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淘豆网网友近日为您收集整理了关于Bongard - A Functional Crossover Operator for ic Programming (2009)的文档,希望对您的工作和学习有所帮助。以下是文档介绍:Bongard - A Functional Crossover Operator for ic Programming (2009) Contents1Functional Crossover 1Josh BongardChapter 1A FUNCTIONAL CROSSOVER OPERATOR IC PROGRAMMINGJosh Bongard11Department puter Science, University of Vermont, josh.bongard@uvm.eduAbstract Practitioners of evolutionary algorithms in general, and of ic programmingin particular, have long sought to develop variation operators that automaticallypreserve bine useful ic substructure. This is often pursued withcrossover operators tha(来源:淘豆网[/p-8585540.html])t swap ic material between genotypes that have sur-vived the selection process. However in ic programming, crossover oftenhas a large phenotypic effect, thereby drastically reducing the probability of abenecial crossover event. In this paper we introduce a new crossover operator,functional crossover (FXO), which swaps subtrees between parents based on thesubtrees’ functional rather than structural similarity. FXO is employed in a ic programming syst(来源:淘豆网[/p-8585540.html])em identication task, where it is shown that FXO oftenoutperforms standard crossover on both simulated and physically-generated datasets.Keywords: homologous crossover, crossover operators, system identication2 IC PROGRAMMING THEORY AND PRACTICE V1. ic programming (Koza, 1992) refers to a family of algorithms that em-ploy various data structures to represent candidate solutions to a given problem.These genotypes either produce behavior directly that(来源:淘豆网[/p-8585540.html]) is then selected, or aredirectly or indirectly transformed into a phenotype that in turn exhibits behav-ior which is subjugated to selection pressure. The choice of ic encoding,the genotype to phenotype mapping, and the variation operators have a sig-nicant impact on the system’s evolvability (Wagner and Altenberg, 1996), orability to continually improve solutions.The choice of variation operators is of particular interest in that they sig-nicantly(来源:淘豆网[/p-8585540.html]) affect how the population moves through the search space. Muta-tion operators are designed to discover better variants crossover operators on the other hand should, when implemented bine useful ic substructure from multiple genotypes. Because ic programming instantiations are tree-based, crossover typically involvesswapping subtrees between two parent trees, and this structural change oftenhas a large phenotypic effect on the r(来源:淘豆网[/p-8585540.html])esulting genotypes. As originally articu-lated by Fischer (Fischer, 1930), the magnitude of the phenotypic effect of ic perturbation is inversely proportional to the probability of that pertur-bation being benecial. For this reason it is often observed that random subtreecrossover can adversely affect the performance of a ic programming sys-tem. It may favor gradual increase in the size of genotypes over evolutionarytime without providing any tness (来源:淘豆网[/p-8585540.html]), a problem known as bloat (Langdonand Poli, 1997), and/or it may slow search by producing offspring that are lesst than their parents.Several crossover operators have been proposed in the GP literature to im-prove their ability bine useful ic substructure from several parentgenotypes. Headless chicken crossover (Jones, 1995) crosses subtreesbetweentwo GP trees in which one tree has survived selection while the second is cre-ated randomly in an atte(来源:淘豆网[/p-8585540.html])mpt to introduce fresh ic material into the pop-ulation. Size fair crossover (Langdon, 1999) crosses subtrees between parenttrees with a probability that is proportional to the size similarity between theselected subtrees. Homologous crossover refers to a family of crossover op-erators that attempt to preserve the context of the two crossed subtrees withintheir parent trees. D’haeseleer (D’haeseleer, 1994) has described deterministicand Langdon (Lan(来源:淘豆网[/p-8585540.html])gdon, 1999) probabilistic homologous crossover operatorsthat swap subtrees based on the similarity of their positions within their par-ent trees. Other homologous crossover operators based on syntactic similarity(Poli and Langdon, 1998; Nordin et al., 1999) have met with limited ess.Several researchers have argued that ic material should binedbased on its semantic, rather than syntactic or structural similarity. Seman-Functional Crossover 3tic cross(来源:淘豆网[/p-8585540.html])over (Beadle and Johnson, 2008) uses standard (random) crossoverbetween two trees and then retains the resulting trees only if they differ se-mantically from their parents. In enzyme ic programming (Lones andTyrrell, 2001), genotypes posed of independent elements that attach toone another based on their input and output characteristics. Crossover is plished by injecting elements from a donor into
ponents will only be incorporat(来源:淘豆网[/p-8585540.html])ed into the new genotype if theycan connect to ponents.In this paper we introduce a crossover operator that swaps subtrees basedon their functional (semantic) rather than structural (syntactic) similarity, inan attempt to reduce the magnitude of the phenotypic effect of the cross. Thenext section describes this functional crossover (FXO) operator and its appli-cation to a system identication task. Section 3 contrasts FXO with standardcrossover and no crossover, and section 4 provides some concluding remarks.2. GP-based system identicationIn previous work (Bongard and Lipson, 2007) ic programming wasapplied to the problem of nonlinear system identication, in which coupled,nonlinear posed of multiple state variables are modeled as sets ofordinary differential equations. The system posed of ponents:a modeling and ponent. The ponent uses icprogramming to evolve a population of models to describe a subset of timeseries data extracted from the system under study. The ponent usesthe model population to derive a new set of initial conditions with which toperturb the system, and thereby generate new useful training data.The algorithm proceeds as follows. Initially, a random set of initial con-ditions is provided to the target system, which generates a short tract of timeseries data in response. The modeling phase mences by creating 15random models and training them against this training data for 200 generations.A model’s tness is determined as its ability to reproduce as closely as possiblethe behavior of the target system when integrated starting with the same set ofinitial conditions.Model evolution is then paused, and the mences bycreating 15 random sets of initial conditions. Each initial condition is providedto each of the current models, and the tness of each set of initial conditionsis determined as the rate of divergence in the models’ predictions about howthe system would respond to these initial conditions. The initial conditionsare optimized for 200 generations, and the most t set of initial conditions isprovided to the target system, which generates a second tract of time seriesdata in response. This second tract is added to the training set, and the mod-ponent mences evolution with the current set of models, and4 IC PROGRAMMING THEORY AND PRACTICE VFigure 1-1. Overview of the GP-based system identication system.re-optimizes them against both time series tracts in the training set for 200 gen-erations. This cycle of system interrogation, modeling and testing is repeateda set number of times during each experiment, and is summarized in Fig. 1-1.During modeling, it was found previously that integrating all of the ODEsdescribing each state variable together, and puting the tness of themodel as a whole has low evolvability: If there is coupling between a well-modeled and a poorly-modeled state variable in a model, then that model willobtain an overall low tness because the poorly-modeled ODE will drag thewell-modeled state variable off course, and this well-ponent willbe lost during evolution. In (Bongard and Lipson, 2007) a technique calledpartitioning was introduced in which each ODE is integrated and evaluatedseparately, even though there may be coupling between the variables. This plished as follows. During each time step of the integration of the ODEFunctional Crossover 5describing a state variable, if there is a reference to another state variable in theGP tree then the value of that state variable generated by the target system, atthat time step, is substituted into the terminal node. At the end of the modelingphase, the newly-optimized ODEs for each model are integrated back togetherto produce a full model. For more details about this methodology, please referto (Bongard and Lipson, 2007).Functional crossoverIn the initial experiments using this system (Bongard and Lipson, 2007),crossover was not used as it was imperative to maintain variation in the popula-tion so that the ponent could induce disagreement amongst the pre-dictions of the models for a given set of initial conditions, and it is well-knownthat crossover can reduce population hetereogeneity without necessarily con-ferring increased evolvability (e.g. (Bongard, 2007)). In order to improve theprobability that crossover will incorporate useful ic substructure into thereceiving GP tree, a crossover operator that relies on semantic similarity be-tween the two subtrees to be crossed was formulated and investigated here:functional crossover (FXO).Given n state variables and 15 models, the population contains a total of15n ODEs encoded as GP trees that are optimized. While each ODE is inte-grated, the minimum and maximum value that is passed upward by each nodeis recorded at that node. This process records the range of values experiencedby each node during integration. After integration, the tness of an ODE puted as the error between the time series produced by the ODE and thetime series produced by the target system for the corresponding state variable.A copy of each evaluated ODE is created, and the copy is mutated using stan-dard GP mutation operators. The child ODE is integrated and evaluated: ifits tness is higher than its parent ODE, the parent is discarded a otherwise, the child is discarded and the parent retained.This experimental regime without crossover was contrasted to a secondregime in which both mutation and standard GP crossover was employed. Af-ter all 15n ODEs are evaluated, they are copied and mutated. Within each ofthe n subgroups of 15 ODEs, a pair of the copies is chosen at random andcrossed: a node is chosen at random in both trees, and the subtrees with thosenodes as roots are swapped between trees. If either of the new trees is more tthan its parent, otherwise, the new tree is discarded.In the third regime, functional crossover is employed. Within each of then subgroups of 15 ODEs, a pair of copies is chosen at random, and a node ischosen at random within the rst tree. The node in the second tree is found thathas the most similar range to that of the chosen node in the rst tree, according6 IC PROGRAMMING THEORY AND PRACTICE VFigure 1-2. Functional crossover. While a GP tree is evaluated (a) the minimum and maximumvalues that pass through each node are recorded (b). If a node in the tree is then selected forcrossover (b; dashed line), a second tree is chosen at random, and the node with the most similarrange is found (c; dashed line), and those subtrees are then crossed as in standard GP crossover.tomintj=1(|imin
jmin| + |imax
jmax|2)where t is the number of nodes in the second tree, i is the index of the nodechosen from the rst tree, and imin and imax are the minimum and maximumvalues passed upward by node i during integration, respectively. After ndingthe most similar node in the second tree, the two subtrees are crossed. Inall other respects the third regime is identical to the rst and second regimes.Functional crossover is illustrated in Fig 1-2.3. ResultsThe three regimes were used to model both synthetic and physical systems.The rst set of synthetic systems is shown in Table 3, and posed ofeighteen coupled, nonlinear systems with from 2 to 7 state variables. For eachsystem, initial conditions for a state variable could range between zero andunity. Two hundred independent trials of the rst regime, 200 trials of thesecond regime and 200 trials of the third regime were applied to each system.Each experiment was conducted for 40 cycles. At the end of each pass throughthe ponent, the objective error of the best model was calcuated:the physical system generates time series that the model was not trained on, andthe error of the model is calculated. The relative errors of the models producedby the three regimes is shown in Fig. 1-3, and the sizes of those models in Fig.1-4.For 6 of the 18 systems, functional crossover led to signcantly more accu-rate models than when either no crossover or standard crossover was employed(Fig. 1-3a,b,d,g,i,j). It can be noted that for these systems, FXO also tended toproduce pact models (Fig. 1-4a,b,d,g,i,j), despite the fact that thereis no explicit selection pressure for smaller models. It is hypothesized thatFXO produces more accurate and pact models in these systems be-Functional Crossover 7System 1 System 2 System 3a b cdx1/dt = 3x1
3x1x2 + 2x2x2 3x1x1 + 3x1x2 + 3x2x2 3x1x1
x2x2dx2/dt = x1x1
2x2x2 3x1x1
2x1x2 + 2x2x2 x1x1 + 3x1x2
x2x2d e fdx1/dt = 3x1x3
3x3x3 x1x2 + x1x3
x2x3 3x1x2 + x1x3
x3x3dx2/dt = 3x1x2 + x1x3
3x2x3 x1x1 + 2x1x2 + 2x2x3 2x1x3 + 3x2x3 + 3x3x3dx3/dt = 3x1x2 + 3x1x3
x2x3 2x1x1 + x1x2
3x2x3 2x1x2
2x2x3g h idx1/dt = x1x1 + 2x2x3 + 2x3x3 x1x4 + x2x4 + x4x4 3x1x1 + 3x1x2 + 3x2x4dx2/dt = x1x2
3x2x3 3x1x2
3x3x4 x1x1
3x4x4dx3/dt = x1x1
x2x4 + 3x4x4 2x1x2
x1x3 + 2x2x2 2x1x4 + x2x2
3x3x4dx4/dt = 3x1x2
3x3x4 x1x3 + 3x2x3
x3x4 x1x2 + 2x1x4
3x3x4j k ldx1/dt = 3x1x5 + 3x2x3
3x2x5 2x2x2 + 3x3x5 + 2x4x5 2x1x4 + 2x2x3
x2x4dx2/dt = 3x1x3
x4x5 3x1x2 + x1x5
2x2x5 x1x3 + 3x1x4 + x2x4dx3/dt = x1x1
3x1x4 + x2x4 x1x2 + 2x2x5 + 2x4x5 2x1x1 + 2x1x2
3x1x3dx4/dt = 3x1x3
3x1x4 + 2x2x2 2x1x2 + 3x1x5
x4x5 3x2x5 + 3x3x4
x3x5dx5/dt = 3x1x4 + 3x3x3 + 3x3x4 2x1x5
2x5x5 x1x1 + x1x5 + x2x3m n odx1/dt = 2x1x6 + x2x4
2x2x6 2x1x3
3x2x4 + 2x3x6 x1x5 + x1x6 + x4x5dx2/dt = x1x4
2x4x4 3x2x4 + x3x4
x3x6 2x2x5
2x2x6 + 2x3x6dx3/dt = 2x2x5
x3x4 + x5x5 x1x2
x1x3 + x4x6 x1x5
2x3x4 + x4x4dx4/dt = 3x4x5
2x4x6 + 2x5x5 x1x4 + x3x5
2x4x6 3x1x2 + 3x2x3
2x4x5dx5/dt = x3x6
3x4x5 3x1x2
x5x5 3x1x5 + x2x2 + 3x2x6dx6/dt = x3x4
x3x6 + 2x4x6 3x1x3
3x4x6 x2x5
3x5x6p q rdx1/dt = x2x2
x1x2 + 3x1x1 2x2x3
3x3x3 3x1x1 + 3x2x2 + 3x1x2dx2/dt = 3x1x2
x2x2 + x1x1 3x1x2
3x2x3 + x1x3 2x1x2
3x1x1 + 2x2x2dx3/dt = 2x6x7
2x4x4 + 3x5x7 3x1x3
x2x3 + 3x1x2 3x4x7
3x3x7 + 3x4x5dx4/dt = x3x7 + 3x3x4
2x4x7 2x5x6
x4x4 + 2x6x6 2x5x6
3x3x5dx5/dt = 2x4x7 + 2x6x7 + x3x4 x4x5
3x4x6 x4x6
3x3x6 + x3x3dx6/dt = 3x3x7
x6x7 + 2x3x4 3x7x7
x4x4 2x4x4 + 3x3x5
3x3x6dx7/dt = 2x3x7
x4x7 3x6x7
3x4x7 3x3x6 + 3x5x5 + 3x5x6Table 1-1. The eighteen coupled nonlinear systems used for initial modeling.cause FXO is able to swap out a large subtree that is an approximation of somefunction that can be expressed using fewer nodes, and therefore has a higherprobability of swapping in a subtree from another tree that represents this func-tion in a pact way. For several of the other systems FXO producedmore accurate models but not signicantly so (Fig. 1-3e,h,n,o,p,q), and for nosystems did the other two regimes signcantly outperform FXO.The three regimes were also applied to four target systems that are manually-derived models of nonlinear mechanical (Pendulum), ecological (Lotka-Volterra)and biological (Lac operon) systems (Table 1-4). The initial values for eachstate variable in each system was restricted to the range [0, 1]. The modelstrained against the pendulum could posed of algebraic and trigonomet- the Lotka-Volterra and high degree models were restricted t and the Lac operon models were allowed algebraic func-tions and the Hill function (x/(x + 1)). Terminal nodes were restricted to statevariable references and oating-point constants. Fig. 1-5 reports the relativeerrors of the best models from 200 independent trials run using each of thethree experimental regimes. Fig. 1-6 reports the relative sizes of these models.8 IC PROGRAMMING THEORY AND PRACTICE V0.00.082StateVariablesa0.010.1ObjectiveErrorb0.010.1 c0.030.13StateVariablesd0.020.1ObjectiveErrore0.050.1 f0.060.14StateVariablesg0.010.1ObjectiveErrorh0.090.1 i0.440.625StateVariablesj0.951.0ObjectiveErrork0.870.99 l0.490.966StateVariablesm0.120.21ObjectiveErrorn0.160.26 o0 5 10 15 20 25 30 35 40Cycle1.652.07StateVariablesp0 5 10 15 20 25 30 35 40Cycle0.140.23ObjectiveErrorq0 5 10 15 20 25 30 35 40Cycle0.490.72 rFigure 1-3. Relative modeling performance against the 18 synthetic systems using nocrossover (blue), random crossover (green), and functional crossover (red). Thick lines indi- thin lines indicate one unit of standard error of the mean.As can be seen in Fig. 1-5, functional crossover signcantly outperforms theother two regimes when employed for modeling the system of high degree andthe Lac operon (Fig. 1-5c,d), provides some advantage for the Lotka-Volterrasystem (Fig. 1-5b), and provides a slight advantage for the pendulum, as doesstandard crossover. Fig. 1-6 indicates that for two of the systems functional播放器加载中,请稍候...
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Bongard - A Functional Crossover Operator for ic Programming (2009) Contents1Functional Crossover 1Josh BongardChapter 1A FUNCTIONAL CROSSOVER OPERATOR IC PROGRAMMINGJosh Bongard11Department puter Science, University of Vermont, josh.bongard@uvm.eduAbstract Practitioners of evolutionary algorithms i...
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