E-251 5.16E-255 2.61E-250 8.13E-249 3.39E-243 0.26627775 0.80406077 0.09801927 0.91925507 0.94288345ReGenSSGAX10 4.68E-248 4.21E-252 2.67E-247 8.95E-246 4.85E-240 0.73618084 0.31284451 0.3961808 0.3961808 0.49602171SSGAX06 0.02222998 0.25184377 0.01232805 0.00330137 6.21E-06 4.20E-255 6.88E-260 3.42E-253 2.08E-259 6.86E-259SSGAX07 0.00465945 0.09801927 0.00231558 0.0005018 4.40E-07 8.21E-257 4.64E-261 5.16E-255 9.71E-261 2.04E-260SSGAX08 0.78430534 0.45468783 0.6529301 0.3961808 0.01834505 8.27E-248 4.45E-253 8.95E-246 1.89E-252 7.02E-252SSGAX09 0.14042521 0.69248564 0.09261653 0.03509616 0.00021879 2.47E-252 2.23E-257 1.97E-250 8.21E-257 2.75E-256SSGAX10 0.15999358 0.01012854 0.22613672 0.41519752 0.59061606 5.77E-241 1.56E-246 7.78E-239 6.80E-246 2.89E-245
Table 4.14.
D4 Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.06311581 - - - - - - - -ReGenSSGAX08 0.3961808 0.34659749 - - - - - - -ReGenSSGAX09 0.03221118 0.80406077 0.22613672 - - - - - -ReGenSSGAX10 0.19171249 0.63324079 0.67268757 0.45468783 - - - - -SSGAX06 1.67E-251 3.51E-258 8.57E-255 4.33E-259 2.17E-256 - - - -SSGAX07 2.63E-253 7.65E-260 1.72E-256 1.53E-260 4.26E-258 0.6529301 - - -SSGAX08 5.62E-244 3.78E-251 2.01E-247 4.21E-252 3.43E-249 0.04934242 0.01232805 - -SSGAX09 1.08E-248 1.43E-255 5.36E-252 1.72E-256 1.13E-253 0.47513258 0.22613672 0.23957048 -SSGAX10 5.89E-237 1.75E-244 1.44E-240 1.60E-245 2.05E-242 0.0001064 1.02E-05 0.08609593 0.00231558 HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.15.
RR Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 1.64E-06 - - - - - - - - -GGAX08 2.21E-27 1.83E-10 - - - - - - - -GGAX09 2.49E-46 1.19E-24 3.78E-05 - - - - - - -GGAX10 4.37E-73 2.66E-48 5.20E-20 2.08E-07 - - - - - -ReGenGGAX06 5.59E-168 1.09E-144 8.81E-112 8.33E-90 3.21E-62 - - - - -ReGenGGAX07 6.43E-169 1.10E-145 7.83E-113 7.12E-91 2.93E-63 0.86524143 - - - -ReGenGGAX08 1.81E-170 2.51E-147 1.42E-114 1.20E-92 5.40E-65 0.6320313 0.7738968 - - -ReGenGGAX09 1.53E-169 2.42E-146 1.57E-113 1.39E-91 5.93E-64 0.7738968 0.90582906 0.86524143 - -ReGenGGAX10 3.67E-170 5.31E-147 3.14E-114 2.70E-92 1.20E-64 0.67967992 0.82408949 0.94563689 0.90582906 -ReGenSSGAX06 1.92E-158 1.27E-134 3.42E-101 4.02E-79 5.72E-52 0.05012867 0.03093773 0.01289906 0.02203385 0.01555171ReGenSSGAX07 4.39E-159 2.69E-135 6.71E-102 7.85E-80 1.22E-52 0.06851925 0.04303471 0.01867431 0.03093773 0.02203385ReGenSSGAX08 2.15E-159 1.25E-135 3.02E-102 3.49E-80 5.67E-53 0.07937734 0.05012867 0.02203385 0.03656145 0.02624799ReGenSSGAX09 3.86E-165 1.07E-141 1.28E-108 1.34E-86 4.11E-59 0.58909905 0.45928081 0.28402906 0.38220426 0.31346893ReGenSSGAX10 1.88E-165 5.02E-142 5.74E-109 5.96E-87 1.87E-59 0.6320313 0.49787684 0.31346893 0.41972909 0.34677055SSGAX06 8.98E-100 4.98E-125 3.73E-157 1.12E-176 5.80E-200 1.07E-270 2.43E-271 3.14E-272 9.53E-272 4.43E-272SSGAX07 6.71E-102 4.27E-127 4.39E-159 1.61E-178 1.10E-201 6.43E-272 2.32E-272 5.71E-273 1.08E-272 5.71E-273SSGAX08 3.58E-85 9.95E-111 1.08E-143 7.14E-164 6.62E-188 1.60E-261 3.24E-262 2.44E-263 1.14E-262 4.01E-263SSGAX09 3.00E-81 7.45E-107 5.08E-140 2.36E-160 1.32E-184 5.25E-259 1.07E-259 7.95E-261 3.83E-260 1.32E-260SSGAX10 8.54E-68 2.34E-93 4.27E-127 5.53E-148 5.79E-173 6.52E-250 1.21E-250 7.68E-252 4.00E-251 1.32E-251
Table 4.16.
RR Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.90582906 - - - - - - - -ReGenSSGAX08 0.86524143 0.94563689 - - - - - - -ReGenSSGAX09 0.18112563 0.22782816 0.25486 - - - - - -ReGenSSGAX10 0.16007994 0.20415385 0.22782816 0.94563689 - - - - -SSGAX06 5.03E-264 1.79E-264 1.10E-264 9.06E-269 5.73E-269 - - - -SSGAX07 2.31E-265 8.37E-266 5.21E-266 4.36E-270 2.80E-270 0.72848115 - - -SSGAX08 1.75E-254 5.86E-255 3.45E-255 1.80E-259 1.07E-259 0.00713327 0.00191234 - -SSGAX09 7.68E-252 2.61E-252 1.52E-252 7.19E-257 4.25E-257 0.00057512 0.00011766 0.49787684 -SSGAX10 1.87E-242 5.96E-243 3.40E-243 1.06E-247 6.13E-248 2.18E-09 1.83E-10 0.00119893 0.01289906
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Multiple pairwise t-test:
Multiple pairwise-comparison between means of EA groupsis performed. In the one-way ANOVA test described above, significant p-values indi-cate that some group means are different. In order to know which pairs of groups aredifferent, multiple pairwise-comparison is performed for Deceptive Order Three (D3), De-ceptive Order Four Trap (D4), and Royal Road (RR) best solutions samples. Tables (4.11,4.12, 4.13, 4.14, 4.15, and 4.16) present Pairwise comparisons using t-tests with pooledstandard deviation (SD) with their respective p-values. The test adjusts p-values withthe Benjamini-Hochberg method. Pairwise comparisons show that only highlighted val-ues in gray between two algorithms are significantly different ( p < . Paired Samples Wilcoxon Test:
For this test, algorithms are grouped per populationreplacement strategy, without taking into account the crossover rates. Wilcoxon signedrank test for generational EAs (GGA and ReGen GGA) and Wilcoxon signed rank testfor steady state EAs (SSGA and ReGen SSGA). The test assesses classic EAs versusEpigenetic EAs. • Deceptive Order Three (D3)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. The
P-value is equal to2 . e −
26, which is less than the significance level alpha (0 . P-value is equalto 2 . e −
26, which is less than the significance level alpha = 0 . • Deceptive Order Four Trap (D4)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. The P-value is equal to2 . e −
26, which is less than the significance level alpha (0 . P-value is equalto 2 . e −
26, which is less than the significance level alpha = 0 . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION • Royal Road (RR)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. The P-value is equal to2 . e −
26, which is less than the significance level alpha = 0 . P-value is equalto 1 . e −
26, which is less than the significance level alpha (0 . . e −
26 (D3 samples), 2 . e −
26 (D4 samples), and2 . e −
26 (RR samples). So, the alternative hypothesis is true.The median fitness of solutions found by classic steady state genetic algorithms (SS-GAs) is significantly different from the median fitness of solutions found by steadystate genetic algorithms with regulated genes (ReGen SSGAs) with p-values equal to2 . e −
26 (D3 sampling fitness), 217806 e −
26 (D4 sampling fitness), and 1 . e − .
05, it may be con-cluded that there are significant differences between the two EAs groups in each WilcoxonTest.
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION The real problems have been encoded as binary strings. The individuals are initializedwith randomized binary strings of ( d · n ), where d is the number of dimensions of theproblem and n the length in bits of the binary representation for a real value. The processto obtain real values from binary strings of 32 bits is done by taking its representation asan integer number and then applying a decoding function. Equation 4.3 and Equation 4.4define the encoding/decoding schema [8].In the general form for an arbitrary interval [a, b] the coding function is defined as: C n , [ a, b ] : [ a, b ] −→ { , } n x (cid:55)−→ bin n (cid:18) round (cid:18) (2 n − · x − ab − a (cid:19)(cid:19) (4.3)where bin n is the function which converts a number from { , ..., n − } to its binaryrepresentation of length n [8]. The corresponding decoding function is defined as follows: (cid:101) C n , [ a, b ] : { , } n −→ [ a, b ] s (cid:55)−→ a + bin − n ( s ) · b − a n − − . , .
11] with n = 32,where the total size of the search space is 2 = 4 . . . { , ..., } ,the 32 bits string is equal to , and the bit stringrepresentation as integer number is 4294967295. The decoding function yields: s (cid:55)−→ − .
12 + 4294967295 · . − ( − . . Experiments using real definition are performed to determine the proposed techniqueapplicability. For the selected problems with real definition, a vector with binary valuesencodes the problem’s solution. The real functions explained in this section are used astestbeds. For all functions, the problem dimension is fixed to n = 10; each real value isrepresented with a binary of 32-bits. HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION The Rastrigin function has several local minima, it is highly multimodal, but locationsof the minima are regularly distributed. Among its features: the function is continuous,convex, defined on n-dimensional space, multimodal, differentiable, and separable. Thefunction is usually evaluated on the hypercube x i ∈ [ − . , .
12] for i = 1 , ..., n . Theglobal minimum f ( x ∗ ) = 0 at x ∗ = (0 , ...,
0) [1, 68]. On an n-dimensional domain, it isdefined by Equation 4.5 as: f ( x, y ) = 10 n + n (cid:88) i =1 ( x i − cos (2 πx i )) (4.5) The Rosenbrock function, also referred to as the Valley or Banana function, is a populartest problem for gradient-based optimization algorithms. Among its features: the functionis continuous, convex, defined on n-dimensional space, multimodal, differentiable, and non-separable. The function is usually evaluated on the hypercube x i ∈ [ − ,
10] for i = 1 , ..., n ,although it may be restricted to the hypercube x i ∈ [ − . , . i = 1 , ..., n . Theglobal minimum f ( x ∗ ) = 0 at x ∗ = (1 , ..., a and b areconstants and are generally set to a = 1 and b = 100 [1, 68]. On an n-dimensional domain,it is defined by: f ( x, y ) = n (cid:88) i =1 [ b ( x i +1 − x i ) + ( a − x i ) ] (4.6) The Schwefel function is complex, with many local minima. Among its features: thefunction is continuous, not convex, multimodal, and can be defined on n-dimensionalspace. The function can be defined on any input domain but it is usually evaluatedon the hypercube x i ∈ [ − , i = 1 , ..., n . The global minimum f ( x ∗ ) = 0at x ∗ = (420 . , ..., . f ( x ) = f ( x , x , ..., x n ) = 418 . n − n (cid:88) i =1 x i sin (cid:16)(cid:112) | x i | (cid:17) (4.7) HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION The Griewank function has many widespread local minima, which are regularly dis-tributed. Among its features: this function is continuous, not convex, can be definedon n-dimensional space, and is unimodal. This function can be defined on any input do-main but it is usually evaluated on x i ∈ [ − , i = 1 , ..., n . The global minimum f ( x ∗ ) = 0 at x ∗ = (0 , ...,
0) [1, 68]. On an n-dimensional domain, it is defined by Equation4.8 as: f ( x ) = f ( x , ..., x n ) = 1 + n (cid:88) i =1 x i − n (cid:89) i =1 cos (cid:18) x i √ i (cid:19) (4.8) Based on the defined configuration, both classic and ReGen GA are compared to identifythe tags’ behavior during individuals’ evolution. Results are tabulated from Table 4.17to Table 4.20, these tables present real defined functions: Rastrigin (RAS), Rosenbrock(ROSE), Schwefel (SCHW), and Griewank (GRIE). Both EA implementations with gen-erational (GGA) and steady state (SSGA) replacements, and five crossover rates per tech-nique. For each rate, the best fitness based on the minimum median performance isreported, following the standard deviation of the observed value, and the iteration wherethe reported fitness is found. The latter is enclosed in square brackets.Graphs from Fig. 4.7 to Fig. 4.10 illustrate the best individuals’ fitness in performedexperiments, reported fitnesses are based on the minimum median performance. Eachfigure shows the tendency of the best individuals per technique. For ReGen GA and ClassicGA, two methods are applied: steady state and generational population replacements. Thefitness evolution of individuals can be appreciated by tracking green and red lines thatdepict the best individual’s fitness for classic GA. Blue and black lines trace the bestindividual’s fitness for ReGen GA. From top to bottom, each figure displays individuals’behavior with crossover rates from 0 . .
0. Figures on the right side show definedmarking periods. Vertical lines in blue depict the starting of a marking period, lines ingray delimit the end of such periods.
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.17.
Results of the experiments for Generational and Steady replacements: Rastrigin
Rate RastriginClassic GGA Classic SSGA ReGen GGA ReGen SSGA . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best
Figure 4.7.
Rastrigin. Generational replacement (GGA) and Steady State replacement (SSGA).From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.18.
Results of the experiments for Generational and Steady replacements: Rosenbrock
Rate RosenbrockClassic GGA Classic SSGA ReGen GGA ReGen SSGA . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - + + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best
Figure 4.8.
Rosenbrock. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.19.
Results of the experiments for Generational and Steady replacements: Schwefel
Rate SchwefelClassic GGA Classic SSGA ReGen GGA ReGen SSGA . ± . . ± . . e − ± . . e − ± . . ± . . ± . . e − ± . . e − ± . . ± . . ± . . e − ± . . e − ± . . ± . . ± . . e − ± . . e − ± . . ± . . ± . . e − ± . . e − ± . - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - - + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best
Figure 4.9.
Schwefel. Generational replacement (GGA) and Steady State replacement (SSGA).From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Table 4.20.
Results of the experiments for Generational and Steady replacements: Griewank
Rate GriewankClassic GGA Classic SSGA ReGen GGA ReGen SSGA . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best - - + + + Iterations F i t ne ss Classic GGA Best
ReGen GGA Best
Classic SSGA Best
ReGen SSGA Best
Figure 4.10.
Griewank. Generational replacement (GGA) and Steady State replacement(SSGA). From top to bottom, crossover rates from 0 . . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION .
0) is not reached, it achieves suitable solutions in general. ReGen GA reportssolutions with local minimum under 1 .
0; in contrast, classic GA solutions are above 5 .
0. InFig. 4.7 is observable that marking periods applied on chromosomes at iterations 200, 500,and 800 produce significant changes in the evolution of individuals. After starting the firstmarking period, the fitness improves to be closer to the optimum, and populations improvetheir performance once tags are added to individuals. Fig. 4.7 also shows that classic GAperformance is under ReGen GA performance in all crossover rates levels. Generationaland steady replacements performed similarly to this problem.Tabulated results in Table 4.18 for Rosenbrock show that ReGen GA accomplishesbetter solutions than the classic GA. However, not much difference is evident in results.ReGen GA solutions are a bit closer to the global minimum (0 .
0) than solutions reportedby the classic GA. In Fig. 4.8 is noticeable that the pressure applied on chromosomesat iteration 200 does cause a change in the evolution of individuals. After starting themarking period, the fitness slightly improves, and populations improve their performanceonce tags are bound. ReGen GA reports better local minima than GA, and generationaland steady replacements have almost similar results for all crossover rates.On the other hand, tabulated results in Table 4.19 for Schwefel, evidence that ReGenGA performs much better than the classic GA for current experiments. ReGen GA reportssuitable solutions nearer the global minimum (0 .
0) than GA solutions. It can be appreci-ated that the best solutions are close to the optima for all crossover rates. On the contrary,individuals’ fitness for classic GA does not reach the same local optima. Fig. 4.9 remarksthat the pressure applied on chromosomes during defined marking periods introduces agreat change in the evolution of individuals. After starting the first marking period, thefitness improves to be closer to the optimum, and populations improve their performanceonce tags are attached. The ReGen GA reaches a variety of good solutions during theevolution process, exposing the ability of the proposed approach to discover novelties thatare not identified by the classic GA. Fig. 4.9 also shows that classic GA performance isbelow ReGen GA performance in all crossover rates levels; the classic GA does not findsuitable solutions for this experiment. Generational replacement performed better thanthe steady replacement for this problem.As well, tabulated results in Table 4.20 for Griewank objective function, show thatboth ReGen GA and classic GA have a small margin of difference on their performances;still, ReGen GA produces better solutions than the classic GA. Both reached local optimaunder 1 .
0. In Fig. 4.10 is evident that the marking process at iteration 200 generates achange in the evolution of individuals. After starting the marking period, fitness improvesand keeps stable for the best individuals. Both generational and steady replacementsperformed slightly similar for all crossover rates.
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION . Table 4.21.
Solutions found by different EAs on real functions
EA Rosenbrock Schwefel Rastrigin Griewank
ReGen GGA 0 . ± .
18 0 . ± .
15 0 . ± .
67 0 . ± . . ± .
20 0 . ± .
420 0 . ± .
72 0 . ± . . ± .
11 2 . ± .
210 0 . ± .
15 0 . ± . . ± .
03 378 . ± . . ± .
01 0 . ± . . ± .
08 0 . ± .
570 0 . ± .
07 0 . ± . . ± .
04 659 . ± . . ± .
80 0 . ± . . .
000 0 . . . .
8; ReGen SSGA with crossover 1 .
0) and Rastrigin (ReGen GGA with crossover1 .
0; ReGen SSGA with crossover 0 .
9) functions. Nevertheless, for Rosenbrock (ReGenGGA with crossover 1 .
0; ReGen SSGA with crossover 0 .
9) and Griewank (with crossover0 . The statistical analysis presented in this subsection follows the same scheme from binaryproblems section; therefore, some descriptions are omitted, refer to subsection 4.2.3 formore details. Three different tests are performed, One-Way ANOVA test, Pairwise Stu-dent’s t-test, and Paired Samples Wilcoxon Test (also known as Wilcoxon signed-ranktest). The data set ReGen EAs Samples in Appendix B is used, the samples containtwenty EAs implementations for each of the following functions: Ratrigin, Rosenbrock,Schwefel, and Griewank. The samples refer to the best fitness of a solution found in eachrun, the number of executions per algorithm is 30. Different implementations involveclassic GAs and ReGen GAs with Generational (G) and Steady State (SS) populationreplacements, and crossover rates from 0 . . Gomez implements four GAs with Single Point Real Crossover (X), Gaussian (G), and Uniform (U)Mutation as genetic operators in order to compare their performance with
HaEa . Two generationalGAs (GGA(XG) and GGA(XU)), and two steady state GAs (SSGA(XG) and SSGA(XU)). The GAsuses a tournament size of four as parent selection method. For steady state implementations, the worstindividual of the population is replaced with the best child generated after crossover and mutation occurs.The reported results are performed with a mutation rate of 0 . . HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.22.
Anova Single Factor: SUMMARY
Rastrigin Rosenbrock Schwefel Griewank
Groups Count Sum Average Variance Sum Average Variance Sum Average Variance Sum Average VarianceGGAX06 30 340.2427 11.3414 19.3266 52.58683 1.75289 13.96380 5859.204 195.3068 30075.83 5.33366 0.17779 0.00578GGAX07 30 342.8903 11.4297 24.1491 46.71745 1.55725 5.79244 5045.616 168.1872 14482.14 6.11428 0.20381 0.00946GGAX08 30 336.9313 11.2310 23.8248 97.34299 3.24477 17.80430 2816.054 93.8685 6880.24 5.85960 0.19532 0.01076GGAX09 30 310.2216 10.3407 19.6901 69.00354 2.30012 11.97435 2676.372 89.2124 13007.02 5.13625 0.17121 0.00449GGAX10 30 260.1163 8.6705 14.5260 42.00588 1.40020 7.17257 2057.618 68.5873 5671.65 4.28734 0.14291 0.00598SSGAX06 30 352.4735 11.7491 19.0427 77.03410 2.56780 11.33058 6353.236 211.7745 13329.70 7.33354 0.24445 0.02425SSGAX07 30 335.8611 11.1954 18.6028 52.04329 1.73478 10.39958 4691.105 156.3702 19022.31 5.45293 0.18176 0.00758SSGAX08 30 303.3327 10.1111 13.3066 48.09139 1.60305 7.03069 4213.878 140.4626 15033.45 6.57597 0.21920 0.01325SSGAX09 30 292.4265 9.7475 11.4686 62.27272 2.07576 11.61922 3689.997 122.9999 10995.33 5.11276 0.17043 0.00483SSGAX10 30 228.0086 7.6003 12.6033 58.61603 1.95387 11.33277 2557.336 85.2445 5461.39 5.83554 0.19452 0.00733ReGenGGAX06 30 21.1354 0.7045 0.5303 11.54441 0.38481 0.16875 344.900 11.4967 559.56 2.29069 0.07636 0.00237ReGenGGAX07 30 19.2442 0.6415 0.6273 9.33206 0.31107 0.03705 601.931 20.0644 1538.00 1.90437 0.06348 0.00186ReGenGGAX08 30 13.3463 0.4449 0.6015 8.84472 0.29482 0.06261 302.420 10.0807 673.19 2.44422 0.08147 0.00326ReGenGGAX09 30 13.9836 0.4661 0.5390 8.97137 0.29905 0.08245 93.521 3.1174 104.54 2.55778 0.08526 0.00372ReGenGGAX10 30 8.2175 0.2739 0.2014 6.59899 0.21997 0.03338 24.872 0.8291 11.52 2.65118 0.08837 0.00197ReGenSSGAX06 30 24.7609 0.8254 1.3584 16.02581 0.53419 0.66559 891.330 29.7110 2116.61 2.14538 0.07151 0.00158ReGenSSGAX07 30 19.3034 0.6434 0.4815 14.03180 0.46773 0.40673 854.721 28.4907 2233.54 1.96305 0.06544 0.00216ReGenSSGAX08 30 17.8044 0.5935 0.6824 11.06604 0.36887 0.20173 210.062 7.0021 505.22 2.08242 0.06941 0.00241ReGenSSGAX09 30 10.9318 0.3644 0.3144 8.10569 0.27019 0.03800 820.391 27.3464 2287.48 2.08257 0.06942 0.00260ReGenSSGAX10 30 9.6430 0.3214 0.2819 9.05231 0.30174 0.09498 0.00824998 0.00027 1.41E-09 2.05718 0.06857 0.00186HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION . .
0, algorithms are twenty in total. Ta-ble 4.22 shows a summary for each algorithm and function; the summary presents thenumber of samples per algorithm (30), the sum of the fitness, the average fitness, andtheir variances. Results of the ANOVA single factor are tabulated in Table 4.23.
Table 4.23.
Anova Single Factor: ANOVA
Rastrigin
Source of Variation SS df MS F P-value F critBetween Groups 14947.3266 19 786.70140 86.3753 4.8815E-155 1.60449Within Groups 5282.60586 580 9.10794Total 20229.9325 599
Rosenbrock
Source of Variation SS df MS F P-value F critBetween Groups 507.02716 19 26.68564 4.8426 1.30194E-10 1.60449Within Groups 3196.1356 580 5.51057Total 3703.1628 599
Schwefel
Source of Variation SS df MS F P-value F critBetween Groups 2832064.8 19 149056.042 20.7038 2.4055E-53 1.60449Within Groups 4175672.6 580 7199.43552Total 7007737.3 599
Griewank
Source of Variation SS df MS F P-value F critBetween Groups 2.26223 19 0.11906 20.2641 2.6291E-52 1.60449Within Groups 3.40786 580 0.00587Total 5.67009 599As P-values for Rastrigin, Rosenbrock, Schwefel, and Griewank functions are less thanthe significance level 0 .
05, results allow concluding that there are significant differencesbetween groups, as shown in Table 4.23. In one-way ANOVA tests, significant P-valuesindicate that some group means are different, but it is not evident which pairs of groups aredifferent. In order to interpret one-way ANOVA test’ results, multiple pairwise-comparisonwith Student’s t-test is performed to determine if the mean difference between specific pairsof the group is statistically significant. Also, paired-sample Wilcoxon tests are computed. HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10 EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA EAs F i t ne ss GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10 EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA EAs F i t ne ss Figure 4.11.
From top to bottom: Rastrigin and Rosenbrock Functions. On the left, EAs with Generational replacement (GGA) and Steady Statereplacement (SSGA) with Crossover rates from 0 . .
0. On the right, EAs grouped by Generational replacement (GGA) and SteadyState replacement (SSGA). HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10 EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA EAs F i t ne ss GGAX06
GGAX08
GGAX10
ReGenGGAX08
ReGenSSGAX06
ReGenSSGAX10
SSGAX08
SSGAX10 . . . . . . . EAs F i t ne ss GGA
ReGenGGA
ReGenSSGA
SSGA . . . . . . . EAs F i t ne ss Figure 4.12.
From top to bottom: Schwefel and Griewank Functions. On the left, EAs with Generational replacement (GGA) and Steady Statereplacement (SSGA) with Crossover rates from 0 . .
0. On the right, EAs grouped by Generational replacement (GGA) and SteadyState replacement (SSGA).
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION . . . . . . . . .
0. Epigenetic EAs for Schwefel achieved median fitness inferior to 0 . . .
1. So, based on thesedata, it seems that Epigenetic GAs find better solutions than classic GAs. However, it isneeded to determine whether these findings are statistically significant. HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.24.
RAS Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 0.96041246 - - - - - - - - -GGAX08 0.95257897 0.92910961 - - - - - - - -GGAX09 0.29168935 0.24549691 0.36794632 - - - - - - -GGAX10 0.0011367 0.00076506 0.0018616 0.05322976 - - - - - -ReGenGGAX06 2.66E-36 9.11E-37 9.78E-36 7.19E-31 2.44E-22 - - - - -ReGenGGAX07 1.24E-36 4.69E-37 4.64E-36 3.45E-31 1.23E-22 0.97139031 - - - -ReGenGGAX08 1.45E-37 6.45E-38 4.69E-37 3.12E-32 1.36E-23 0.90599934 0.92910961 - - -ReGenGGAX09 1.80E-37 7.59E-38 6.04E-37 4.03E-32 1.71E-23 0.91303244 0.92910961 0.98343714 - -ReGenGGAX10 2.71E-38 1.36E-38 7.82E-38 3.65E-33 1.96E-24 0.79383597 0.83932039 0.92910961 0.92910961 -ReGenSSGAX06 1.10E-35 4.00E-36 4.54E-35 3.19E-30 9.30E-22 0.95195906 0.92910961 0.83697676 0.83932039 0.67474664ReGenSSGAX07 1.24E-36 4.69E-37 4.65E-36 3.48E-31 1.24E-22 0.97139031 0.99797965 0.92910961 0.92910961 0.83932039ReGenSSGAX08 7.13E-37 2.77E-37 2.66E-36 1.92E-31 7.23E-23 0.95257897 0.97139031 0.94200353 0.95195906 0.85814598ReGenSSGAX09 6.72E-38 2.71E-38 1.91E-37 1.13E-32 5.46E-24 0.85648226 0.89695052 0.96340589 0.95659485 0.96041246ReGenSSGAX10 4.29E-38 1.88E-38 1.27E-37 6.62E-33 3.36E-24 0.83697676 0.85814598 0.95195906 0.94200353 0.97139031SSGAX06 0.81568977 0.85814598 0.70748155 0.11090763 0.00015681 3.31E-38 1.88E-38 3.11E-39 3.33E-39 1.14E-39SSGAX07 0.94200353 0.91303244 0.97375344 0.39322514 0.00216188 1.50E-35 7.11E-36 7.10E-37 8.88E-37 1.14E-37SSGAX08 0.17605857 0.14080147 0.22979948 0.91303244 0.10208473 1.16E-29 5.50E-30 5.08E-31 6.54E-31 6.38E-32SSGAX09 0.06643715 0.05168347 0.09169682 0.63832768 0.24857143 9.17E-28 4.36E-28 4.09E-29 5.18E-29 5.31E-30SSGAX10 3.71E-06 2.16E-06 7.19E-06 0.00082833 0.25058299 2.04E-17 1.09E-17 1.49E-18 1.84E-18 2.56E-19
Table 4.25.
RAS Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.92910961 - - - - - - - -ReGenSSGAX08 0.91303244 0.97139031 - - - - - - -ReGenSSGAX09 0.76325659 0.89695052 0.91303244 - - - - - -ReGenSSGAX10 0.71850131 0.85814598 0.89710033 0.97139031 - - - - -SSGAX06 1.14E-37 1.88E-38 1.36E-38 1.37E-39 1.14E-39 - - - -SSGAX07 7.11E-35 7.14E-36 4.03E-36 2.86E-37 1.80E-37 0.67474664 - - -SSGAX08 4.98E-29 5.55E-30 3.15E-30 1.92E-31 1.15E-31 0.05841924 0.24627159 - -SSGAX09 3.92E-27 4.41E-28 2.45E-28 1.53E-29 9.20E-30 0.01743086 0.10081497 0.83932039 -SSGAX10 6.80E-17 1.11E-17 6.77E-18 6.50E-19 4.18E-19 2.73E-07 8.82E-06 0.00227926 0.01015915 HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.26.
ROSE Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 0.97208658 - - - - - - - - -GGAX08 0.04881414 0.02555667 - - - - - - - -GGAX09 0.57153304 0.36485797 0.20856668 - - - - - - -GGAX10 0.83562136 0.97544713 0.01549184 0.23648078 - - - - - -ReGenGGAX06 0.06582199 0.10908846 7.08E-05 0.01126426 0.16925516 - - - - -ReGenGGAX07 0.05376269 0.08887445 5.27E-05 0.00799463 0.13573914 0.99094875 - - - -ReGenGGAX08 0.05286811 0.08631238 5.27E-05 0.00788007 0.13321453 0.99094875 0.9964496 - - -ReGenGGAX09 0.05286811 0.08664538 5.27E-05 0.00788007 0.13387032 0.99094875 0.9964496 0.9964496 - -ReGenGGAX10 0.04274603 0.07223188 5.27E-05 0.00609823 0.10737367 0.97544713 0.99094875 0.99094875 0.99094875 -ReGenSSGAX06 0.09586036 0.16641953 0.00017713 0.01918161 0.26058028 0.97544713 0.96054387 0.95886064 0.95886064 0.86336478ReGenSSGAX07 0.08069305 0.13573914 0.00011926 0.01600757 0.21502063 0.99094875 0.97544713 0.97544713 0.97544713 0.953995ReGenSSGAX08 0.06294053 0.10522403 7.08E-05 0.01068613 0.16328371 0.9964496 0.9964496 0.99094875 0.99094875 0.97544713ReGenSSGAX09 0.04910282 0.08069305 5.27E-05 0.00745631 0.12296574 0.99094875 0.9964496 0.9964496 0.9964496 0.9964496ReGenSSGAX10 0.05286811 0.08664538 5.27E-05 0.00788007 0.13387032 0.99094875 0.9964496 0.9964496 0.9964496 0.99094875SSGAX06 0.29886179 0.17046693 0.42748594 0.93354867 0.10908846 0.00362422 0.00256288 0.00256288 0.00256288 0.00206334SSGAX07 0.9964496 0.97544713 0.04576565 0.55629676 0.8428942 0.06943874 0.05516519 0.05376269 0.05376269 0.04560677SSGAX08 0.97544713 0.9964496 0.02812045 0.41044716 0.97208658 0.09586036 0.08069305 0.07931275 0.07947888 0.06294053SSGAX09 0.85566196 0.60652592 0.10908846 0.96054387 0.42748594 0.02555667 0.01918161 0.01918161 0.01918161 0.01508228SSGAX10 0.97208658 0.77377559 0.08069305 0.83665025 0.56743955 0.03752499 0.02812045 0.02790432 0.02790432 0.0218973
Table 4.27.
ROSE Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.99094875 - - - - - - - -ReGenSSGAX08 0.97544713 0.99094875 - - - - - - -ReGenSSGAX09 0.93354867 0.97208658 0.99094875 - - - - - -ReGenSSGAX10 0.95886064 0.97544713 0.99094875 0.9964496 - - - - -SSGAX06 0.00745631 0.00569756 0.00347469 0.00256288 0.00256288 - - - -SSGAX07 0.10151582 0.085759 0.06582199 0.05236195 0.05376269 0.2855976 - - -SSGAX08 0.14452907 0.12182725 0.09212144 0.07258523 0.07947888 0.19702636 0.99094875 - -SSGAX09 0.04186549 0.03244222 0.02448551 0.01789277 0.01918161 0.63931284 0.83884205 0.66236082 -SSGAX10 0.05531127 0.04910282 0.03549543 0.02555667 0.02790432 0.4973954 0.96054387 0.83562136 0.99094875 HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.28.
SCHW Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 0.29349563 - - - - - - - - -GGAX08 1.32E-05 0.00154568 - - - - - - - -GGAX09 4.96E-06 0.00074085 0.88288414 - - - - - - -GGAX10 4.79E-08 1.91E-05 0.32852579 0.44530689 - - - - - -ReGenGGAX06 5.04E-15 1.89E-11 0.00042857 0.00089786 0.01566357 - - - - -ReGenGGAX07 7.65E-14 2.05E-10 0.00165974 0.00325874 0.04229624 0.76682348 - - - -ReGenGGAX08 3.60E-15 1.27E-11 0.00034065 0.0007295 0.01320684 0.9634304 0.73372792 - - -ReGenGGAX09 4.10E-16 1.56E-12 0.00010421 0.00023203 0.00539223 0.76682348 0.52852644 0.80585652 - -ReGenGGAX10 2.03E-16 8.63E-13 6.76E-05 0.00015732 0.00394782 0.71707208 0.47538279 0.75214023 0.93656485 -ReGenSSGAX06 1.38E-12 2.53E-09 0.00634508 0.01164887 0.11267669 0.50102081 0.74186251 0.46632728 0.30322679 0.26447332ReGenSSGAX07 9.78E-13 1.88E-09 0.00541319 0.00994238 0.10132833 0.52852644 0.76682348 0.49805897 0.32852579 0.28531851ReGenSSGAX08 1.33E-15 5.10E-12 0.0002037 0.00043581 0.00889768 0.88404959 0.6425634 0.91726021 0.89709331 0.8306861ReGenSSGAX09 7.48E-13 1.46E-09 0.00465967 0.00863338 0.09089022 0.55086676 0.8031202 0.52488929 0.35275945 0.30322679ReGenSSGAX10 1.72E-16 7.34E-13 5.90E-05 0.00013811 0.00351558 0.69085485 0.4561716 0.73372792 0.91726021 0.96983594SSGAX06 0.53740475 0.07160523 3.70E-07 1.27E-07 7.75E-10 3.25E-17 4.38E-16 2.31E-17 2.82E-18 1.77E-18SSGAX07 0.11267669 0.68333237 0.00796687 0.00427897 0.00017385 4.93E-10 4.37E-09 3.38E-10 4.80E-11 2.49E-11SSGAX08 0.02007775 0.28531851 0.05230058 0.0308658 0.0021947 2.82E-08 2.09E-07 2.03E-08 3.37E-09 1.88E-09SSGAX09 0.00207074 0.06067189 0.26108727 0.17920944 0.02103374 1.54E-06 9.73E-06 1.15E-06 2.33E-07 1.38E-07SSGAX10 2.07E-06 0.00039175 0.76682348 0.89709331 0.53459108 0.00165974 0.00551935 0.00136046 0.0004371 0.00030782
Table 4.29.
SCHW Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.9634304 - - - - - - - -ReGenSSGAX08 0.39089825 0.4227465 - - - - - - -ReGenSSGAX09 0.93656485 0.9634304 0.45073509 - - - - - -ReGenSSGAX10 0.25082493 0.27097547 0.80585652 0.29040453 - - - - -SSGAX06 8.06E-15 5.65E-15 9.17E-18 4.38E-15 1.77E-18 - - - -SSGAX07 4.79E-08 3.65E-08 1.47E-10 2.82E-08 2.01E-11 0.01900709 - - -SSGAX08 1.80E-06 1.40E-06 9.20E-09 1.11E-06 1.58E-09 0.00237449 0.55086676 - -SSGAX09 6.52E-05 5.27E-05 5.75E-07 4.25E-05 1.16E-07 0.00014806 0.18460857 0.52185554 -SSGAX10 0.0188548 0.01622979 0.00083026 0.01419946 0.00026794 4.85E-08 0.00242003 0.0193023 0.1247525 HA P T E R . R E G E N G A : B I NA R YAN D R E A L C O D I F I C A T I O N Table 4.30.
GRIE Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs GGAX06 GGAX07 GGAX08 GGAX09 GGAX10 ReGenGGAX06 ReGenGGAX07 ReGenGGAX08 ReGenGGAX09 ReGenGGAX10GGAX07 0.29214033 - - - - - - - - -GGAX08 0.510415 0.78361424 - - - - - - - -GGAX09 0.83155928 0.15842627 0.32989657 - - - - - - -GGAX10 0.12649056 0.00396273 0.01463119 0.24076026 - - - - - -ReGenGGAX06 1.06E-06 1.54E-09 1.51E-08 4.91E-06 0.00153207 - - - - -ReGenGGAX07 5.06E-08 4.12E-11 4.61E-10 2.45E-07 0.0001398 0.6486839 - - - -ReGenGGAX08 3.48E-06 5.70E-09 5.65E-08 1.57E-05 0.00365564 0.87028325 0.49701859 - - -ReGenGGAX09 8.27E-06 1.67E-08 1.42E-07 3.59E-05 0.00666481 0.77062026 0.3850767 0.90053134 - -ReGenGGAX10 1.67E-05 3.75E-08 2.93E-07 6.94E-05 0.01072871 0.66683146 0.31512052 0.82281877 0.9185665 -ReGenSSGAX06 3.45E-07 4.14E-10 3.85E-09 1.56E-06 0.00063804 0.87589027 0.79842526 0.73481827 0.62597133 0.53081354ReGenSSGAX07 7.80E-08 7.51E-11 7.69E-10 3.82E-07 0.00020698 0.70346634 0.94620992 0.55156577 0.44359153 0.35545711ReGenSSGAX08 2.09E-07 2.29E-10 2.20E-09 9.91E-07 0.00043492 0.82281877 0.84437496 0.66683146 0.55156577 0.46623314ReGenSSGAX09 2.09E-07 2.29E-10 2.20E-09 9.91E-07 0.00043492 0.82281877 0.84437496 0.66683146 0.55156577 0.46623314ReGenSSGAX10 1.72E-07 1.91E-10 1.86E-09 8.31E-07 0.00038006 0.80431313 0.87028325 0.6486839 0.53081354 0.44359153SSGAX06 0.00151754 0.06686587 0.02281821 0.00045194 1.04E-06 4.59E-15 1.92E-16 2.84E-14 1.04E-13 2.96E-13SSGAX07 0.89755203 0.3797239 0.62951206 0.7142955 0.08207869 4.14E-07 1.76E-08 1.36E-06 3.37E-06 6.82E-06SSGAX08 0.06140684 0.56499111 0.33338764 0.02626616 0.00026054 1.87E-11 3.06E-13 8.84E-11 2.54E-10 5.91E-10SSGAX09 0.81757796 0.14718982 0.31512052 0.9735815 0.25697268 5.86E-06 2.93E-07 1.84E-05 4.22E-05 8.13E-05SSGAX10 0.53081354 0.75872072 0.9735815 0.3471931 0.01630752 1.78E-08 5.73E-10 6.80E-08 1.72E-07 3.51E-07
Table 4.31.
GRIE Student T-tests pairwise comparisons with pooled standard deviation. Benjamini Hochberg (BH) as p-value adjustment method.
EAs ReGenSSGAX06 ReGenSSGAX07 ReGenSSGAX08 ReGenSSGAX09 ReGenSSGAX10 SSGAX06 SSGAX07 SSGAX08 SSGAX09GGAX07 - - - - - - - - -GGAX08 - - - - - - - - -GGAX09 - - - - - - - - -GGAX10 - - - - - - - - -ReGenGGAX06 - - - - - - - - -ReGenGGAX07 - - - - - - - - -ReGenGGAX08 - - - - - - - - -ReGenGGAX09 - - - - - - - - -ReGenGGAX10 - - - - - - - - -ReGenSSGAX06 - - - - - - - - -ReGenSSGAX07 0.84437496 - - - - - - - -ReGenSSGAX08 0.94566043 0.89755203 - - - - - - -ReGenSSGAX09 0.94566043 0.89755203 0.99979789 - - - - - -ReGenSSGAX10 0.92073066 0.9185665 0.9735815 0.9735815 - - - - -SSGAX06 8.07E-16 2.13E-16 4.22E-16 4.22E-16 4.22E-16 - - - -SSGAX07 1.37E-07 2.96E-08 7.80E-08 7.80E-08 6.65E-08 0.00298689 - - -SSGAX08 3.72E-12 5.71E-13 1.90E-12 1.90E-12 1.62E-12 0.31027932 0.09591503 - -SSGAX09 1.87E-06 4.56E-07 1.15E-06 1.15E-06 9.91E-07 0.00039995 0.69048565 0.02377523 -SSGAX10 4.78E-09 9.57E-10 2.67E-09 2.67E-09 2.20E-09 0.02056055 0.64946895 0.31848689 0.32989657
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION Multiple pairwise t-test:
Multiple pairwise-comparison between means of groups isperformed. In the one-way ANOVA test described above, significant p-values indicatethat some group means are different. In order to know which pairs of groups are differ-ent, multiple pairwise-comparison is performed for Rastrigin (RAS), Rosenbrock (ROSE),Schwefel (SCHW), and Griewank (GRIE) best solutions samples. Tables (4.24, 4.25, 4.26,4.27, 4.28, 4.29, 4.30, and 4.31) present Pairwise comparisons using t-tests with pooledstandard deviation (SD) with their respective p-values. The test adjusts p-values withthe Benjamini-Hochberg method. Pairwise comparisons show that only highlighted val-ues in gray between two algorithms are significantly different ( p < . Paired Samples Wilcoxon Test:
For this test, algorithms are grouped per populationreplacement strategy, ignoring crossover rates. Wilcoxon signed rank test for generationalEAs (GGA and ReGen GGA) and Wilcoxon signed rank test for steady state EAs (SSGAand ReGen SSGA). The test assesses classic EAs versus Epigenetic EAs. In the results, V represents the total of the ranks assigned to differences with a positive sign, and P-value refers to the probability value. In statistical hypothesis testing, the p-value corresponds tothe probability of obtaining test results as evidence to reject or confirm the null hypothesis. • Rastrigin (RAS)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. V = 11325, P-value isequal to 2 . e −
26, which is less than the significance level alpha (0 . V = 11325, P-value isequal to 2 . e −
26, which is less than the significance level alpha = 0 . • Rosenbrock (ROSE)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. V = 10368, P-value isequal to 1 . e −
18, which is less than the significance level alpha (0 . V = 10114, P-value isequal to 6 . e −
17, which is less than the significance level alpha = 0 . HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION • Schwefel (SCHW)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. V = 11121, P-value isequal to 1 . e −
24, which is less than the significance level alpha = 0 . V = 10913, P-value isequal to 6 . e −
23, which is less than the significance level alpha (0 . • Griewank (GRIE)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from GGAs and ReGen GGAs. V = 10438, P-value isequal to 3 . e −
19, which is less than the significance level alpha = 0 . V = 10975, P-value isequal to 2 . e −
23, which is less than the significance level alpha (0 . . e −
26 (RAS samples), 1 . e −
18 (ROSE samples),1 . e −
24 (SCHW samples), and 3 . e −
19 (GRIE samples). So, the alternativehypothesis is true.The median fitness of solutions found by classic steady state genetic algorithms (SS-GAs) is significantly different from the median fitness of solutions found by steadystate genetic algorithms with regulated genes (ReGen SSGAs) with p-values equalto 2 . e −
26 (RAS sampling fitness), 6 . e −
17 (ROSE sampling fitness),6 . e −
23 (SCHW sampling fitness), and 2 . e −
23 (GRIE sampling fitness).As p-values are less than the significance level 0 .
05, it may be concluded that there aresignificant differences between the two EAs groups in each Wilcoxon Test.
The epigenetic technique is implemented on GAs to solve both binary and real encodingproblems. For real encoding, the search space must be discretized by using a binaryrepresentation of real values. A decoding schema from binary to real value is performed inorder to evaluate individuals’ fitness. Results have shown that the marking process doesimpact the way the population evolves, and the fitness of individuals considerably improvesto the optimum. The use of epigenetic tags revealed that they help the ReGen GA to findbetter solutions (although the optimum is not always reached). A better exploration andexploitation of the search space is evident; in addition,
Tags are transmitted through
HAPTER 4. REGEN GA: BINARY AND REAL CODIFICATION
HAPTER ReGen HAEA: Binary and Real Codification
The Hybrid Adaptive Evolutionary Algorithm with Regulated Genes (ReGen
HaEa ) isthe implementation of the proposed epigenetic model on the standard
HaEa . This imple-mentation is meant to address real and binary encoding problems. Experimental functionswith binary and real encoding have been selected for determining the model applicabil-ity. In section 5.1, general settings for all experiments are described. In section 5.2, twobinary experiments are presented, performing Deceptive order three and Deceptive orderfour trap functions to evidence tags effect on populations’ behavior. Also, some experi-mental results and their analysis are exhibited in subsection 5.2.1 and subsection 5.2.2. Insection 5.3, three Real encoding problems are presented, implementing Rastrigin, Schwe-fel, and Griewank functions. Additionally, some experimental results and their analysisare exhibited in subsections 5.3.1 and 5.3.2. In section 5.4, the statistical analysis of theresults is described. At the end of this chapter, a summary is given in section 5.5.Gomez in [30, 31] proposed an evolutionary algorithm that adapts operator rates whileit is solving the optimization problem.
HaEa is a mixture of ideas borrowed from evolu-tionary strategies, decentralized control adaptation, and central control adaptation. Al-gorithm 4 presents the pseudo-code of
HaEa with the embedded epigenetic components.As can be noted,
HaEa does not generate a parent population to produce the nextgeneration. Among the offspring produced by the genetic operator, only one individual ischosen as a child (lines 16 and 18) and will take the place of its parent in the next popu-lation (line 28). In order to be able to preserve competent individuals through evolution,
HaEa compares the parent individual against the offspring generated by the operator,for steady state replacement. For generational replacement, it chooses the best individualamong the offspring (lines 15 and 17).At line 11, the marking period function has been embedded to initiate the markingprocess on individuals when defined periods are activated. Then, the epigenetic growing82
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION Algorithm 4
Hybrid Adaptive Evolutionary Algorithm (
HaEa ) HaEa (fitness , µ, terminationCondition) t = 0 P = initPopulation ( µ ) evaluate( P , fitness) while (cid:0) terminationCondition( t, P t , fitness) is false (cid:1) do P t +1 = ∅ for each ind ∈ P t do rates = extracRatesOper(ind) oper = OpSelect (operators, rates) parents = ParentsSelection (cid:0) P t , ind, arity(oper) (cid:1) offspring = apply(oper, parents) if markingPeriodON (t) then applyMarking (offspring) end if offspring ← decode ( epiGrowingFunction (offspring)) if steady then child = Best (offspring, ind) else child =
Best (offspring) end if δ = random(0 ,
1) // learning rate if (cid:0) fitness(child) > fitness(ind) (cid:1) then rates[oper] = (1 . δ ) ∗ rates[oper] // reward else rates[oper] = (1 . − δ ) ∗ rates[oper] // punish end if normalizeRates(rates) setRates(child, rates) P t +1 = P t +1 ∪ { child } end for t = t + 1 end while function (line 14) interprets markers on the chromosome structure of individuals with thepurpose of generating phenotypes that will be evaluated by the objective function.For all experiments in this chapter, three marking periods have been defined. Thereis not a particular reason why this number of periods has been chosen. Marking periodscould be between different ranges of iterations, defined periods are just for testing purposes.Marking periods can be appreciated in figures of reported results delineated with verticallines. Vertical lines in blue depict the starting point of marking periods and gray lines,the end of them. HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION Following experimental settings apply to binary and real experiments reported in sections5.2 and 5.3. For the standard
HaEa , a population size of 100 is used and 1000 iterations.A tournament size of 4 is implemented to select the parent of crossover. Reported resultsare the median over 30 different runs. For current tests,
HaEa only uses one geneticoperator combination: the single-point mutation and the single-point crossover. Themutation operator always modifies the genome by randomly changing only one singlebit with uniform distribution. The single-point crossover splits and combines parents’chromosome sections (left and right) using a randomly selected cutting point. The set upfor the standard
HaEa also includes: generational (G
HaEa ) and steady state (SS
HaEa )replacements to choose the fittest individuals for the new population.The ReGen
HaEa setup involves the same defined configuration for the standard
HaEa with an additional configuration for the epigenetic process as follows: a markingprobability of 0 .
02 (the probability to add a tag is 0 .
35, to remove a tag is 0 .
35, and tomodify a tag is 0 .
3) and three marking periods.
Binary encoding experiments have been performed in order to determine the proposed ap-proach applicability. In binary encoding, a vector with binary values encodes the problem’ssolution.
Two well known binary problems deceptive order three and deceptive order four trapfunctions developed by Goldberg in 1989 [29] have been selected. The genome lengthfor each function is 360, the global optimum for Deceptive order three is 3600, and 450for Deceptive order four trap. A complete definition of these functions can be found inprevious chapter 4, section 4.2. Also, in chapter 4, a more in-depth explanation can befound regarding the implemented binary to real decoding mechanism.
Based on the defined configuration, both
HaEa and ReGen
HaEa are compared to iden-tify tags’ behavior during individuals’ evolution. Results are tabulated in Table 5.1, the ta-ble presents binary functions: Deceptive order three and four with generational (G
HaEa )and steady state (SS
HaEa ) replacements for standard and epigenetic implementations.Also, the table shows the best fitness based on the maximum median performance, fol-
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION
HaEa and ReGen
HaEa , twomethods are applied: steady state and generational population replacements. The fitnessevolution of individuals can be appreciated by tracking green and red lines that depict bestindividuals’ fitness for the standard
HaEa . Blue and black lines trace best individuals’fitness for ReGen
HaEa . Figures on the right side show defined marking periods. Verticallines in blue depict the starting of a marking period, lines in gray delimit the end of suchperiods.
Table 5.1.
Results of the experiments for Generational and Steady state replacements
EA Deceptive Order 3 Deceptive Order 4 G HaEa ± . ± . HaEa ± . ± . HaEa ± . ± . HaEa ± . ± . Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best
Figure 5.1.
Deceptive Order 3. Generational replacement (G
HaEa ) and Steady state replace-ment (SS
HaEa ). Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best
Figure 5.2.
Deceptive Order 4. Generational replacement (G
HaEa ) and Steady state replace-ment (SS
HaEa ). Tabulated results in Table 5.1 for Deceptive Order Three and Deceptive Order FourTrap, show that ReGen
HaEa performs better than standard
HaEa implementations.ReGen
HaEa is able to discover varied optimal solutions until achieving the total ofconfigured iterations, even though, it did not find the global optimum in performed ex-
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION
Experiments using real definition are performed for determining the proposed techniqueapplicability. For the selected problems with real coded definition, a vector with binaryvalues encodes the problem’s solution.
Real functions shown in Table 5.2 are used as testbeds. Each real value is representedwith a binary string of 32-bits, for each function, the dimension of the problem is fixedto n = 10. A complete definition of these functions can be found in previous chapter4, section 4.3. Also, a detailed description of the encoding/decoding scheme to obtainreal values from binary strings of 32 bits and its representation as integer numbers arepresented in the same section. Table 5.2.
Real functions tested
Name Function Feasible RegionRastrigin f ( x ) = 10 n + (cid:80) ni =1 ( x i − cos (2 πx i )) -5.12 ≥ x i ≤ f ( x ) = 418 . d − (cid:80) ni =1 x i sin ( (cid:112) | x i | ) -500 ≥ x i ≤ f ( x ) = 1 + (cid:80) ni =1 x i − (cid:81) ni =1 cos ( x i √ i ) -600 ≥ x i ≤ Results are tabulated in Table 5.3, the table presents real encoded functions: Rastrigin,Schwefel, and Griewank with generational (G
HaEa ) and steady state (SS
HaEa ) replace-ments for standard and ReGen implementations. Additionally, the table includes the bestfitness based on the minimum median performance, following the standard deviation ofthe observed value, and the iteration where the reported fitness is found, which is enclosedin square brackets. The last row displays
HaEa (XUG) implementation with the best re-sults reported by Gomez [30, 31] using three different genetic operators: Single real pointcrossover, Uniform mutation, and Gaussian mutation (XUG).
HaEa (XUG) performedexperiments with real encoding.
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION
HaEa and ReGen
HaEa , two methods are applied: steady state and generational populationreplacements. The fitness evolution of individuals can be noted by tracking green and redlines, which depict best individuals’ fitness for the standard
HaEa . Blue and black linestrace best individuals’ fitness for ReGen
HaEa . Figures on the right side show definedmarking periods. Vertical lines in blue depict the starting of a marking period, lines ingray delimit the end of such periods.
Table 5.3.
Results of the experiments for Generational and Steady state replacements
EA Rastrigin Schwefel Griewank
ReGen G
HaEa . ± . . ± . . ± . HaEa . ± . . ± . . ± . HaEa . ± . . ± . . ± . HaEa . ± . . ± . . ± . HaEa (XUG) 0 . ± . . ± . . ± . - - + + + Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best - - + + + Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best
Figure 5.3.
Rastrigin. Generational replacement (G
HaEa ) and Steady state replacement(SS
HaEa ). - - + + Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best - - + + Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best
Figure 5.4.
Schwefel. Generational replacement (G
HaEa ) and Steady state replacement(SS
HaEa ). HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION - - + + + Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best - - + + + Iterations F i t ne ss Classic GHAEA Best
ReGen GHAEA Best
Classic SSHAEA Best
ReGen SSHAEA Best
Figure 5.5.
Griewank. Generational replacement (G
HaEa ) and Steady state replacement(SS
HaEa ). Based on tabulated results in Table 5.3 it can be noted that ReGen
HaEa imple-mentations perform better than standards
HaEa implementations, including results from
HaEa (XUG), which used real encoding for experiments. ReGen
HaEa is able to dis-cover suitable candidate solutions. In Fig. 5.3, Fig. 5.4, and Fig. 5.5 is observable thatmarking periods applied on chromosomes at iterations 200, 500, and 800 does cause agreat change on the evolution of individuals. After starting the first marking period,populations improve their performance once tags are added, especially for Rastrigin andSchwefel functions. It is remarkable how in every defined marking period (delimited withvertical blue lines), individuals improve their fitness. For the Griewank function, there isa small margin of difference between the two implementation performances, even thoughReGen
HaEa accomplishes better results. In Fig. 5.5 is evident that the pressure appliedon chromosomes at iteration 200 affects the evolution of individuals, the fitness improves,and keeps stable for best individuals until the evolution process finishes. ReGen
HaEa found a variety of good solutions during the evolution process, exposing the ability of theproposed approach to discover local minima that are not identified by standard
HaEa implementations.
Three different tests are performed, One-Way ANOVA test, Pairwise Student’s t-test, andPaired Samples Wilcoxon Test (also known as Wilcoxon signed-rank test). The data setReGen EAs Samples in Appendix B is used. The samples contain four
HaEa imple-mentations for each of the following functions: Deceptive Order Three, Deceptive OrderFour Trap, Rastrigin, Schwefel, and Griewank. The samples refer to the best fitness ofa solution found in each run, the number of executions per algorithm is 30. Differentimplementations involve standard
HaEa and ReGen
HaEa with Generational (G) andSteady State (SS) population replacements.
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION Table 5.4.
Anova Single Factor: SUMMARY
Deceptive Order Three
Groups Count Sum Average VarianceGHAEA 30 103146 3438.2 102.993103SSHAEA 30 103052 3435.066667 111.650575ReGenGHAEA 30 107554 3585.133333 93.4298851ReGenSSHAEA 30 107650 3588.333333 78.3678161
Deceptive Order Four Trap
Groups Count Sum Average VarianceGHAEA 30 11815 393.8333333 9.385057471SSHAEA 30 11737 391.2333333 20.73678161ReGenGHAEA 30 13410 447 4.75862069ReGenSSHAEA 30 13390 446.3333333 3.471264368
Rastrigin
Groups Count Sum Average VarianceGHAEA 30 329.0666924 10.96888975 20.2816323SSHAEA 30 403.9728574 13.46576191 26.1883055ReGenGHAEA 30 4.911251327 0.163708378 0.14340697ReGenSSHAEA 30 3.576815371 0.119227179 0.09299807
Schwefel
Groups Count Sum Average VarianceGHAEA 30 1344.597033 44.81990111 7258.77527SSHAEA 30 4439.27726 147.9759087 13144.3958ReGenGHAEA 30 30.50459322 1.016819774 31.002189ReGenSSHAEA 30 218.4970214 7.283234047 527.103699
Griewank
Groups Count Sum Average VarianceGHAEA 30 6.520457141 0.217348571 0.01290201SSHAEA 30 7.181644766 0.239388159 0.02090765ReGenGHAEA 30 1.624738713 0.054157957 0.0015871ReGenSSHAEA 30 1.61989951 0.05399665 0.000493Based on ReGen EAs Samples in Appendix B, the analysis of variance is computed toknow the difference between evolutionary algorithms with different implementations thatinclude standard
HaEa and ReGen
HaEa with generational and steady state replacementstrategies. Algorithms are four in total, in Table 5.4 a summary of each function andalgorithm is shown. The summary presents the number of samples per algorithm (30), thesum of fitnesses, the average fitness, and their variances. Results of the ANOVA singlefactor is tabulated in Table 5.5.
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION Table 5.5.
Anova Single Factor: ANOVA
Deceptive Order Three
Source of Variation SS df MS F P-value F critBetween Groups 676201.16 3 225400.38 2333.0875 1.7577E-103 2.6828Within Groups 11206.8 116 96.6103Total 687407.96 119
Deceptive Order Four Trap
Source of Variation SS df MS F P-value F critBetween Groups 88020.6 3 29340.2 3060.1179 3.2412E-110 2.6828Within Groups 1112.2 116 9.5879Total 89132.8 119
Rastrigin
Source of Variation SS df MS F P-value F critBetween Groups 4468.33 3 1489.44 127.5582 1.39871E-36 2.6828Within Groups 1354.48 116 11.6765Total 5822.8196 119
Schwefel
Source of Variation SS df MS F P-value F critBetween Groups 415496.57 3 138498.85 26.4294 4.22517E-13 2.6828Within Groups 607877.03 116 5240.3192Total 1023373.61 119
Griewank
Source of Variation SS df MS F P-value F critBetween Groups 0.918607 3 0.306202 34.1270 6.92905E-16 2.6828Within Groups 1.040803 116 0.008972Total 1.959410 119As P-values for Deceptive Order Three, Deceptive Order Four Trap, Rastrigin, Schwe-fel, and Griewank functions are less than the significance level 0 .
05, results allow concludingthat there are significant differences between the groups as shown in Table 5.5. In one-wayANOVA tests, significant P-values indicate that some of the group means are different, butit is not evident which pairs of groups are different. In order to interpret one-way ANOVAtest’ results, multiple pairwise-comparison with Student’s t-test is performed to determineif the mean difference between specific pairs of the group is statistically significant. Also,paired-sample Wilcoxon tests are computed. HA P T E R . R E G E NHA E A : B I NA R YAN D R E A L C O D I F I C A T I O N GHAEA
ReGenGHAEA
ReGenSSHAEA
SSHAEA
EAs F i t ne ss GHAEA
ReGenGHAEA
ReGenSSHAEA
SSHAEA
EAs F i t ne ss GHAEA
ReGenGHAEA
ReGenSSHAEA
SSHAEA EAs F i t ne ss GHAEA
ReGenGHAEA
ReGenSSHAEA
SSHAEA EAs F i t ne ss Figure 5.6.
EAs with Generational replacement (G
HaEa ) and Steady State replacement (SS
HaEa ). On top: Deceptive Order Three and DeceptiveOrder Four Trap Functions. On the bottom: Rastrigin and Schwefel functions.
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION GHAEA
ReGenGHAEA
ReGenSSHAEA
SSHAEA . . . . . . . . EAs F i t ne ss Figure 5.7.
EAs with Generational replacement (G
HaEa ) and Steady State replacement(SS
HaEa ). Griewank function.
Box plots in Fig. 5.6 and Fig. 5.7 depict the median fitness of EAs’ best solutions(ReGen EAs Samples in Appendix B). Four EAs are illustrated, epigenetic EAs in Or-ange (ReGen G
HaEa ) and Blue (ReGen SS
HaEa ), standard EAs in Gray (G
HaEa )and White (SS
HaEa ). For Deceptive Order Three function, the median fitness for eachEpigenetic EA is close to the global optimum (3600), while median fitnesses for classic
HaEa are under the local optimum (3450). On the other hand, Deceptive Order FourTrap median fitness surpasses 440 for all Epigenetic implementations; in contrast, for stan-dard
HaEa , the median fitness does not reach 400. For Rastrigin function, the medianfitness for each Epigenetic EA is lower than the local minima (1 . HaEa are over the local optimum (10). Next in order, epigenetic EAs forSchwefel achieved median fitness inferior to 0 . HaEa median fitnessesare greater than 0 . . HaEa are above 0 .
2. So, based on these data, it seemsthat Epigenetic
HaEa versions find better solutions than classic
HaEa implementations.However, it is needed to determine whether these findings are statistically significant.
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION Table 5.6.
Student T-tests pairwise comparisons with pooled standard deviation. BenjaminiHochberg (BH) as p-value adjustment method.
Deceptive Order Three
EAs GHAEA ReGenGHAEA ReGenSSHAEAReGenGHAEA 2.92E-87 - -ReGenSSHAEA 3.65E-88 0.2194596 -SSHAEA 0.2194596 3.65E-88 1.02E-88
Deceptive Order Four Trap
EAs GHAEA ReGenGHAEA ReGenSSHAEAReGenGHAEA 6.52E-94 - -ReGenSSHAEA 2.04E-93 0.40607501 -SSHAEA 0.00180068 8.80E-96 1.72E-95
Rastrigin
EAs GHAEA ReGenGHAEA ReGenSSHAEAReGenGHAEA 1.83E-22 - -ReGenSSHAEA 1.83E-22 0.959877985 -SSHAEA 0.006585082 1.27E-28 1.27E-28
Schwefel
EAs GHAEA ReGenGHAEA ReGenSSHAEAReGenGHAEA 0.03120645 - -ReGenSSHAEA 0.05632517 0.73803211 -SSHAEA 4.21E-07 1.29E-11 3.65E-11
Griewank
EAs GHAEA ReGenGHAEA ReGenSSHAEAReGenGHAEA 1.35E-09 - -ReGenSSHAEA 1.35E-09 0.99474898 -SSHAEA 0.44325525 2.87E-11 2.87E-11
Multiple pairwise t-test:
Multiple pairwise-comparison between means of groups isperformed. In the one-way ANOVA test described above, significant p-values indicatethat some group means are different. In order to know which pairs of groups are different,multiple pairwise-comparison is performed for Deceptive Order Three (D3), DeceptiveOrder Four Trap (D4), Rastrigin (RAS), Schwefel (SCHW), and Griewank (GRIE) bestsolutions samples. Table 5.6 presents Pairwise comparisons using t-tests with pooledstandard deviation (SD) with their respective p-values. The test adjusts p-values withthe Benjamini-Hochberg method. Pairwise comparisons show that only highlighted valuesin gray between two algorithms are significantly different ( p < . HaEa and
HaEa with regulated genes), theWilcoxon test is conducted.
HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION Paired Samples Wilcoxon Test:
For this test, algorithms are grouped per populationreplacement strategy. Wilcoxon signed rank test for generational EAs (G
HaEa and Re-Gen G
HaEa ) and Wilcoxon signed rank test for steady state EAs (SS
HaEa and ReGenSS
HaEa ). The test assesses standard
HaEa versus Epigenetic
HaEa implementations. • Deceptive Order Three (D3)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from G
HaEa and ReGen G
HaEa implementations. V =0, P-value is equal to 1 . e −
06, which is less than the significance levelalpha (0 . HaEa and ReGen SS
HaEa algorithms. V = 0, P-value is equal to 1 . e −
06, which is less than the significance level alpha = 0 . • Deceptive Order Four Trap (D4)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from G
HaEa and ReGen G
HaEa implementations. V =0, P-value is equal to 1 . e −
06, which is less than the significance levelalpha (0 . HaEa and ReGen SS
HaEa versions. V = 0, P-value is equal to 1 . e −
06, which is less than the significance level alpha = 0 . • Rastrigin (RAS)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from G
HaEa and ReGen G
HaEa implementations. V =465, P-value is equal to 1 . e −
09, which is less than the significance levelalpha (0 . HaEa and ReGen SS
HaEa algorithms. V = 465, P-value is equal to 1 . e −
09, which is less than the significance level alpha =0 . • Schwefel (SCHW)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from G
HaEa and ReGen G
HaEa implementations. V =450, P-value is equal to 2 . e −
07, which is less than the significance level alpha = 0 . HAPTER 5. REGEN HAEA: BINARY AND REAL CODIFICATION
HaEa and ReGen SS
HaEa versions. V = 452, P-value is equal to 1 . e −
07, which is less than the significance level alpha(0 . • Griewank (GRIE)1. Wilcoxon signed rank test with continuity correction for generational EAs usesall data-set samples from G
HaEa and ReGen G
HaEa implementations. V =462, P-value is equal to 9 . e −
09, which is less than the significance level alpha = 0 . HaEa and ReGen SS
HaEa algorithms. V = 465, P-value is equal to 1 . e −
09, which is less than the significance level alpha(0 . HaEa ) are significantly differentfrom median fitnesses of solutions found by generational
HaEa with regulated genes (Re-Gen G
HaEa ) with p-values equal to 1 . e −
06 (D3 samples), 1 . e −
06 (D4samples), 1 . e −
09 (RAS samples), 2 . e −
07 (SCHW samples), and 9 . e −
09 (GRIEsamples). So, the alternative hypothesis is true.The median fitness of solutions found by classic steady state Hybrid Adaptive Evolu-tionary Algorithms (SS
HaEa ) is significantly different from the median fitness of solutionsfound by steady state
HaEa with regulated genes (ReGen SS
HaEa ) with p-values equalto 1 . e −
06 (D3 sampling fitness), 1 . e −
06 (D4 sampling fitness), 1 . e − . e −
07 (SCHW sampling fitness), and 1 . e −
09 (GRIEsampling fitness). As p-values are less than the significance level 0 .
05, it may be con-cluded that there are significant differences between the two EAs groups in each WilcoxonTest.
The epigenetic technique is implemented on
HaEa to solve both binary and real encodingproblems. Results have shown that the marking process did impact the way populationsevolve, and the fitness of individuals considerably improves to the optimum. It is importantto point out that only two operators are used: single point Crossover and single bitMutation. This thesis intends to avoid giving too many advantages to implemented EAsin terms of parametrization and specialized operators in order to identify the applicabilityof the proposed epigenetic model. The statistical analysis helps to conclude that epigeneticimplementations performed better than standard versions.
HAPTER Concluding Remarks
Epigenetics has proven to be a useful field of study to extract elements for improving theframework of evolutionary algorithms. Primarily because epigenetic encompasses mecha-nisms that support inheritance and prolongation of experiences, so future generations haveenough information to adapt to changing environments. Based on the preceding, someelements are used to bring into life the ReGen EA technique. This research abstractsepigenetics fundamental concepts and introduces them as part of standard evolutionaryalgorithms’ elements or operations.Modeling epigenetic evolutionary algorithms is not easy, mainly because epigenetic in-volves too many elements, concepts, principles, and interactions to describe what is knownabout the epigenetic landscape today. The process of designing epigenetic algorithms re-quires well-defined abstractions to simplify the epigenetic dynamics and its computationalimplementation. Epigenetic strategies for EAs variations have been designed during thelast decade; those strategies have reported improvement in EAs performance and reduc-tion in the computational cost when solving specific problems. Nevertheless, almost allstrategies use the same idea of switching genes off and on (gene activation mechanism),or silencing chromosome sections in response to a changing environment. This approachis correct, but epigenetic goes beyond on and off states. The ReGen EA approach focuseson developing interactions by affecting genetic codes with tags that encode epigeneticinstructions.One thing ReGen EA has in common with other strategies is the use of epigeneticmechanisms such as DNA Methylation. This research characterizes DNA Methylationalong with the Histone Modification mechanism. These epigenetic mechanisms are thebest characterization among all epigenetic modifications, the most studied, and offer adescription that is easy to understand and represent computationally. Most epigenetic ap-96
HAPTER 6. CONCLUDING REMARKS .Compared to other approaches, emerging interactions from the dynamic of marking andreading processes is beneficial to combine multiple schemes and build varied phenotypes;avoiding to create such large genomes and regulate them by activation and deactivationmechanisms. The epigenetic mechanisms aforementioned have been useful for the design ofthe markers, but it has been too complicated to define what tags encode today. Tags designinvolves encoding, structure, and meaning. These properties have helped in proposing tagssections and rules to be interpreted by the reader function.In this thesis, gene regulation is accomplished by adding, modifying, and removingepigenetic tags from individuals genome. The complete regulation process produces phe-notypes that, in most cases, become feasible solutions to a problem. Tags structure con-tains binary operations; defining operations has implied a process of trial and error. Theoperations do not represent any biological mechanism, this fact may be miss-interpreted;it is clear that the decision to include binary operations to build the instructions may beseen as advantageous, but it is not the case. The operations have been selected taking intoaccount many factors, three of the most prominent are: first, designed tags are meant toonly solve binary and real defined problems since it is the scope of this research; second,the idea has always been to avoid giving advantages to the marking process and followsome basic principles that biological epigenetic mechanisms offer when attaching and in-terpreting tags, based on this, simpler operations, that do not cause abrupt changes tothe phenotype generation process are chosen; and third, chemical tags in biology containepigenetic codes that are interpreted to maintain the dynamic of many natural processes,in this case, defined operations are considered plausible to solve binary and real codedproblems. Even though other operations must be explored, it is conceivable that betteroperations have not been taken into consideration yet.
HAPTER 6. CONCLUDING REMARKS
HAPTER 6. CONCLUDING REMARKS
CpG islands ). The hardest activity has been focused on testingthe entire framework through the marking and reading processes. The positive thing isthe ReGen EA architecture is simple, the epigenetic elements do not add complexity inits implementation, ReGen EA follows a generic idea of an individual with a genomeand a well-defined epigenome structure that is shaped during the evolution process. Thisapproach obeys to a population-based bio-inspired technique that adapts while signals fromthe environment influence the epigenome; the phenotype is configured from interactionsbetween the genotype and the environment; tags are transmitted through generations tomaintaining a notion of memory between generations.It is important to mention that, EA implementations are also statistically analyzed;the median analysis of samples allows graphically depict groups of data and explain themvisually to identify the distribution of the samples’ median fitnesses. For all functions,except by Max Ones function, median values outpoint the samples from EA standardversions. The analysis to find differences statistically significant between EA groups alsoconfirms there are remarkable differences between the algorithms. This process has evi-denced an improvement in epigenetic implementations performance compared to standardversions as reported in experimental results; differences between epigenetic algorithms andstandard EAs fitnesses vary significantly, leading to conclude that introducing epigeneticfactors to classical versions of EAs do accelerate the search process. These analyses alsodemonstrate it is not needed to increase or vary mutations rate -for classical mutations-;experiments have used the same rate, inversely, ReGen EA takes advantage of the recom-bination operator -to promote inheritance-; this operator is a powerful element to evaluateresults. For many strategies mutation operator introduces diversity, this thesis does notdeny it but instead embraces the idea that mutations can be used with a low rate to havea closer occurrence as seen in biology, and contemplates epigenetic assimilation -fixedchanges- to influence the fitness.Epigenetic components presented in this thesis for the Evolutionary Algorithms frame-work, describe a way to model Epigenetic Evolutionary Algorithms. ReGen EvolutionaryAlgorithms involve populations of individuals with genetic and epigenetic codes. Thisresearch mainly focuses on those experiences that individuals could acquire during theirlife cycle and how epigenetic mechanisms lead to learn and adapt for themselves underdifferent conditions. So, such experiences can be inherited over time, and populationswould evidence a kind of power of survival. To validate the technique applicability, onlyproblems with real definition and binary encoding schemes were selected. Designed oper-ations are meant to exclusively cover problems with these kinds of encoding, even though,problems with different encoding should be addressed by transforming their domain setinto a binary representation; the performance and possible results of such implementa-tions are unknown since no experiments of that kind have been conducted, but ReGen EAmust allow them to be performed. The journey with this research allows concluding that
HAPTER 6. CONCLUDING REMARKS
Epigenetic mechanisms offer a variety of elements to extend this work and improve theadaptation of individuals in population-based methods to identify novelties during theevolution process. From a biological point of view, some ideas are linked to the factthat there are mechanisms that make epigenetic tags keep fixed and maintained over along time. It seems to be a process that let tags be bound without changing, just beingpreserved in the same location under specific environmental conditions or in certain lifestages of an individual to avoid their degradation. This hypothesis supports the idea ofmemory consolidation and adds another dimension to describe a kind of intelligence at amolecular level.It is intended to extend this model to cover a wider set of optimization problems,different from binary and real encoding problems. The plan is to design a mechanismto create dynamic tags during the evolution process, tags that use a generic encodingand do not depend on specific encoding problems. Problems with numbers-forms, chars,instructions, permutations, commands, expressions among others encoding schemes thatcan be influenced by generic-defined tags. Base on the former, designed operations needto be redefined, the reading process might be extended, any domain-specific problem mustkeep its encoding and not be transformed into a binary representation, as it is the currentcase, and also the marking process may be expanded.Currently, the application of the marking actions by the ReGen EA is mutually ex-clusive; marking actions are not happening at the same time. The proceeding opens thepossibility to think about another marking process enhancement so that adding, removing,and modifying actions are applied independently based on their distributed probabilities.Performed experiments changing this configuration reveal that by having the possibility ofapplying them simultaneously with their probability rate, they can produce more suitableindividuals with scores closer to the global optimum. However, experiments are required tosee individuals’ behavior in problems with different encoding from binary and real defined,such as a permutation problem.From the computational point of view, attempts are made to facilitate this model’sreplicability in the evolutionary algorithms community. It is expected to elaborate moretests by designing a complete benchmark to continue assessing the ReGen EA performanceand improving what it integrates today.
PPENDIX A Examples of Individuals with Tags
Table A.1.
Individual representation for Binary functions, D = 20. epiGrowingFunction ). The second row presents thegenotype code; the third row exhibits the bit string generated by the epigenetic growthfunction. Finally, the fourth row shows the phenotype representation of the individual.For Real defined functions, binary strings of -bits encode real values.101 PPE N D I XA . E XA M P L E S O F I N D I V I D UA L S W I T H T A G S Table A.2.
Individual representation for Real functions, D = 2. PPENDIX B Standard and ReGen EAs Samples for StatisticalAnalysis
Standard and ReGen GAs Samples are tabulated from Appendices B.1 to B.8 by function.Tables present fitness samples of twenty implementations with crossover rates (X) from 0 . .
0, generational (G), and steady state (SS) population replacements. Columns reportEA implementations and rows contain the runs per algorithm, in total there are thirtyruns. Algorithms are represented by numbers from one to twenty, for example, GGAX06refers to a standard generational GA with crossover rate 0 . .On the other hand, HaEa implementations in Appendix B.9 are grouped by function,each function reports four implementations: standard generational
HaEa (GHAEA (1)),steady state
HaEa (SSHAEA (2)), ReGen generational
HaEa (ReGenGHAEA (3)), andReGen steady state
HaEa (ReGenSSHAEA (4)). Columns report EA implementationsand rows contain the runs per algorithm, in total there are thirty runs. Algorithms arerepresented by numbers from one to four. GGAX06 (1), GGAX07 (2), GGAX08 (3), GGAX09 (4), GGAX10 (5), SSGAX06 (6), SSGAX07(7), SSGAX08 (8), SSGAX09 (9), SSGAX10 (10), ReGenGGAX06 (11), ReGenGGAX07 (12), ReGenG-GAX08 (13), ReGenGGAX09 (14), ReGenGGAX10 (15), ReGenSSGAX06 (16), ReGenSSGAX07 (17),ReGenSSGAX08 (18), ReGenSSGAX09 (19), ReGenSSGAX10 (20).
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.1.
Deceptive Order Three Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitnessvalue per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.2.
Deceptive Order Four Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness valueper run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.3.
Royal Road Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness value per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.4.
Max Ones Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness value per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.5.
Rastrigin Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness value per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.6.
Rosenbrock Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness value per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.7.
Schwefel Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness value per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.8.
Griewank Fitness Sampling: Ten Classic GAs and Ten ReGen GAs with different crossover rates and 30 runs. Best fitness value per run.
PPE N D I X B . S T AN D A R D AN D R E G E N E A SS A M P L E S F O R S T A T I S T I C A L ANA L Y S I S Table B.9.
Fitness Sampling: Two standard
HaEa and Two ReGen
HaEa implementations with 30 runs. Best fitness value per run.
Deceptive Order Three Deceptive Order Four Ratrigin Schwefel Griewank1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 43430 3440 3596 3576 392 387 447 447 5.8084 5.6855 0.0337 0.0298 1.489E+02 9.149E+01 2.284E-04 2.636E-04 0.1294 0.3327 0.0609 0.06193456 3438 3586 3584 391 393 446 446 17.8005 18.9239 0.0147 0.0223 1.184E+02 2.979E+02 2.672E-04 2.459E-04 0.0805 0.1648 0.1170 0.02613444 3446 3592 3594 390 387 446 443 5.3688 19.2643 0.0198 0.0160 9.149E+01 4.131E-04 2.870E-04 3.050E+01 0.2237 0.3112 0.0248 0.05593454 3436 3590 3590 395 393 448 449 14.7865 10.5025 0.0000 0.0099 3.515E-04 1.496E+02 2.284E-04 2.284E-04 0.2458 0.1858 0.1020 0.09853436 3450 3570 3590 393 382 441 447 6.6905 14.0264 0.0246 0.0259 6.099E+01 6.099E+01 2.591E-04 1.184E+02 0.0965 0.4865 0.0517 0.08223430 3434 3588 3574 391 394 449 444 12.0701 20.2973 0.0239 0.0099 4.130E-04 4.438E-04 2.365E-04 2.591E-04 0.1813 0.0913 0.0228 0.06073438 3426 3578 3598 389 395 449 449 15.1366 6.5026 1.3411 0.0296 6.099E+01 2.099E+02 2.284E-04 2.591E-04 0.0622 0.4339 0.0614 0.02833438 3444 3590 3586 394 392 446 446 4.1784 13.2038 0.0149 0.0048 4.131E-04 3.109E+02 3.096E-04 7.655E+00 0.3097 0.2075 0.0494 0.00993422 3442 3576 3590 392 405 442 446 12.1856 6.9979 0.0148 0.0187 3.823E-04 2.099E+02 2.283E-04 2.899E-04 0.2064 0.1752 0.1728 0.04263426 3428 3598 3596 395 392 449 446 5.7429 22.4438 0.0248 0.0298 3.050E+01 3.321E+02 3.044E-04 3.515E-04 0.5050 0.1125 0.0142 0.03563432 3418 3588 3590 393 391 449 446 8.9663 16.6233 0.0112 0.0248 3.823E-04 2.935E+02 2.283E-04 2.364E-04 0.1039 0.3101 0.0007 0.05673430 3422 3596 3592 396 394 448 446 16.6328 12.2330 0.0209 0.0149 6.074E+01 3.050E+01 2.284E-04 2.591E-04 0.2305 0.0818 0.0173 0.06173444 3420 3582 3570 399 388 447 444 9.4359 16.1912 0.0198 0.0273 3.823E-04 1.220E+02 2.591E-04 2.591E-04 0.2643 0.3328 0.0466 0.07383450 3442 3594 3596 394 389 449 446 16.1239 10.1660 0.0248 0.0177 3.823E-04 2.674E+02 3.188E-04 2.591E-04 0.2036 0.2282 0.0497 0.04953442 3426 3596 3590 395 394 446 448 7.2021 5.3528 0.0389 0.0284 6.099E+01 4.462E+02 2.591E-04 2.364E-04 0.1272 0.1607 0.0522 0.02963426 3430 3566 3596 393 391 446 448 13.8471 12.2382 0.0112 0.0112 4.368E+02 1.184E+02 2.591E-04 2.445E-04 0.2504 0.1152 0.0780 0.03423422 3432 3578 3588 389 394 442 449 13.9872 16.4488 0.0099 1.0186 3.207E-04 1.185E+02 2.283E-04 3.196E-04 0.2690 0.3538 0.0745 0.06843438 3436 3588 3588 397 395 448 447 2.0198 7.3038 1.0236 1.0225 4.130E-04 1.492E+02 2.592E-04 2.283E-04 0.1056 0.0869 0.0484 0.05313430 3420 3570 3586 394 387 447 446 19.4519 3.8559 0.0198 0.0099 3.050E+01 4.131E-04 2.284E-04 3.050E+01 0.1198 0.2346 0.0314 0.03073446 3448 3576 3594 396 385 447 445 6.8260 16.2976 0.0197 0.0198 3.050E+01 3.050E+01 2.591E-04 2.592E-04 0.2196 0.2147 0.0261 0.05913452 3418 3592 3592 390 395 449 449 11.1479 15.3320 0.0274 0.0149 3.050E+01 1.490E+02 2.283E-04 2.899E-04 0.3161 0.4101 0.0746 0.05593446 3448 3576 3562 398 392 448 447 7.6809 12.3748 0.0294 0.0298 3.050E+01 6.099E+01 2.283E-04 2.591E-04 0.2613 0.1142 0.0404 0.02983440 3442 3588 3586 388 391 447 448 11.1480 13.3377 0.0182 0.0177 4.131E-04 2.711E+02 3.050E+01 2.591E-04 0.3308 0.1421 0.0370 0.09693446 3432 3596 3596 395 387 449 444 16.5726 17.9414 0.0197 1.0143 4.130E-04 3.050E+01 2.899E-04 2.284E-04 0.2313 0.0914 0.1219 0.05303436 3426 3598 3590 397 391 449 444 9.8528 20.9124 0.0099 0.0098 1.222E+02 1.489E+02 2.592E-04 2.283E-04 0.5083 0.2356 0.1160 0.04253456 3434 3578 3600 401 394 448 447 12.8651 6.3322 0.0112 0.0246 3.515E-04 3.050E+01 2.672E-04 2.284E-04 0.1499 0.0751 0.0680 0.06933446 3430 3592 3594 393 387 446 449 6.4923 16.9398 0.0099 0.0147 3.050E+01 1.801E+02 2.592E-04 3.050E+01 0.1251 0.7126 0.0270 0.07463432 3460 3586 3588 396 396 449 442 13.9234 17.4008 0.0215 0.0198 3.207E-04 1.191E+02 2.284E-04 2.591E-04 0.3905 0.3456 0.0044 0.04473436 3440 3566 3598 394 383 448 446 14.8389 15.8254 1.0198 0.0198 4.746E-04 1.490E+02 2.672E-04 9.049E-01 0.1547 0.1656 0.0188 0.03873422 3444 3584 3576 395 393 445 446 10.2835 13.0179 1.0322 0.0198 4.131E-04 6.099E+01 2.283E-04 2.591E-04 0.1183 0.2691 0.0149 0.0960 ibliography [1] Mazhar Ansari Ardeh,
Benchmarkfcns , The page is publicly available athttp://benchmarkfcns.xyz/fcns, 2020.[2] Christian Arnold, Peter Stadler, and Sonja Prohaska,
Chromatin computation: Epi-genetic inheritance as a pattern reconstruction problem , Journal of theoretical biology (2013), 61–74.[3] Thomas Back,
Evolutionary algorithms in theory and practice: evolution strategies,evolutionary programming, genetic algorithms , Oxford university press, 1996.[4] Mark Bedau,
Artificial life , Handbook of the Philosophy of Science. Volume 3: Phi-losophy of Biology (Mohan Matthen and Christopher Stephens, eds.), Elsevier BV,2007.[5] Adrian Bird,
Dna methylation patterns and epigenetic memory , Genes & development (2002), no. 1, 6–21.[6] Serdar Birogul, Epigenetic algorithm for optimization: Application to mobile networkfrequency planning , Arabian Journal for Science and Engineering (2016), no. 3,883–896.[7] Subhankar Biswas and Chamallamudi Rao, Epigenetic tools (the writers, the read-ers and the erasers) and their implications in cancer therapy , European Journal ofPharmacology (2018), 0.[8] Ulrich Bodenhofer,
Genetic algorithms: theory and applications , 2003, Lecture NotesThird Edition—Winter 2003/2004.[9] Paige Bommarito and Rebecca Fry,
The role of dna methylation in gene regulation ,pp. 127–151, Elsevier, 01 2019.[10] Warren Burggren,
Epigenetic inheritance and its role in evolutionary biology: Re-evaluation and new perspectives , Biology (2016), 24.[11] Edmund K Burke and Graham Kendall, Search methodologies: Introductory tutorialsin optimization and decision support techniques , Springer Science & Business Media,2013.[12] Marco Cavazzuti,
Optimization methods: from theory to design scientific and techno-logical aspects in mechanics , Springer Science & Business Media, 2012.113
IBLIOGRAPHY
The complex role of the znf224 transcription factor in cancer , vol. 107,pp. 191–222, Elsevier, 12 2017.[14] Oliver Chikumbo, Erik Goodman, and Kalyanmoy Deb,
Approximating a multi-dimensional pareto front for a land use management problem: A modified moea withan epigenetic silencing metaphor , 2012 IEEE congress on evolutionary computation,IEEE, 2012, pp. 1–9.[15] ,
Triple bottomline many-objective-based decision making for a land use man-agement problem , Journal of Multi-Criteria Decision Analysis (2015), no. 3-4,133–159.[16] B.J. Clark and Carolyn Klinge, Control of gene expression , pp. 51–69, Elsevier, 122010.[17] Dipankar Dasgupta and Douglas R McGregor, sga: A structured genetic algorithm ,Citeseer, 1993.[18] Richard Dawkins,
The selfish gene oxford university press , New York, New York, USA(1976), 1976.[19] Carrie Deans and Keith Maggert,
What do you mean, ”epigenetic”? , Genetics (2015), 887–96.[20] Jason Digalakis and Konstantinos G. Margaritis,
An experimental study of bench-marking functions for genetic algorithms , Int. J. Comput. Math. (2002), 403–416.[21] AE Eiben and JE Smith, Evolutionary computing: The origins , Introduction to Evo-lutionary Computing, Springer, 2015, pp. 13–24.[22] Marie-Anne F´elix and Andreas Wagner,
Robustness and evolution: concepts, insightsand challenges from a developmental model system , Heredity (2008), no. 2, 132–140.[23] Richard Festenstein,
Epigenetics and epigenomics in human health and disease ,pp. 51–74, Elsevier, 12 2016.[24] Alessandro Fontana,
Epigenetic tracking: biological implications , European Confer-ence on Artificial Life, Springer, 2009, pp. 10–17.[25] Stephanie Forrest and Melanie Mitchell,
Relative building-block fitness and thebuilding-block hypothesis , Foundations of genetic algorithms, vol. 2, Elsevier, 1993,pp. 109–126.[26] Genetic Home Reference GHR,
Help me understand genetics , How Genes Work, Thechapter is publicly available at https://ghr.nlm.nih.gov/primer, 2018.[27] Jeffrey Gilbert,
Epigenetics in the developmental origin of cardiovascular disorders ,pp. 127–141, Elsevier, 12 2016.[28] Aaron Goldberg, C. Allis, and Emily Bernstein,
Epigenetics: A landscape takes shape ,Cell (2007), 635–8.
IBLIOGRAPHY
Messy genetic algorithms: Motivation analysis, and first results ,Complex systems (1989), 415–444.[30] Jonatan Gomez, Self Adaptation of Operator Rates for Multimodal Optimization ,CEC2004 Congress on Evolutionary Computation, vol. 2, 2004, pp. 1720–1726.[31] ,
Self Adaptation of Operator Rates in Evolutionary Algorithms , Genetic andEvolutionary Computation - GECCO 2004, Part I, Lecture Notes in Computer Sci-ence, vol. 3102, Springer, 2004, pp. 1162–1173.[32] Crina Grosan and Ajith Abraham,
Hybrid evolutionary algorithms: Methodologies,architectures, and reviews , pp. 1–17, Springer Berlin Heidelberg, Berlin, Heidelberg,2007.[33] John H Holland,
Adaptation in natural and artificial systems. an introductory anal-ysis with application to biology, control, and artificial intelligence , Ann Arbor, MI:University of Michigan Press (1975), 439–444.[34] Robin Holliday,
Epigenetics: an overview , Developmental genetics (1994), no. 6,453–457.[35] ROBIN Holliday, Is there an epigenetic component in long-term memory? , Journal ofTheoretical Biology (1999), no. 3, 339–341.[36] Eva Jablonka and Marion Lamb,
The changing concept of epigenetics , Annals of theNew York Academy of Sciences (2003), 82–96.[37] William R. Jeffery,
Chapter 12 - astyanax mexicanus: A vertebrate model for evolu-tion, adaptation, and development in caves , Encyclopedia of Caves (Third Edition)(William B. White, David C. Culver, and Tanja Pipan, eds.), Academic Press, thirdedition ed., 2019, pp. 85 – 93.[38] Gunnar Kaati, Lars O Bygren, and Soren Edvinsson,
Cardiovascular and diabetesmortality determined by nutrition during parents’ and grandparents’ slow growth pe-riod , European journal of human genetics (2002), no. 11, 682–688.[39] Gunnar Kaati, Lars Olov Bygren, Marcus Pembrey, and Michael Sj¨ostr¨om, Trans-generational response to nutrition, early life circumstances and longevity , EuropeanJournal of Human Genetics (2007), no. 7, 784–790.[40] Slawomir Koziel and Xin-She Yang, Computational optimization, methods and algo-rithms , vol. 356, Springer, 2011.[41] Ursula Kyle and Claude Pichard,
The dutch famine of 1944-1945: A pathophysiolog-ical model of long-term consequences of wasting disease , Current opinion in clinicalnutrition and metabolic care (2006), no. 4, 388–94.[42] William La Cava, Thomas Helmuth, Lee Spector, and Kourosh Danai, Genetic pro-gramming with epigenetic local search , Proceedings of the 2015 Annual Conference onGenetic and Evolutionary Computation, 2015, pp. 1055–1062.[43] William La Cava and Lee Spector,
Inheritable epigenetics in genetic programming ,Genetic Programming Theory and Practice XII, Springer, 2015, pp. 37–51.
IBLIOGRAPHY
Evolving dif-ferential equations with developmental linear genetic programming and epigenetic hillclimbing , Proceedings of the Companion Publication of the 2014 Annual Conferenceon Genetic and Evolutionary Computation, 2014, pp. 141–142.[45] Ingrid Lobo,
Environmental influences on gene expression
Genetic algorithms: An overview , Complexity (1995), no. 1, 31–39.[47] , An introduction to genetic algorithms , MIT press, 1998.[48] Melanie Mitchell and Stephanie Forrest,
Genetic algorithms and artificial life , Artifi-cial Life (1993), 267–289.[49] Pablo Moscato, On evolution, search, optimization, genetic algorithms and martialarts: Towards memetic algorithms , Caltech concurrent computation program, C3PReport (1989), 1989.[50] National Human Genome Research Institute NHGRI,
Epigenomics fact sheet
Roadmapepigenomics program
The new ge-netics , NIH Publication No.10-662, The publication is publicly available athttps://publications.nigms.nih.gov/thenewgenetics, 2010.[53] Jorge Nocedal and Stephen Wright,
Numerical optimization , Springer Science & Busi-ness Media, 2006.[54] Arnold L. Patton, Terrence Dexter, Erik D. Goodman, and William F. Punch,
On theapplication of cohort-driven operators to continuous optimization problems using evo-lutionary computation , Evolutionary Programming VII (Berlin, Heidelberg) (V. W.Porto, N. Saravanan, D. Waagen, and A. E. Eiben, eds.), Springer Berlin Heidelberg,1998, pp. 669–681.[55] Marcus E Pembrey, Lars Olov Bygren, Gunnar Kaati, S¨oren Edvinsson, Kate North-stone, Michael Sj¨ostr¨om, and Jean Golding,
Sex-specific, male-line transgenerationalresponses in humans , European journal of human genetics (2006), no. 2, 159–166.[56] David Penny, Epigenetics, darwin, and lamarck , Genome biology and evolution (2015), no. 6, 1758–1760.[57] Sathish Periyasamy, Alex Gray, and Peter Kille, The epigenetic algorithm , 2008 IEEECongress on Evolutionary Computation (IEEE World Congress on ComputationalIntelligence), IEEE, 2008, pp. 3228–3236.
IBLIOGRAPHY
The role of methylation in gene expression
Artificial life
Understanding epi-genetic changes in aging stem cells–a computational model approach , Aging cell (2014), no. 2, 320–328.[61] Esteban Ricalde, A genetic programming system with an epigenetic mechanism fortraffic signal control , arXiv preprint arXiv:1903.03854 (2019), 183.[62] Robert Santer,
Cellular mechanisms of aging , Brocklehurst’s Textbook of GeriatricMedicine and Gerontology (2010), 42–50.[63] Mehdi Shafa and Derrick Rancourt,
Stem cell epigenetics and human disease , Epige-netics in Human Disease (2012), 481–501.[64] Jorge Sousa and Ernesto Costa,
Epial-an epigenetic approach for an artificial lifemodel , Proceedings of the 2nd International Conference on Agents and Artificial In-telligence - Volume 1: ICAART,, INSTICC, SciTePress, 2010, pp. 90–97.[65] Jorge AB Sousa and Ernesto Costa,
Designing an epigenetic approach in artificiallife: The epial model , International Conference on Agents and Artificial Intelligence,Springer, 2010, pp. 78–90.[66] Daniel H. Stolfi and Enrique Alba,
Epigenetic algorithms: A new way of building gasbased on epigenetics , Information Sciences (2018), 250–272.[67] Daniel Hector Stolfi Rosso et al.,
Bio-inspired computing and smart mobility , Univer-sidad de Malaga (2018), 152–175.[68] Sonja Surjanovic and Derek Bingham,
Virtual library of simulation experiments: Testfunctions and datasets
How adaptive learning affects evolu-tion: Reviewing theory on the baldwin effect , Evolutionary biology (2012), 301–310.[70] Lukas Tamayo, Ontogenia y fisionomia del paisaje epigenetico: Un modelo generalpara explicar sistemas en desarrollo , Acta Biologica Colombiana (2013), 3–18.[71] Ivan Tanev and Kikuo Yuta, Epigenetic programming: Genetic programming incorpo-rating epigenetic learning through modification of histones , Information Sciences (2008), no. 23, 4469–4481.[72] Alexander P Turner, Michael A Lones, Luis A Fuente, Susan Stepney, Leo SD Caves,and Andy M Tyrrell,
The incorporation of epigenetics in artificial gene regulatorynetworks , BioSystems (2013), no. 2, 56–62.[73] University of Utah UTAH,
Epigenetics , Genetic Science Learning Center, The in-formation is publicly available at http://learn.genetics.utah.edu/content/epigenetics,2013.
IBLIOGRAPHY
The epigenotype , Endeavour (1942), 18–20.[75] Bob Weinhold, Epigenetics: The science of change , Environmental health perspec-tives (2006), A160–7.[76] Iva B Zovkic, Mikael C Guzman-Karlsson, and J David Sweatt,