Approximating the Genetic Diversity of Populations in the Quasi-Equilibrium State

This paper analyzes an evolutionary algorithm in the quasi-equilibrium state, i.e., when the population of chromosomes fluctuates around a single peak of the fitness function. The analysis is aimed at approximating the genetic variance of the population when chromosomes are real-valued. The infinite population model is considered which allows the quasi-equilibrium state to be defined as the state when the density of chromosomes contained by the population remains unchanged over consecutive generations. This paper provides formulas for genetic diversity in the quasi-equilibrium state for fitness proportionate, tournament, and truncation selection types, with and without elitism, with Gaussian mutation, and with and without arithmetic crossover. The formulas are experimentally validated.

[1]  T. Nomura,et al.  An analysis on linear crossover for real number chromosomes in an infinite population size , 1997, Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC '97).

[2]  Adam Prügel-Bennett Modeling Finite Populations , 2002, FOGA.

[3]  Zbigniew Michalewicz,et al.  Parameter Setting in Evolutionary Algorithms , 2007, Studies in Computational Intelligence.

[4]  T. Mahnig,et al.  Evolutionary algorithms: from recombination to search distributions , 2001 .

[5]  Tatsuya Nomura An Analysis on Crossovers for Real Number Chromosomes in an Infinite Population Size , 1997, IJCAI.

[6]  Steven M. Gustafson An analysis of diversity in genetic programming , 2004 .

[7]  Nicholas J. Radcliffe,et al.  Forma Analysis and Random Respectful Recombination , 1991, ICGA.

[8]  E. Baake,et al.  Ancestral processes with selection: Branching and Moran models , 2007, q-bio/0702002.

[9]  Francesco Palmieri,et al.  Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation , 1994, IEEE Trans. Neural Networks.

[10]  Michael D. Vose,et al.  The simple genetic algorithm - foundations and theory , 1999, Complex adaptive systems.

[11]  Sewall Wright,et al.  Evolution and the Genetics of Populations. I, Genetic and Biometric Foundations. , 1969 .

[12]  Yong Gao,et al.  Comments on "Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. I. Basic properties of selection and mutation" [and reply] , 1998, IEEE Trans. Neural Networks.

[13]  Hans-Paul Schwefel,et al.  How to analyse evolutionary algorithms , 2002, Theor. Comput. Sci..

[14]  Kenneth A. De Jong,et al.  Measurement of Population Diversity , 2001, Artificial Evolution.

[15]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[16]  Iwona Karcz-Duleba,et al.  Dynamics of infinite populations evolving in a landscape of uni and bimodal fitness functions , 2001, IEEE Trans. Evol. Comput..

[17]  Dario Floreano,et al.  Measures of Diversity for Populations and Distances Between Individuals with Highly Reorganizable Genomes , 2004, Evolutionary Computation.

[18]  S. Gould,et al.  Punctuated equilibria: an alternative to phyletic gradualism , 1972 .

[19]  D. Fogel Evolutionary algorithms in theory and practice , 1997, Complex..

[20]  Heinz Mühlenbein,et al.  The Science of Breeding and Its Application to the Breeder Genetic Algorithm (BGA) , 1993, Evolutionary Computation.

[21]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[22]  Mengjie Zhang,et al.  Another investigation on tournament selection: modelling and visualisation , 2007, GECCO '07.

[23]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Comparing Review , 2006, Towards a New Evolutionary Computation.

[24]  Kalyanmoy Deb,et al.  A Comparative Analysis of Selection Schemes Used in Genetic Algorithms , 1990, FOGA.

[25]  Lothar Thiele,et al.  A Mathematical Analysis of Tournament Selection , 1995, ICGA.

[26]  Christopher R. Stephens,et al.  Schemata Evolution and Building Blocks , 1999, Evolutionary Computation.

[27]  D. Pollard Convergence of stochastic processes , 1984 .

[28]  Samir W. Mahfoud A Comparison of Parallel and Sequential Niching Methods , 1995, ICGA.

[29]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[30]  H. Georgii,et al.  Mutation, selection, and ancestry in branching models: a variational approach , 2006, Journal of mathematical biology.

[31]  Tobias Blickle,et al.  Theory of evolutionary algorithms and application to system synthesis , 1997 .