The Equation for Response to Selection and Its Use for Prediction

The Breeder Genetic Algorithm (BGA) was designed according to the theories and methods used in the science of livestock breeding. The prediction of a breeding experiment is based on the response to selection (RS) equation. This equation relates the change in a population's fitness to the standard deviation of its fitness, as well as to the parameters selection intensity and realized heritability. In this paper the exact RS equation is derived for proportionate selection given an infinite population in linkage equilibrium. In linkage equilibrium the genotype frequencies are the product of the univariate marginal frequencies. The equation contains Fisher's fundamental theorem of natural selection as an approximation. The theorem shows that the response is approximately equal to the quotient of a quantity called additive genetic variance, VA, and the average fitness. We compare Mendelian two-parent recombination with gene-pool recombination, which belongs to a special class of genetic algorithms that we call univariate marginal distribution (UMD) algorithms. UMD algorithms keep the genotypes in linkage equilibrium. For UMD algorithms, an exact RS equation is proven that can be used for long-term prediction. Empirical and theoretical evidence is provided that indicates that Mendelian two-parent recombination is also mainly exploiting the additive genetic variance. We compute an exact RS equation for binary tournament selection. It shows that the two classical methods for estimating realized heritabilitythe regression heritability and the heritability in the narrow sensemay give poor estimates. Furthermore, realized heritability for binary tournament selection can be very different from that of proportionate selection. The paper ends with a short survey about methods that extend standard genetic algorithms and UMD algorithms by detecting interacting variables in nonlinear fitness functions and using this information to sample new points.

[1]  Calyampudi R. Rao,et al.  Linear statistical inference and its applications , 1965 .

[2]  Thomas Bäck,et al.  An Overview of Evolutionary Algorithms for Parameter Optimization , 1993, Evolutionary Computation.

[3]  Karl Pearson Mathematical Contributions to the Theory of Evolution. XII. On a Generalised Theory of Alternative Inheritance, with Special Reference to Mendel's Laws , 1904 .

[4]  H. Mühlenbein,et al.  Gene Pool Recombination in Genetic Algorithms , 1996 .

[5]  G. Bortolan,et al.  The problem of linguistic approximation in clinical decision making , 1988, Int. J. Approx. Reason..

[6]  Prügel-Bennett,et al.  Analysis of genetic algorithms using statistical mechanics. , 1994, Physical review letters.

[7]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[8]  M. Bulmer The Mathematical Theory of Quantitative Genetics , 1981 .

[9]  J. Haldane,et al.  The Causes of Evolution , 1933 .

[10]  H. Tietze,et al.  Über das Schicksal gemischter Populationen nach den Mendelschen Vererbungsgesetzen , 2022 .

[11]  R. Fisher XV.—The Correlation between Relatives on the Supposition of Mendelian Inheritance. , .

[12]  Larry J. Eshelman,et al.  Crossover's Niche , 1993, International Conference on Genetic Algorithms.

[13]  Thomas Bäck,et al.  Generalized Convergence Models for Tournament- and (mu, lambda)-Selection , 1995, ICGA.

[14]  T. Nagylaki Introduction to Theoretical Population Genetics , 1992 .

[15]  Colin R. Reeves,et al.  Epistasis in Genetic Algorithms: An Experimental Design Perspective , 1995, ICGA.

[16]  Alden H. Wright,et al.  Simple Genetic Algorithms with Linear Fitness , 1994, Evolutionary Computation.

[17]  D. Falconer,et al.  Introduction to Quantitative Genetics. , 1961 .

[18]  J. Besag Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[19]  Dirk Thierens,et al.  Convergence Models of Genetic Algorithm Selection Schemes , 1994, PPSN.

[20]  David E. Goldberg,et al.  SEARCH, Blackbox Optimization, And Sample Complexity , 1996, FOGA.

[21]  L. Baum,et al.  An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology , 1967 .

[22]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[23]  Magnus Rattray The Dynamics of a Genetic Algorithm under Stabilizing Selection , 1995, Complex Syst..

[24]  R. B. Robbins Some Applications of Mathematics to Breeding Problems III. , 1917, Genetics.

[25]  John H. Holland,et al.  Adaptation in natural and artificial systems , 1975 .

[26]  Gilbert Syswerda Simulated Crossover in Genetic Algorithms , 1992, FOGA.

[27]  Paul A. Viola,et al.  MIMIC: Finding Optima by Estimating Probability Densities , 1996, NIPS.

[28]  Lee Altenberg The Schema Theorem and Price's Theorem , 1994, FOGA.

[29]  Alan Hastings,et al.  Aspects of Optimality Behavior in Population Genetics Theory , 1995, Evolution and Biocomputation.

[30]  Heinz Mühlenbein,et al.  On the Mean Convergence Time of Evolutionary Algorithms without Selection and Mutation , 1994, PPSN.

[31]  B. Arnold,et al.  A first course in order statistics , 1994 .

[32]  Colin R. Reeves,et al.  An Experimental Design Perspective on Genetic Algorithms , 1994, FOGA.

[33]  Rich Caruana,et al.  Removing the Genetics from the Standard Genetic Algorithm , 1995, International Conference on Machine Learning.

[34]  Emile H. L. Aarts,et al.  Simulated annealing and Boltzmann machines - a stochastic approach to combinatorial optimization and neural computing , 1990, Wiley-Interscience series in discrete mathematics and optimization.

[35]  Heinz Mühlenbein,et al.  Estimating the Heritability by Decomposing the Genetic Variance , 1994, PPSN.

[36]  H. Muhlenbein,et al.  Gene pool recombination and utilization of covariances for the Breeder Genetic Algorithm , 1995, Proceedings of 1995 IEEE International Conference on Evolutionary Computation.

[37]  W J Ewens,et al.  An interpretation and proof of the Fundamental Theorem of Natural Selection. , 1989, Theoretical population biology.

[38]  Larry J. Eshelman,et al.  Productive Recombination and Propagating and Preserving Schemata , 1994, FOGA.

[39]  T Nagylaki Error bounds for the fundamental and secondary theorems of natural selection. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[40]  Heinz Mühlenbein,et al.  Predictive Models for the Breeder Genetic Algorithm I. Continuous Parameter Optimization , 1993, Evolutionary Computation.

[41]  Lothar Thiele,et al.  A Comparison of Selection Schemes Used in Evolutionary Algorithms , 1996, Evolutionary Computation.

[42]  Heinz Mühlenbein,et al.  Fuzzy Recombination for the Breeder Genetic Algorithm , 1995, ICGA.

[43]  N. Barton,et al.  Genetic and statistical analyses of strong selection on polygenic traits: what, me normal? , 1994, Genetics.

[44]  Gilbert Syswerda,et al.  Uniform Crossover in Genetic Algorithms , 1989, International Conference on Genetic Algorithms.

[45]  David E. Goldberg,et al.  Genetic Algorithms, Tournament Selection, and the Effects of Noise , 1995, Complex Syst..

[46]  Bernard Manderick,et al.  The Genetic Algorithm and the Structure of the Fitness Landscape , 1991, ICGA.

[47]  H. Geiringer On the Probability Theory of Linkage in Mendelian Heredity , 1944 .

[48]  R. B. Robbins Some Applications of Mathematics to Breeding Problems. , Genetics.

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

[50]  R. A. Fisher,et al.  The Genetical Theory of Natural Selection , 1931 .

[51]  S. Baluja,et al.  Using Optimal Dependency-Trees for Combinatorial Optimization: Learning the Structure of the Search Space , 1997 .