Erroneous Truncation Selection - A Breeder's Decision Making Perspective

Based on experiences from livestock breeding we introduce erroneous truncation selection for the Breeder Genetic Algorithm BGA The decision behavior of the breeder is given by a simple model It is shown that there is no bene t to the BGA by using erroneous selection though the variance of the parent population is increased by increasing the decision error variance Introduction Selection is one of the most fundamental operators in Evolutionary Algorithms i e it is inherent to Genetic Algorithms Evolution Strategies Evolutionary Pro gramming and Genetic Programming Generally for all these algorithms a di rected selection scheme is used which favors individuals having smaller for minimization or larger for maximization tness values With respect to Evolutionary Algorithms we will use the notion of tness trait and character synonymously to characterize one measurable variable or function Correspond ing to the great variety of algorithms we also have a great variety of selection operators This ranges from proportionate selection tournament selection selection Boltzmann selection linear and exponential ranking selection to truncation selection Perhaps the most widespread used selection schemes in modern Evolutionary Algorithms are tournament selection and truncation or selection With this paper we consider selection from a breeder s perspective The ulti mate selection scheme in livestock breeding is truncation selection The outward results of selection to a population is given by the response to selection R t f t f t The amount of selection is measured by the selection di erential S t fs t f t where fs t is the average tness of the selected parents The equation for the response to selection relates R and S by R t b t S t HMV is also with the Technical University of Berlin b t is called the realized heritability For many tness functions and selection schemes the selection di erential can be expressed as a function of the pheno typic standard deviation For truncation selection selecting the T N best individuals one obtains S t t I I is called the selection intensity With these notions the famous equation for the response to selection is obtained R t I b t t Usually the response to selection equation is considered for a population having a normal tness distribution But it holds also for other unimodal tness distri butions It should be noted that the response to selection equation does not explain how selection changes the genetic composition of a population Truncation selection in real breeding is hard to realize because of the precise truncation threshold which implies a perfect measurement of the trait under con sideration Therefore breeders are well aware of possible errors in selection The kind of selection pictured in Fig left corresponds to that actually practiced for important traits in stock breeding where many di erent traits must be considered Some animals which are mediocre or even inferior in the character istic pictured are saved because they are unusually desirable in several other characteristics or because the breeder is careless or confused 0 -0.5 F re q u e n c y Fitness Selection with Errors for Minimization Selected Parents Population 1 -0.5 F re q u e n c y Fitness Distribution of Decision Errors Fig Parent population selected from a population with erroneous truncation selection left and distribution of decision errors made by a breeder around the truncation threshold fT right In this paper we consider only one trait which is measured and therefore selected with errors The case of considering more then one trait is closely con nected with multiobjective and or subjective decision problems and will be an alyzed in a forthcoming paper A usual way breeders cope with such problems is the application of selection indices Furthermore we assume that selection is done without replacement The questions to be raised are How does selection errors in uence the behav ior of an Evolutionary Algorithm How can the decision behavior of a breeder be modeled Is the response to selection equation su cient for predicting the convergence of an Evolutionary Algorithm To give some rst answers to these questions we proceed in the following way Based on a very simple but comprehensive model for erroneous breeder decisions we present a general method for computing the selection intensity and the standard deviation of a parent population with respect to decision errors This will be compared for selected tness functions with empirical observations from simulation Based on these results rst conclusions concerning decision errors in truncation selection are presented Truncation Selection with Errors The analysis of truncation selection with errors has to be based on a decision model for the breeder As a rst attempt we use the following model De nition Breeder decision model The decision behavior of a breeder is given by a probability density function pdf pe f with the expectation E f fT and the variance V f e fT is the truncation threshold and e the vari ance of the decision or selection error This model implies that decision errors of the breeder are most probable at the truncation threshold That is a rational behavior because at this point it is most di cult to decide whether an individual will be saved as a parent or not The shrewdness of the breeder or the precision of measurement of the considered trait is given by the selection error variance The smaller this variance is the less is the decision error For lim e we have a one point distribution at the truncation threshold fT which corresponds to the usual truncation selection An example for such a decision model is shown in Fig right for a truncation threshold fT Obviously the selection scheme will be in uenced by the variance of the selection error Having a population with a tness probability density function p f with expectation E f f and variance V f we will analyze the dependency of the selection intensity I fp and the standard deviation of the selected parents p on the decision error standard deviation e We consider the standardized and normalized probability density function of the population tness distribution which is given by p z with