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GPLAB A Genetic Programming Toolbox for MATLAB

1. 8 Graphical output produced by the plotfitness option in the graphics parameter 2 2 20000000000 000 36 Graphical output produced by the plotdiversity option in the graphics parameter es e a we a eai a a 36 Graphical output produced by the plotcomplexity option in the graphics parameter sa cs iy kaus boror ki Cupar 37 Graphical output produced by the plotoperators option in the graphics parameter Serios rota a Ei e a dad 37 Graphical output produced by the function accuracy_complexity 45 Graphical output produced by the function plotpareto 46 Graphical output produced by the function desired_obtained 46 Graphical output produced by the function operator_evolution 47 Graphical output produced by the function drawtree 47 Chapter 1 Introduction MATLAB 1 is a widely used programming environment available for a large number of computer platforms Its programming language is simple and easy to learn yet fast and powerful in mathematical calculus Furthermore its ex tensive and straightforward data visualization tools make it a very appealing programming environment Toolboxes are collections of optimized application specific functions which extend the MATLAB environment and provide a solid foundation on which to build GPLAB is a genetic programming toolbox for MATLAB Versatile general ist and easily extendable it can be used by all types of users from the layman to the advanced
2. operator2 replacing any operator previously declared operator1 needs two parents and produces two children operator2 also needs two parents but produces only one child Any number of genetic opera tors can be declared at one time by adding more arguments to the function The second function accepts the same arguments but adds the declared operators to the already defined set keeping the previously declared operators untouched These functions have the same effect as directly setting the parameter variables operatornames operatornparents operatornchildren operator and operator2 are the names of the new MATLAB functions that implement the new operators The only rules these functions must follow concern their input and output arguments Please see functions crossover and mutation for examples on how to correctly build genetic operators A set of tree manipulation functions is available use help lt function name gt in the MATLAB prompt for usage e maketree level functions arities exactlevel depthnodes this function returns a new random tree no deeper bigger than level us ing the functions with respective arities If exactlevel is true the new tree will be initialized using the Full method otherwise it will be initialized using the Grow method see Sect 3 1 depthnodes indicates whether restrictions are to be applied in tree depth or tree size number of nodes e tree2str tree returns the str
3. the children replace the parent population completely even if they are worse individuals than their parents This option is not elitist at all e keepbest the best individual from both parents and children is kept for the new population independently of being a parent or a child The remaining places in the new population are occupied by children only If not all children produced can be used in the new population due to size constraints the worst are discarded e halfelitism half of the new population will be occupied by the best individuals chosen from both parents and children The remaining places will be occupied by the best children still available e totalelitism the best individuals from both parents and children are chosen to fill the new population The survival module is in fact elitist even when the non elitism option is chosen If GPLAB is operating in batch mode see Sect 3 10 the best children are always chosen and the worst discarded 30 3 12 Operator probabilities in runtime operatorprobstype adaptwindowsize numbackgen percentback adaptinterval percentchange minprob GPLAB implements an automatic adaptation procedure for the genetic op erator probabilities of occurrence based on 5 This procedure can be turned on by setting the parameter variable operatorprobstype to variable and turned off by setting the same variable with fixed What follows is a brief descript
4. a 1 0 5 10 15 20 25 generation Figure 5 1 Graphical output produced by the function accuracy_complexity 45 Pareto front 40 35 best for nodes pareto front current population 307 test fitness 25 3 20 F 15F 107 BL 0 f f f f f 1 0 5 10 15 20 25 30 nodes Figure 5 2 Graphical output produced by the function plotpareto Desired versus Obtained to approximate on generation 0 on generation 1 on generation 3 3 on generation 6 on generation 9 on generation 13 desired y approximation y s 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 Figure 5 3 Graphical output produced by the function desired_obtained 46 e crossover Operators Evolution mutation e T e N T o D gt a T o ES probabilities of occurence e w o o T 0 1 1 0 20 40 60 80 100 120 140 160 generation Figure 5 4 Graphical output produced by the function operator_evolution Figure 5 5 Graphical output produced by the function drawtree 47 Chapter 6 Summary of toolbox functions The more than one hundred functions provided in the toolbox GPLAB can be divided into different functional groups What follows is a list of the functions included in each group The same function may be listed in more than one
5. plus a random number between 0 and 1 generated in runtime as the function rand with null arity The declaration of terminals is done similarly to the declaration of functions by using friendly substitutes to directly setting the parameter variable terminals For example to declare the constant 1 and the random number generator as members of the set of terminals use help setterminals in the MATLAB prompt for usage params setterminals params rand 1 Unlike in setfunctions there is no need to indicate the arity which is always null To add a new terminal to an already declared set of terminals use help addterminals in the MATLAB prompt for usage params addterminals params new_terminal Any number of terminals can be declared or added at one time by adding more input arguments The terminals available for artificial ant problems are antright antleft antmove 20 Table 3 3 Protected and logical functions for use with GPLAB Protected MATLAB Input 1 A Output argument function function arguments ai ee a if b 0 Division mydivide a b a b Cotherwise 0 if a lt 0 Square root mysqrt a sart a oth rwis Power ab a if a is a valid non complex number pr mypower i 0 otherwise Natural a 2 0 if a 0 logarithm ps log abs a otherwise Base 2 is t 0 if a 0 logarithm ree log2 abs a otherwise Base 10 0 if a 0 logarithm my
6. see Sect 3 13 initialprobstype fixed variable fixed initialvarprobs list of probability values see Sect 3 13 initpoptype fullinit growinit rampedinit rampedinit keepevalssize integer gt 0 see Sect 3 8 lowerisbetter 0 1 1 minprob gt Oand lt 1 0 1 numbackgen integer gt 0 3 numvars or integer gt 0 see Sect 3 3 operatornames or oa crossover mutation see Sect 3 4 operatornchildren list of number of children produced 2 1 operatornparents list of number of parents needed 2 1 operatorprobstype fixed variable fixed output silent normal verbose normal percentback gt Oand lt 1 0 25 percentchange gt 0and lt 1 0 25 precision integer gt 0 12 j 2 realmaxlevel integer gt 0 a PEREI b 5 reproduction gt Oand lt 1 0 1 E roulette sus p A sampling A A i lexictour tournament lexictour savedir name for a new directory user provided see Sect 3 15 16 continued on next page Table 3 2 continued Parameter Possible values Default value never firstlast every10 every100 always smalldifference gt 0and lt 1 J see Sect 3 13 replace keepbest savetofile never survival halfelitism totalelitism Teplace terminals see Sect 3 3 testdatafilex name of a valid input file user provided see Sect 3 8 testdatafiley name of a valid input file us
7. A e ee 3 12 Operator probabilities in runtime 3 13 Initial operator probabilities o o 3 14 Stop conditions 0 ee ee 3 15 Saving results to fle o o 3 16 Runtime textual output e e 3 17 Runtime graphical output o o State 4 1 Population Lx dd a des h 4 2 Tree depth siZe coco Go o a A Doe eee eee 4 3 Functions and terminals e 4 4 Operator probabilities and frequencies 4 5 Population fitness e steure sse a Daoa o 4 6 Fitness statistics o wee le be ee ee e Af Best Individual cocinada a ke ee he EAR ed 2 48 Control ota ace se ee Gin oY dot he Phot edhe oo Aba ed 4 9 Complexity and diversity statistics history Offline graphical output 5 1 Accuracy versus Complexity 0 0 2 0 00 004 5 2 Pareto rom so akon Sk ele a ad dl ad 5 3 Desired versus Obtained o e o 5 4 Operator Evolution ooa 20000200000 0s 5 5 Tree visualization 0 0 0020 ee eee eee Summary of toolbox functions 6 1 Demonstration functions 2000 004 6 2 Running the algorithm and testing result 6 3 Parameter and state setting 6 4 Automatic variable checking 0 6 5 Description of parameter and state variables 6 6 Creation of new generations e 6 7 Creation of new
8. GPLAB uses it 3 5 Validating new individuals filters fixedlevel dynamiclevel depthnodes After a new individual is produced by any of the genetic operators it must be validated in terms of depth size before being considered as a candidate for the new population Several validation functions or filters are provided in GPLAB and others may be built and integrated as plug and play functions see Sect 2 4 The filters parameter is simply a list of those functions by the order in which they should be applied It should not however be set by the user but instead be automatically set in the beginning of the run depending on the parameters fixedlevel dynamiclevel and depthnodes Bellow is a list of available filter functions along with the description of their purpose see Sect 3 2 for more details e strictdepth this filter rejects an individual that is deeper than the strict maximum allowed depth does nothing otherwise e strictnodes this filter rejects an individual that is bigger contains more nodes than the strict maximum allowed size does nothing other wise e dyndepth this filter measures the fitness of an individual that is deeper than the dynamic maximum allowed depth if the individual is better than the best so far the dynamic depth is increased and the new individual is accepted otherwise it is rejected The filter does nothing if the individual is no deeper than the limit e d
9. generations completed so far 4 7 Best individual bestsofar bestsofarhistory bestfithistory bestnodeshistory bestintronshistory bestlevelhistory Ultimately the result of a genetic programming algorithm is one individ ual the best individual found during the whole run bestsofar is a struc ture like each individual in pop and stores the individual with better fitness found since the beginning of the run bestsofarhistory stores a list of all the individuals that have once been considered the best so far Each time a new individual updates the variable bestsofar the same individual is added to bestsofarhistory bestfithistory bestnodeshistory bestintrons history and bestlevelhistory contain respectively the fitness number of nodes number of introns and depth of the parse trees of all the individuals in 42 bestsofarhistory bestfithistory may also contain the test fitness cross validation in a different data set of the best individual in a separate column in case the usetestdata parameter was on see Sect 3 8 4 8 Control generation maxgen GPLAB runs until either a stop condition Sect 3 14 or the maximum generation indicated by the user see Sect 2 3 is reached The state variable generation indicates which generation is currently running and maxgen indi cates the maximum number of generations allowed 4 9 Complexity and diversity statistics history avgnodeshistory avgintronshistory
10. individuals 04 6 8 Filtering of new individuals o o 6 9 Protected and logical functions o o 6 10 Artificial ant functions o o 6 11 Tree manipulation e 6 12 Data manipulation 0 2 0 4 ek ea es 6 13 Expected number of children o o 6 14 Sampling 44 4 4 eA A A BM 6 15 Genetic operators 2 i a ee 6 16 Fitness eens as Sa oie ee oe ae Gah ek eee 6 17 Diversity measures e su eos a e Re ee as 6 18 Automatic operator probability adaptation 6 19 Runtime graphical output 2048 6 20 Offline graphical output o e 6 21 Utilitarian functions e e 6 22 Text input files e 6 23 License fil e ua a abe ba ee POS eee Se A ey 38 39 40 41 41 41 42 42 43 43 44 44 44 44 45 45 List of Tables 3 1 3 1 3 2 3 2 3 3 3 4 4 1 4 1 Location of parameters in this manual 14 continued lacas She de is lid rd 15 Possible and default values of the parameters 16 CONTINUCO mi A E E PE AA RE DE O 17 Protected and logical functions for use with GPLAB 21 List of filters for each combination of parameters 25 Location of state variables in this manual 38 CONUNUCA A A e RSE A ae 39 List of Figures 2 1 3 1 3 2 3 3 3 4 5 1 5 2 5 3 5 4 5 5 Operational structure of the GPLAB toolbox
11. number of individuals are initialized with sizes ranging from 2 to inicmaxlevel As in the standard procedure for each size half of the trees are initialized with the Full method and the other half with the Grow method 3 2 Tree depth and size limits fixedlevel realmaxlevel dynamiclevel inicdynlevel depthnodes Trees in GPLAB may be subject to a set of restrictions on depth or size number of nodes by setting appropriate parameters These restrictions are meant to avoid bloat a phenomenon consisting of an excessive code growth without the corresponding improvement in fitness The standard way of avoiding bloat is by setting a maximum depth on trees being evolved whenever a genetic operator produces a tree that breaks this limit one of its parents enters the new population instead 8 GPLAB implements this strict limit on depth as well as a dynamic limit similar to the first but with two important differences it is initially set with a low value it is increased when needed to accommodate an individual that is deeper than the dynamic limit but is better than any other individual found during the run Both limits can be used in conjunction For each new individual produced by a genetic operator there are three possible scenarios The individual does not exceed the dynamic maximum depth it can be used freely because no constraints have been violated The individual is deeper than the dynamic maximum depth but
12. runs where the user chooses to measure the amount of introns of the generated trees see Sect 3 9 particularly in problems like the artificial ant where every tree branch is repeated many times throughout the population and takes the same amount of time steps to evaluate 3 9 Measuring complexity and diversity calccomplexity calcdiversity During the run it may be useful to gather more information about the evolution ary process namely the structure complexity and diversity of the population When the parameter calccomplexity is turned on calccomplexity 1 GPLAB stores information regarding the number of nodes and intron nodes of the trees depth level and balancing between branches tree fill rate see 12 Obtaining some of this information is extremely time consuming particularly the number of introns so it must not be used unless absolutely necessary GPLAB may also store information regarding the population diversity Two different diversity measures are provided uniquegen and hamming use help uniquegen and help hamming in the MATLAB prompt for details and the user can add more as plug and play functions see Sect 2 4 Several diversity measures may be calculated at the same time and calcdiversity contains the list of measures to be used it is a list like the one for graphics Sect 6 19 Measuring diversity may be more or less time consuming depending on the measure s chosen 3
13. the randomly created trees on the initial generation see Sect 3 1 and maxlevel indicates the current updated every generation maximum depth size allowed for any parse tree this is the dynamic depth size see Sect 3 2 levelhistory stores all the past settings of maxlevel one row per generation 40 4 3 Functions and terminals functions terminals arity The state variables functions and terminals are similar to the parameter variables with the same names see Sect 3 3 but they present some important differences terminals includes not only the constants or null arity functions specified in the parameters but also all the variables needed to evaluate the in dividuals in the current data set generated automatically before the run starts see Sect 3 3 functions includes not only the functions specified in the pa rameters but also all the terminals included in the state variable terminals arity contains the second column of the state variable functions i e the number of input arguments of all the functions and terminals used This seem ingly redundant organization of variables increases the efficiency of the algorithm when creating new parse trees 4 4 Operator probabilities and frequencies operatorprobs ophistory operatorfreqs opfreqhistory reproductions reproductionhistory clonings cloninghistory adaptwindow lastadaptation The state variable operatorprobs contains the current operator proba bil
14. the tournament methods is chosen the number of individuals participating in each tournament is determined by the parameter variable tour namentsize Like gengap see Sect 3 10 the value of this parameter can represent either the absolute number of individuals tournamentsize gt 1 or a proportion of the population size otherwise When the tournament method is chosen and tournamentsize is left blank tournamentsize GPLAB sets it with the default value 10 of the population size or 2 which one is larger in the beginning of the run If tournamentsize equals 1 the selection of parents is random if tournamentsize equals the population size only the best individual in the population is chosen to produce all the offspring The tournament method does not need to know the expected number of children of each individual unlike the other two methods Alternative sampling methods may be built and easily used in GPLAB as plug and play devices to module SAMPLING see Fig 2 1 All the user has to do is build a new function that implements the sampling method respecting the input and output arguments and set the parameter variable sampling with the name of the new function params sampling new_sampling_ method The new function must accept as input arguments the current population pa rameters and state vars pop vars params vars state the number of indi viduals to draw and a list of identifiers of individuals that must not be
15. yy EXPECTED generation gap OPERATOR FITNESS ae 9 probs type ADAPT PROBS MV WINDOW ADD CREDIT UPDATE SURVIVAL Figure 2 1 Operational structure of the GPLAB toolbox fitness value the better the individual This is the standard for symbolic re gression and parity problems regfitness in Fig 2 1 but GPLAB accepts any other plug and play function to calculate fitness like the function for artificial ant problems also provided antfitness GEN POP is called by the user It starts by requesting some parameter initializations to SET VARS and finishes by passing the execution to GENER ATION If the user only requests the creation of the initial generation GEN ERATION is not used 2 1 2 GENERATION This module creates a new generation of individuals by applying the genetic operators to the previous population OPERATORS Standard tree crossover and tree mutation are the two genetic operators available as plug and play functions They must have a pool of parents to choose from created by a SAM PLING method which may or may not base its choice on the EXPECTED number of offspring of each individual Four sampling methods Roulette 6 SUS 3 Tournament 4 Lexicographic Parsimony Pressure Tournament 9 and three methods for calculating the expected number of offspring Absolute 7 Rank85 2 Rank89 10 are available as plug and play functions and any combination of the two can be us
16. 10 Generation gap gengap The number of new individuals necessary to create a new GPLAB generation is determined by the parameter variable gengap Like tournamentsize see Sect 3 6 the value of this parameter can represent either the absolute number of individuals gengap gt 1 or a proportion of the population size otherwise 29 When gengap is left blank gengap GPLAB sets it with the default value in the beginning of the run The default value is the population size which corresponds to using the algorithm in the generational mode of operation If gengap is set to a very low value like 2 it clearly corresponds to a steady state mode of operation but there is no frontier between both modes in GPLAB In fact gengap may even be set to a value higher than the population size which corresponds to what may be called a batch mode of operation many more individuals are produced than the ones needed for the new population but the SURVIVAL module see Fig 2 1 discards the worst of them independently from the elitism level chosen see Sect 3 11 3 11 Survival survival After producing gengap new individuals for the new population see Sect 3 10 GPLAB enters the SURVIVAL module see Fig 2 1 where from the current population plus all the new children a number of individuals is chosen to form the new population One of four elitism levels may be used indicated in the parameter variable survival e replace
17. 2 bestlevelhistory 4 7 42 bestnodeshistory 4 7 42 bestsofar 4 7 42 bestsofarhistory 4 7 42 cloninghistory 4 4 41 clonings 4 4 41 depthnodes 4 2 40 diversityhistory 4 9 43 continued on next page 38 Table 4 1 continued State variable Section Page fithistory 4 6 42 functions 4 3 41 generation 4 8 43 iniclevel 4 2 40 keepevals 4 5 41 lastadaptation 4 4 41 lastid 4 1 39 levelhistory 4 2 40 maxfitness 4 6 42 maxgen 4 8 43 maxlevel 4 2 40 medianfitness 4 6 42 minfitness 4 6 42 operatorfreqs 4 4 41 operatorprobs 4 4 41 opfreqhistory 4 4 41 ophistory 4 4 41 pop 4 1 39 popexpected 4 5 41 popfitness 4 5 41 popnormfitness 4 5 41 popranking 4 5 41 popsize 4 1 39 reproductionhistory 4 4 41 reproductions 4 4 41 stdfitness 4 6 42 terminals 4 3 41 varsvals 4 5 41 4 1 Population pop popsize maxlevel levelhistory lastid The variable that holds the information concerning the current population the algorithm is using in each moment is pop a one dimensional array of indi viduals Each individual is a structure with fields e id a unique identifier If an individual survives from one generation to the next its identifier will not be changed If two individuals are identical but were generated independently their identifiers will be different 39 e origin the name of the operator that generated this individual or ran dom if it was randomly generated for the initial population e tree th
18. GPLAB A Genetic Programming Toolbox for MATLAB Sara Silva ECOS Evolutionary and Complex Systems Group University of Coimbra Portugal Version 2 1 October 2006 Contents 1 Introduction 1 1 Update from version 2 0 0 0 00002000004 1 2 Acknowledgements 2 0 0 0 000000 eee Operational structure 2 1 Main modiil s r d gaen Ssh 4 do a PR ee 2d 211 AGEN POP 2 5 jee ae Owe pee A he hoe ee eS 2 12 GENERATION epa 4404 BAe Boe ee a Gee 21 35 DELVARS cee ean A ae ee as 2 2 Working variables 2 o o o 209 USAge uta sia BS te Bete hee th ae ta a e a e AS E bo ae Poe de EE Sa SS 2 3 2 The regular user o a 2 3 3 The advanced researcher o 2 4 Plugvand pl do ey ee ee A a Pt 2 4 1 Building plug and play functions 2 4 2 Using new plug and play functions 2 4 3 Integrating new plug and play functions in GPLAB Parameters 3 1 Tree initialization 2 2 ee ee 3 2 Tree depth and size limits 04 3 3 Functions and terminals 200 3 4 Genetic operators 2 2 ee 3 5 Validating new individuals o o a 3 6 Selection for reproduction e e 3 7 Expected number of children o 3 8 Measuring fitness 3 9 Measuring complexity and diversity 3 10 Generation gap ee IL Survival ar A
19. _ x txt and exp_y txt e quartic_x txt and quartic_y txt parity3bit_x txt and parity3bit_y txt parity5bit_x txt and parity5bit_y txt santafetrail txt and santafepellets txt 6 23 License file e license txt 54 Bibliography 1 The MathWorks 2003 10 http www mathworks com products matlab Baker J E Adaptive selection methods for genetic algorithms In Grefen stette J editor Proceedings of the First International Conference on Ge netic Algorithms and Their Applications Hillsdale NJ Erlbaum 1985 101 111 Baker J E Reducing bias and inefficiency in the selection algorithm In Grefenstette J editor Proceedings of the Second International Conference on Genetic Algorithms Hillsdale NJ Erlbaum 1987 14 21 Blickle T Tournament selection In Back T Fogel D B Michalewicz Z Handbook of Evolutionary Computation Bristol UK and New York NY Institute of Physics Publishing and Oxford University Press 1997 C2 3 1 4 Davis L Adapting operator probabilities in genetic algorithms In Schaf fer J D editor Proceedings of the Third International Conference on Ge netic Algorithms San Mateo CA Morgan Kaufmann 1989 61 69 Goldberg D E Genetic algorithms in search optimization and machine learning Reading MA Addison Wesley 1989 Holland J H Adaptation in natural and artificial systems Ann Arbor MI University of Michigan Press 1975 Koza J R Genetic programmi
20. alidation in a different data set the exponential It draws all the available plots in runtime see Sect 6 19 and finishes with several additional output plots see Sect 5 including the pareto front and the drawing of the best individual found e demoparity runs the parity 3 problem with 50 individuals for 20 gen erations with fixed operator probabilities drawing some of the available runtime plots and finishing by drawing the best individual found e demoant runs the artificial ant problem in the Santa Fe food trail with 20 individuals for 10 generations drawing half of the available plots in runtime and finishing by drawing the best individual found 2 3 2 The regular user This is the type of user who knows what the parameters mean and wants to test different sets of values besides the defaults To set the parameters the available functions are params resetparams params setparams params parami valuel param2 value2 etc vars b gplab g n params where param1 param2 are the names of parameters and valuel value2 the values pretended The first function initializes and returns the parameters structured variable with the default values and the second alters some of the parameters according to the list given as argument The third acts like the first function described for the layman except that it uses the parameter values previously set instead of initializing them with the default values The parameter setti
21. ameters using the function setparams will ensure minimum range checking whereas setting the fields of the variable vars params directly will not Some parameter settings are automatically corrected to allowed values with a warning to the user in case they are set incorrectly Others are automatically set when left empty Generally speaking the only parameters that must be set by the user are the maximum number of generations the population size and the names of the files that contain the data set see Sect 2 3 Table 3 1 Location of parameters in this manual Parameter Section Page adaptinterval 3 12 31 adaptwindowsize 3 12 31 autovars 3 3 19 calccomplexity 3 9 29 calcdiversity 3 9 29 calcfitness 3 8 27 datafilex 3 8 27 datafiley 3 8 27 depthnodes 3 1 3 2 3 5 17 18 24 dynamiclevel 3 2 3 5 18 24 expected 3 7 26 files2data 3 8 27 filters 3 5 24 continued on next page 14 Table 3 1 continued Parameter Section Page fixedlevel 3 2 3 5 18 24 functions 3 3 19 gengap 3 10 29 graphics 3 17 34 hits 3 14 32 inicdynlevel 3 2 18 inicmaxlevel 3 1 17 initialfixedprobs 3 13 32 initialprobstype 3 13 32 initialvarprobs 3 13 32 initpoptype 3 1 17 keepevalssize 3 8 27 lowerisbetter 3 8 27 minprob 3 12 31 numbackgen 3 12 31 numvars 3 3 19 operatornames 3 4 22 operatornchildren 3 4 22 operatornparents 3 4 22 operatorprobstype 3 12 31 output 3 16 33 percentback 3 12 31 percentchange 3 12 31 precision 3 8 27 realm
22. ange of the operator with minimum probability percentchange times minprob divided by the number of genetic operators It is increased 10 in each iteration of the process to avoid an excessive wait time for the stabilization of the initial operator probabilities 3 14 Stop conditions hits GPLAB will run until the maximum generation indicated by the user is reached see Sect 2 3 or until a stop condition is reached Stop conditions are defined by setting the parameter variable hits One hit is a tuple f d where f is the percentage of fitness cases that must obey the stop condition and d is the definition of the stop condition itself meaning that the result obtained by the best individual in the population must be no lower than the expected result minus d of the expected result and no higher that the expected result plus d The default value of hits is 100 0 which means stop if the best individual produces exact results in all fitness cases 50 10 would mean stop if the best individual produces results within minus or plus 10 of the expected results in at least 50 of the fitness cases Several stop conditions can be used by adding rows to the hits variable If the two previous stop conditions were to be used concurrently hits should be set 32 to 100 0 50 10 GPLAB tests each stop condition starting with the first row until one is satisfied or all have been tested It is possible not to use any stop co
23. arameters will be refused for the new plug and play function until it is fully integrated as part of GPLAB 2 4 3 Integrating new plug and play functions in GPLAB Integrating a new function in GPLAB is done by editing the toolbox file avail ableparams This file contains the declarations of the fields that constitute the structure variable params as well as their possible and default values where the possible values may be the names of the plug and play functions This file is divided in three parts the first specifies which variables form the structure params the second specifies the possible values for each variable the third specifies the default values for each variable As an example to integrate a new plug and play function called newfitness that implements a new way of measuring fitness a new line should be added myparams calcfitness regfitness antfitness newfitness where regfitness and antfitness are the standard procedures for calculat ing fitness already provided by GPLAB This line tells GPLAB that all three procedures can be used for calculating fitness When the algorithm begins regfitness is still the default fitness procedure but the user can change it before starting the run Because newfitness is already declared as a standard GPLAB function this change can be made with the setparams function params setparams params calcfitness newfitness To make newfitn
24. avglevelhistory avgtreefillhistory diversityhistory When complexity and diversity is measured during the run see Sect 3 9 the results are stored in state variables The average number of tree nodes and intron nodes per generation are kept in the variables avgnodeshistory and avgintronshistory The average tree depth and fill rate unbalanced trees have lower fill rates than balanced trees per generation are kept in the variables avglevelhistory and avgtreefillhistory Diversity measures per generation are kept in the variable diversityhistory one column per measure used see Sect 3 9 43 Chapter 5 Offline graphical output After completing a run the user has some specialized functions available for visualization of different aspects of the evolution and results obtained by the algorithm Some of them provide arguments to define the size of the plot and whether it should be drawn in color or black and white 5 1 Accuracy versus Complexity This plot is drawn by the function accuracy_complexity use help accu racy_complexity in the MATLAB prompt for usage It draws lines repre senting the evolution of the fitness the depth and the number of nodes of all the best individuals found during the run Fig 5 1 5 2 Pareto front This plot is drawn by the function plotpareto use help plotpareto in the MATLAB prompt for usage It shows the best fitness found for each tree size the pareto front i e the se
25. axlevel 3 2 18 reproduction 3 4 22 sampling 3 6 25 savedir 3 15 33 savetofile 3 15 33 smalldifference 3 13 32 survival 3 11 30 terminals 3 3 19 testdatafilex 3 8 27 testdatafiley 3 8 27 tournamentsize 3 6 25 usetestdata 3 8 27 15 Table 3 2 Possible and default values of the parameters Parameter Possible values Default value adaptinterval integer gt 0 see Sect 6 18 adaptwindowsize integer gt 0 see Sect 6 18 autovars 0 71 a la calccomplexity 0d o calcdiversity list o a 9 calcfitness regfitness antfitness regfitness datafilex name of a valid input file user provided see Sect 3 8 datafiley name of a valid input file user provided see Sect 3 8 depthnodes 14 0 4 dynamiclevel 10 10 2 a ll expected absolute rank85 rank89 rank85 files2data xy2inout anttrail xy2inout list of filter functions filters see Sect 3 5 O fixedlevel 2077712 ple functions see Sect 3 3 pias minuse times sin cos mylog gengap integer gt 0 see Sect 3 10 list of plot names grepnics l see Sect 3 17 l O list of stop conditions hits see Sect 3 14 100 0 Pa 4 6 if depthnodes 1 inicdynlevel integer gt 0 28 if depthnodes 2 j 2 inicmaxlevel integer gt 0 e ass 7 initialfixedprobs list of probability values
26. cond the expected or desired output value one row for each fitness case For artificial ant problems the first file should contain the food trail in the form of a binary matrix and the second file should contain the number of food pellets in it After importing the data stored in these files to the algorithm s variables according to the procedure specified in the parameter files2data GPLAB saves its names with complete path in the parameter variables datafilex and datafiley The parameter usetestdata may be used to indicate whether the best in dividual found so far should have its fitness measured in a different data set 27 usetestdata 1 or not usetestdata 0 If yes this extra measurement will be done in every generation and the user must provide the names of the two input and desired output extra data files to be stored in testdatafilex and testdatafiley When restarting a run the user does not have to provide any file names again Two different methods for importing the text files into the algorithm s vari ables are available in GPLAB not shown in Fig 2 1 e xy2inout for symbolic regression and parity problems e anttrail for artificial ant problems Accordingly there are also two methods for calculating fitness in GPLAB im plemented as plug and play functions see Fig 2 1 e regfitness calculates for each individual the sum of the absolute difference between the expected output va
27. ctical due to differences in the range of possible values e plotcomplexity Figure 3 3 This plot shows the evolution of tree depth and size and the percentage of introns during the run If the calccomplexity parameter is on see Sect 3 9 the plot will show the values concerning the best individual found so far and the population av erage otherwise only the values concerning the best so far will be shown The bold line shows the dynamic limit on depth or size depending on 34 the parameter depthnodes see Sect 3 2 If calccomplexity is on the mean tree fill rate see 12 will also be shown e plotoperators Figure 3 4 This plots shows the evolution of the op erators probabilities in bold and cumulative frequencies of occurrence Both the plots and the legends showing the current values are updated every generation even if the operators probabilities of occurrence are up dated more or less often Also shown are the number of reproductions see Sect 3 4 and clonings resulting from failed genetic operators see Sect 3 2 of the current generation Other examples of possible graphics settings e plotfitness plotcomplexity this setting draws both fitness and complexity plots on the top right corner and bottom right corner of the screen respectively e plotfitness plotdiversity plotoperators this draws the fitness diversity and operators plots leaving only t
28. does not exceed the strict maximum depth stored in realmaxlevel its fitness is measured If the individual proves to be better than the best individual found so far the dynamic maximum depth is increased and the new in dividual is allowed into the population otherwise the new individual is rejected and one of its parents enters the population instead 18 The individual is deeper than the strict maximum depth stored in real maxlevel it is rejected and one of its parents enters the population instead The dynamic maximum tree depth technique is a recent technique that has shown to effectively control bloat in two different types of problems see 11 for details The parameter dynamiclevel can be used to turn it on dynamiclevel 1 the default or off dynamiclevel 0 When on its initial value is determined by the parameter inicdynlevel This should not be confounded with the maximum depth of the initial random trees inicmaxlevel see Sect 3 1 The strict depth limit can also be turned on fixedlevel 1 or off fixedlevel 0 When on the strict maximum depth of trees is de termined by the parameter realmaxlevel Even more recently two variations on the dynamic limit technique have been introduced a heavy dynamic limit dynamiclevel 2 where the dynamic limit can unlike the original one fall back to a lower value in case the new best individual allows it and the dynamic limit on size number o
29. drawn This last input argument is not being used in the current version of GPLAB but the available sampling procedures contemplate this possibility The function must output the identifiers of the parents chosen their indices in the current population the expected number of children of all individuals in the popula tion and the normalized fitness of all individuals in the population The last two output arguments may be left blank if the sampling procedure does not calculate them Please see functions roulette sus and tournament for a prototype 3 7 Expected number of children expected As described in Sect 3 6 some sampling procedures choose the parents based on their expected number of children while others only need to know which are better than which Likewise the calculation of the expected number of children may use the actual fitness values or simply their rank in the population The 26 parameter variable expected determines with method is used for calculating the expected number of children for each individual This calculation is performed only if the selection for reproduction so requires Three different methods are available in GPLAB e absolute the expected number of children for each individual is pro portional to its absolute fitness value it is equal to its normalized or relative fitness 7 e rank85 the expected number of children for each individual is based on its rank in the populat
30. e parse tree e str the translation of the parse tree into a valid MATLAB expression e parents the list of identifiers of the parents that produced this individual or the empty list if the individual has a random origin e xsites the numbers of the nodes where the genetic operator split the parent trees This field is merely informative e nodes the number of nodes that constitute the parse tree This field remains empty until needed e introns the number of nodes on the parse tree that are considered introns This field remains empty until needed e level the depth of the parse tree This field remains empty until needed e fitness the fitness of the individual in the current data set data see Sect 2 2 e result the results obtained by the individual in each fitness case of the current data set e testfitness the fitness of the individual in a different data set for cross validation The state variable popsize indicates how many individuals are in pop and lastid contains the last unique identifier generated and used in the last indi vidual created 4 2 Tree depth size depthnodes iniclevel maxlevel levelhistory The state variable depthnodes is the same as the parameter with the same name determining whether restrictions on the shape of the trees allowed into the population are related to depth or size number of nodes iniclevel specifies the initial maximum depth size allowed for
31. e settings that are not sup posed to vary during the course of a run although the user can continue a GPLAB run with a different set of parameters from how it started Chapter 3 is dedicated to the different running aspects of GPLAB each subsection refers to one or more parameters related to that aspect e state stores all the variables that represent the current state of the algo rithm These settings are constantly updated during the run and should not be modified by the user Chapter 4 is dedicated to describing the meaning of the various state variables e pop stores the current population This variable is constantly updated as the population evolves It can be considered a state variable and accordingly its description can be found in Chap 4 e data is the data set s used by the algorithm to guide the evolutionary process and optionally perform cross validation imported from files in the beginning of each run Because it is stored along with the other algorithm s variables continuing a previously started run does not require the user to provide the data files again 2 3 Usage The large amount of available control parameters may lead to the wrong con clusion that it will take a long time before one can start using the toolbox comfortably and that only expert users will ever be able to use it properly On the contrary GPLAB is very easy to use and suits even the unknowledgeable users due to the automatic parametrization
32. ed The genetic operators create new individu als until a new population is filled a number determined by the generation gap see Sect 3 10 Calculating fitness is followed by the SURVIVAL module where the indi viduals that enter the new generation are chosen according to the elitism level parameter The GENERATION module repeats itself until the stop condition is fulfilled or when the maximum generation is reached Several stop conditions can be used simultaneously see Sect 3 14 This module can be called either by the user or by GEN POP 2 1 3 SET VARS This module either initializes the parameters with the default values or updates them with the user settings Besides the parameters directly related to the execution of the algorithm other parameters affect the output of its results see Chap 3 for a description of all parameters SET VARS can be called either by the user or by a request for parameter initialization from GEN POP 2 2 Working variables GPLAB uses a vast number of working variables organized in a structure ac cording to its role in the algorithm This structure will be referred to as vars throughout this manual It is composed of four fields params state pop data Saving vars to a file stores all the information GPLAB uses produces and will ever need to continue a previously started run e params stores all the variables that determine different ways of running the different parts of the algorithm These ar
33. edures for realizing a module and they behave as plug and play devices 2 1 1 GEN POP This module generates the initial population INIT POP and calculates its fit ness FITNESS The individuals in GPLAB are tree structures initialized with one of three available initialization methods Full Grow Ramped Half and Half 8 The functions available to build the trees include the if then else statement and some protected functions plus any MATLAB function that verifies closure The terminals include a random number generator and all the variables neces sary created in runtime Fitness is by default the sum of absolute differences between the obtained and expected results in all fitness cases The lower the Legend operation module main operation module operation module subdivided 9 parameter optional operation module 1 depending on parameter optional operation module depending on parameter 9 parameter parameter loop depending on parameter user function plug and play Users layman regular advanced cas session gt GENERATION A roulette element A i crossover regfitness antfitness o F E EA ies SAMPLING SET VARS VARS SETTING O initial probs type INITIAL PROBS GEN POP GENERATION GEN POP o INIT POP FITNESS n probs pe lt ADAPT PROBS A stabilize condition MV WINDOW sampling type oleo
34. ement them as MATLAB functions and make sure the input arguments can be either scalars or vectors see MATLAB user s manual and declare them using one of the toolbox functions use help setfunctions and help addfunctions in the MATLAB prompt for usage 19 params setfunctions params func1 2 func2 1 params addfunctions params func1 2 func2 1 setfunctions defines the set of available functions as containing functions func1 and func2 replacing any other functions previously declared func1 has arity 2 it needs two input arguments func2 has arity 1 Any num ber of functions can be declared at one time by adding more arguments to setfunctions addfunctions accepts the same arguments but adds the de clared functions to the already defined set keeping the previously declared functions untouched setfunctions and addfunctions are friendly substitutes to directly setting the parameter variable functions The declaration of genetic operators is done similarly see Sect 3 4 Some examples of MATLAB functions that verify closure fit for use with GPLAB e plus minus times e sin cos e and or not xor e ceil floor e min max e eq equal gt greater than le less than or equal GPLAB also includes some functions for artificial ant problems namely antif antprogn2 antprogn3 arities 2 2 3 respectively Terminals GPLAB can use any constant as a terminal
35. eplaced by a value pro portional to the operator s performance Operators that have been performing well see their probability values increased operators that have been produc ing individuals worse than the population from which they were born see their probability values decreased Operators that haven t been able to produce any children since the last adaptation will receive a substantial increase of proba bility as if their performance was twice as good as the performance of the best operator This will provide them with a chance to produce children again The parameter variable minprob can be used to impose a lower limit on each oper ator s probability of occurrence The default minprob value is 0 01 divided by the number of genetic operators used All the parameter variables described here can be set by the user but when left blank automatic parameterization will occur The adaptation interval adaptinterval is set to every generation as defined by the generation gap see Sect 3 10 the length of the moving window adaptwindowsize is set with numbackgen times the population size or numbackgen times the generation gap which one is larger The remaining default values are the ones indicated in the availableparams file and can also be consulted in Table 3 2 31 3 13 Initial operator probabilities initialprobstype initialfixedprobs initialvarprobs smalldifference Regardless of the operator probabilities in run
36. er provided see Sect 3 8 tournamentsize or gt 0 integer if gt 1 see Sect 3 6 usetestdata 0 1 o 3 1 Tree initialization inicmaxlevel depthnodes initpoptype The initial population of trees created in runtime in the beginning of a GPLAB run is done by choosing random functions and terminals from the respective sets Sect 3 3 The initial maximum depth size of the new trees determined by the parameter inicmaxlevel must not be violated but besides this rule there is still room for different options that may influence the structure of the initial trees These options constitute what is called the generative method specified by the parameter initpoptype There are three different methods available in GPLAB used in the plug and play fashion described in Sect 2 4 and each of them uses either the standard procedure based on depth 8 or the new variation based on size i e number of nodes 12 depending on the parameter depthnodes 1 for depth 2 for size see Sect 3 2 e fullinit this is the Full method In the standard procedure the new tree receives non terminal internal nodes until the initial tree depth inicmaxlevel parameter is reached the last depth level is limited to terminal nodes As a result trees initialized with this method will be perfectly balanced with all the branches of the same length If size is used instead of depth internal nodes are chosen until the size
37. ess the default procedure without the need to change the setting before running the algorithm the line defaults calcfitness regfitness in the availableparams file should be replaced with defaults calcfitness newfitness 6699 Names of functions have to be accompanied by the triple settings can be made like in this example but numeric defaults hits 100 0 90 10 The exceptions to the description above are all the cases when the new plug and play function is to be used along with other plug and play functions like the genetic operators and the functions and terminals Please see file availableparams for examples Similarly the file availablestates may also be edited to include new fields in the structure variable state This may have to be done if a new plug and play function intends to use state variables other than the ones available This is an advanced action that should not be attempted without proper care 13 Chapter 3 Parameters The next sections describe aspects related to the parameters used by GPLAB what are the parameters involved in each part of the algorithm and how their modification affects its behavior Each subsection concerns one or more param eter variables and each parameters may appear in more than one subsection Table 3 1 indicates the location of each parameter in this manual and Table 3 2 specifies their possible and default values When setting par
38. f nodes regardless of depth see 12 for details The parameter depthnodes is used to switch between depth depthnodes 1 and size depthnodes 2 restrictions Any combination of fixedlevel dynamiclevel and depthnodes can be used The default initial values for realmaxlevel and inicdynlevel depend on the setting of depthnodes see Table 3 2 The dynamic limits are turned on in the demo functions of the toolbox and the original dynamic limit on depth is even used as default along with the strict limit because this combination seems to be very effective in controlling bloat Nevertheless the user should keep in mind that they are still experimental techniques 3 3 Functions and terminals functions terminals numvars autovars As any genetic programming algorithm GPLAB needs functions and terminals to create the population in this case the parse trees that represent individuals Functions GPLAB can use any MATLAB function that verifies closure plus some protected and logical functions and the if then else statement also avail able as part of the toolbox The user indicates which functions the algorithm should use by setting the parameter variable functions Table 3 3 contains information on the available toolbox functions All the functions described in Table 3 3 are used in the plug and play fash ion described in Sect 2 4 The advanced users who want to build and use their own functions only have to impl
39. group For help on a particular function use help lt function name gt in the MATLAB prompt 6 1 Demonstration functions e demo e demoparity e demoant 6 2 Running the algorithm and testing result e gplab e testind 6 3 Parameter and state setting e setparams e resetparams e resetstate e setoperators 48 e addoperators e setfunctions e addfunctions e setterminals e addterminals 6 4 Automatic variable checking These are called by gplab and should not be called by the user e checkvarsparams e checkvarsstate e checkvarsdata 6 5 Description of parameter and state variables e availableparams e availablestate 6 6 Creation of new generations e genpop e generation e pickoperator e applyoperator e pickparents e applysurvival e updatestate e stopcondition 49 6 7 Creation of new individuals e initpop e fullinit e growinit e rampedinit e newind e maketree 6 8 Filtering of new individuals e validateinds e strictdepth e strictnodes e dyndepth e dynnodes e heavydyndepth e heavydynnodes 6 9 Protected and logical functions e mydivide e mylog e mylog2 e mylogi0 e mysqrt e mypower e myif e kozadivide e kozasqrt e nand e nor 50 6 10 Artificial ant functions demoant antmove antleft antright antprogn2 antprogn3 antif antfoodahead antnewpos anteval antfitness anttrail antsim antpath 6 11 Tree manipulation maketree treelevel nodes int
40. h of the four different possible plots will be shown in runtime It is a list of plot names it can be empty graphics in which case there will be no runtime graphical output or it can contain either or all of the plots described below The order of the plot names inside the graphics list is respected when positioning the figures on the screen beginning on the top right corner of the screen followed by the bottom right corner the top left corner and finally the bottom left corner the idea is to also keep the textual output visible for as long as possible Each plot may contain more or less information depending on other parameter settings e plotfitness Figure 3 1 This plot shows the evolution of the maxi mum best of current generation median average and average std dev values of fitness In bold it also shows the fitness of the best individual found so far if the usetestdata parameter is true see Sect 3 8 it will also show the evolution of the test fitness cross validation in a different data set of the best individual found so far When the survival param eter is set to other than replace see Sect 3 11 the maximum and best so far fitness values are always the same e plotdiversity Figure 3 2 This plot shows the evolution of the population diversity measures indicated in the parameter calcdiversity see Sect 3 9 Showing more than one diversity measure at the same time may not be very pra
41. he bottom left corner of the screen empty Every generation the plots are updated with the values of the current gener ation The legends of the plots show the last values plotted they may indicate absolute instead of relative values whatever seemed to be more useful When a previously stopped algorithm is run for some additional generations all the previous history values are drawn and the plots continued as if the algorithm was never interrupted 35 log10 fitness E a o 1 5 0 Fitness maximum 1 6673 median 1 9273 average 3 1852 avg std 0 40825 avg std 6 7787 best so far 1 6666 test fitness 15 3186 5 10 15 20 25 generation Figure 3 1 Graphical output produced by the plotfitness option in the graphics parameter population diversity 100 90 80 70 60 50 40 30 Population diversity uniquegen 66 20 0 5 10 15 20 25 generation Figure 3 2 Graphical output produced by the plotdiversity option in the graphics parameter 36 Structural complexity 120 maximum size 38 bestsofar size 37 IEE I e FE SR ae ee a avg size 35 1 bestsofar introns 0 avg introns 0 02 bestsofar depth 11 avg depth 10 72 avg tree fill 23 5591 100 80 60 40 tree depth 10 tree size introns 20 SI II III III III IIE 0 5 10 15 20 25 generation Figure 3 3 Graphical output pr
42. ideas on how to solve the bugs including the people on the MATLAB newsgroup link Many other users have steadily provided a wealth of comments suggestions and useful code thank you all particularly Mehrashk Meidani Ali Nazemi Wo Chiang Lee Vladimir Crnojevic and Peter J Acklam Please forgive me if I have forgotten someone Chapter 2 Operational structure The architecture of GPLAB follows a highly modular and parameterized struc ture which different users may use at various levels of depth and insight What follows is a visual description of this structure along with brief explanations of some operation details and control parameters algorithm s variables a sum mary of three usage profiles appropriate for different types of users and details on how to deal with plug and play functions 2 1 Main modules Figure 2 1 shows the operational structure of GPLAB There are three main operation modules namely SET VARS GEN POP and GENERATION and each represents an interaction point with the user Inside each main module the sub modules are executed from top to bottom the same happening inside INITIAL PROBS and ADAPT PROBS The description of these two can be found in Sects 3 13 and 3 12 respectively Any module with a question mark can be skipped depending on the parameter indicated above it Each module may use one or more parameters and one or more user functions User functions implement alternative or collective proc
43. ing that tree represents e findnode tree x returns the subtree of tree with root on node num ber x The nodes are numbered depth first e swapnode tree x node returns the result of swapping node number x in tree for node e tree2str tree returns the translation of tree into a string e treelevel tree returns the depth of tree e nodes tree returns the number of nodes of tree e intronnodes tree params data state returns the number of introns of tree Needs the variables params data and state Unlike in previous versions of GPLAB the genetic operators do not need to return offspring that conform to the tree depth size restrictiona being applied see Sect 3 2 this is now performed afterwards by applying validation also called filter functions see Sect 3 5 Of all the fields an individual contains only origin parents tree and str must be filled id should be left empty to be filled by the validation functions mentioned above All the other fields xsites fitness result testfit ness nodes introns level can also be left empty if not needed by the genetic 23 operator because they will be calculated and stored as needed by other proce dures xsites is the exception as a merely informative field that may contain information concerning the nodes where the parent trees were split to create the child tree if left empty it will remain so as no other other function in the current version of
44. ion 2 e rank89 the expected number of children for each individual is based on its rank in the population and on the state of the algorithm how far it is from the maximum allowed generation The differentiation between individuals increases in later generations 10 Alternative methods for calculating the expected number of children may be built and used as plug and play devices to module EXPECTED see Fig 2 1 by simply implementing the new method in a MATLAB function and declaring it in the parameter variable expected params expected new_expected_number_of_children_method The new function must accept as input arguments the current population and state vars pop vars state and output the expected number of children of all individuals in the population and the normalized fitness of all individuals in the population The last output argument may be left blank if its calculation is not needed Please see functions absolute rank85 and rank89 for a prototype 3 8 Measuring fitness files2data datafilex datafiley testdatafilex testdatafiley usetestdata calcfitness precision lowerisbetter keepevalssize When starting a GPLAB run the user is required to indicate the names of the files where the fitness cases are stored The files should be in a format readily importable to MATLAB like Tab delimited text For symbolic regression and parity problems the first file should contain the input values and the se
45. ion of this procedure along with the parameters that affect its behavior The algorithm keeps track of some information regarding each child pro duced like which operator was used and which individuals were the parents The first children to enter this information repository are also the first to leave it so only the younger children are tracked This repository of information is like a moving window on the individuals created and its capacity or length is controlled by the parameter variable adaptwindowsize Another information stored in this repository for each child is how good its fitness is when com pared to the best and worst fitness values of the population preceding it Each child receives a credit value based on this information and a percentage of this credit is attributed to its ancestors The number of back generations receiving credit is indicated in the parameter variable numbackgen and the percentage of credit that is passed from each generation back to its ancestors is indicated by percentback Every adaptinterval individuals the performance for each genetic opera tor is calculated by summing the credits of all individuals currently inside the moving window created by that operator and dividing the sum by the number of individuals currently inside the moving window created by that operator Each operator probability value is then adapted to reflect its performance A percentage of the probability value percentchange is r
46. ities one value for each operator and ophistory contains the past set tings of operatorprobs one row per generation and one column per operator The cumulative absolute frequency of occurrence of each operator is stored in operatorfreqs and opfreqhistory stores the past settings of operatorfreqs one row per generation and one column per operator Also stored are the current number of reproductions see reproduction parameter in Sect 3 4 and its past settings reproductionhistory The cur rent number of clonings resulting from failed genetic operators see Sect 3 2 is also stored one column per operator as well as its past settings in cloning history one row per generation and one column per operator When the operator probabilities are automatically adapted adaptwindow is the moving window that stores the information about past produced children see Sect 3 12 and lastadaptation stores the last identifier generated when the last adaptation occurred 4 5 Population fitness popfitness popnormfitness popexpected popranking keepevals varsvals Although each individual in pop stores its own fitness value the state variable popfitness also stores a list of the fitness values of all individuals Depending on the sampling procedure used see Sect 3 6 the normalized fitness expected 41 number of children and ranking may also need to be calculated These are stored in the state variables popnormfitness popexpected a
47. lgorithm will refuse the name given by the user to avoid the over writing of files from a previous run Each file will be named after the current generation 3 16 Runtime textual output output During the run GPLAB may output more or less textual information con cerning the state of the algorithm The amount is determined by setting the parameter variable output e silent this setting produces the minimum amount of textual output during the run Only what are considered important messages will be dis played like the beginning and ending of the algorithm automatic setting of some parameters and overriding of settings made by the user 33 e normal this setting produces textual additional output during the run of the algorithm like the identification fitness depth and size of the best individual found so far If the usetestdata parameter is true see Sect 3 8 it also shows the test fitness cross validation in a different data set of the best individual found so far e verbose this setting will produce the same output as normal plus the parameter and state variables lists in the beginning of the run 3 17 Runtime graphical output graphics GPLAB can represent some of the algorithm s state variables graphically as plots that are updated in runtime every generation Additionally some specialized functions are available for offline use Sect 5 The graphics parameter indicates whic
48. log10 4 log10 abs a otherwise Ifthen else ig ne eval c if eval a 0 statement ES SN eval b otherwise Negation of AND nand a b not and a b Negation of OR nor a b not or a b 1 sqrt log log2 log10 abs eval not and or are MATLAB functions eval x returns the result of evaluating the expression x 21 Variables needed to evaluate the fitness cases are also part of the set of avail able terminals for the algorithm to work with and these can only be generated automatically in the beginning of the run according to the settings of the parameters numvars and autovars e numvars and autovars 0 the parameter numvars is automatically filled with O and no variables are generated This setting is appropriate for artificial ant problems e numvars and autovars 1 the parameter numvars is automatically filled with the number of columns of the input data set and these many variables are generated This setting is appropriate for symbolic regression and parity problems e numvars x customized setting where x is the number of variables gen erated corresponding to the x first columns of the input data set 3 4 Genetic operators reproduction operatornames operatornparents operatornchildren GPLAB may use any number of genetic operators to create new individuals A proportion of individuals specified in parameter reproduction may also be copied into the next generation without
49. lue and the value returned by the individual on all fitness cases The best individuals are the ones that return values less different than the expected values the ones with a lower fitness This function should be used with the parameter lowerisbetter set to 1 e antfitness calculates for each individual the number of food pellets eaten in the artificial ant food trail during 400 time steps the best in dividuals are the ones who eat more pellets meaning they have higher fitness This function should be used with the parameter lowerisbetter set to 0 When regfitness is used all the fitness values stored in the algorithm s vari ables are rounded to a certain number of decimal places given by the parameter precision This is meant to avoid rounding errors that affect the comparison of two different individuals who have the same fitness For example in sym bolic regression problems it is common to see individuals with fitness values like 5 9674e 016 and 1 0131e 015 Without using the precision parameter the first individual would be chosen as the best even when the second one is smaller because these two values are not the same just because of the rounding error since they are in fact both null By default precision is set to 12 but the user can give it any integer number higher that 0 To use an alternative method for calculating fitness all the user has to do is build a new function respecting the input and ou
50. nd popranking Evaluating an individual for its fitness may be a time consuming task so previous evaluations may be stored in memory in case they are needed again see Sect 3 8 in the state variable keepevals with the following fields e inds the string of the individual e fits the fitness of the individual e ress the result of the evaluation in each fitness case e used how many times this evaluation has been used The memory used by this variable is cleared when the run ends Because a great part of the time consumed in the evaluation of individu als consists on the assignment of the fitness cases to the variables particularly when in presence of several inputs a string containing all the inputs ready for assignment is also kept as the varsvals state variable This string is con structed every time the fitness cases change e only once in the beginning of the evolutionary process in this version of the toolbox 4 6 Fitness statistics maxfitness minfitness avgfitness stdfitness medianfitness fithistory Every time a new generation is completed the maximum minimum average std dev and median fitness found in the population are stored in the state vari ables maxfitness minfitness avgfitness medianfitness and stdfitness Additionally every time these variables are updated a new row is added to the variable fithistory which contains five columns one for each fitness measure and as many rows as
51. ndition hits in which case GPLAB will only stop when reaching the maximum number of generations allowed 3 15 Saving results to file savetofile savedir During a run GPLAB can save all of the algorithm s variables vars see Sect 2 2 to file periodically according to the parameter variable savetofile e never this setting never saves the results to file e firstlast this setting causes the variables of the algorithm to be saved after the initial generation has been created and after a stop condition or the maximum generation indicated by the user is reached e every10 this setting causes the variables to be saved to file in the first and last generations as in firstlast plus every 10 generations e every100 this setting behaves like every10 but saves the results every 100 generations instead of every 10 e always this setting causes the variables to be saved to file after every new generation created Disk space may become a problem if this option is often used Except for the never option all the settings will cause GPLAB to request from the user the name of the directory where to save the variables before the algorithm begins unless it was already stored in the parameter variable savedir The new directory is created inside GPLAB s working directory and its complete path stored in savedir If a directory with the same name already exists the a
52. new release solves both and also contains a simulation function for watching the best ant move on the trail The main change in the code is that the artificial ant is now evaluated by its tree not by its string This implied passing the tree as an argument to all the functions that evaluate an individual ant or not and that is why many functions had to be modified Here is a list of modified and new functions of this new release Modified demoant antmove antleft antright antprog2 antprog3 antif antfoodahead antfitness regfitness calcfitness calcpopfitness automaticoperatorprobs stopcondition intronnodes dynnodes dyndepth heavydynnodes heavydyndepth desired_obtained plotpareto mypower New anteval antnewpos antsim antpath 1 2 Acknowledgements I would like to address a big thank you to Henrik Schumann Olsen Jens Thiele mann and Oddvar Kloster at SINTEF http www sintef no for the exten sive additional code they have provided for GPLAB version 1 I have integrated most of its goodies in version 2 but some are still missing like the powerful fea ture of building multiple trees I intend to integrate it also no promises when though Thank you so much to Marc Schoenauer s students Flavien Billard Aurlien Boffy and Thomas De Soza for spotting the nasty artificial ant bugs and to Matthew Clifton for the fruitful exchange of ideas and for providing most of the ant simulation code Thank you all for providing
53. ng on the programming of computers by means of natural selection Cambridge MA MIT Press 1992 Luke S Panait L Lexicographic parsimony pressure In Langdon W B Cant Paz E Mathias K Roy R Davis D Poli R Balakrish nan K Honavar V Rudolph G Wegener J Bull L Potter M A Schultz A C Miller J F Burke E Jonoska N editors Proceedings of GECCO 2002 San Francisco CA Morgan Kaufmann 2002 829 836 Montana D J Davis L Training feedforward neural networks using ge netic algorithms In Proceedings of the International Joint Conference on Artificial Intelligence 1989 762 767 55 11 Silva S Almeida J Dynamic maximum tree depth a simple tech 12 nique for avoiding bloat in tree based GP In E Cant Paz Foster J A Deb K Davis L D Roy R O Reilly U M Beyer H G Standish R Kendall G Wilson S Harman M Wegener J Dasgupta D Pot ter M A Schultz A C Dowsland K A Jonoska N Miller J editors Proceedings of GECCO 2003 Berlin Springer Verlag 2003 1776 1787 Silva S Costa E Dynamic limits for bloat control variations on size and depth To appear in Proceedings of GECCO 2004 Berlin Springer Verlag 2004 56
54. ng functions correspond to the module SET VARS in Fig 2 1 resetparams is appropriate for symbolic regression problems For different types of problems one should see the parameter settings used on the demo functions demoparity and demoant Please see Chap 6 for a complete list of functions There are also functions dedicated to setting some specific parameters like the genetic operators and the functions and terminals used to build the trees see Sects 3 4 and 3 3 for details 11 2 3 3 The advanced researcher Here is the user who wants to build and test new sampling methods new genetic operators in short new user functions as shown in Fig 2 1 without having to construct a new toolbox from the beginning GPLAB allows this with a minimum amount of effort thanks to its plug and play operational structure As an example the user who wants to test a new genetic operator only has to build a new function that implements it using the tree manipulation functions provided This function should use the same input and output arguments as the other genetic operators a template for building new genetic operators is provided in Sect 3 4 To tell the algorithm about the new genetic operator the available function is params addoperators params newoperator nparents nchildren where newoperator is the name of the new function nparents is the number of parents the new operator needs and nchildren is the number of offspring it prod
55. oduced by the plotcomplexity option in the graphics parameter Genetic operators prob crossover 0 98131 prob mutation 0 018689 cum freq crossover 501 cum freq mutation 154 reproductions 0 clones crossover 2 clones mutation 0 e pS operator probability frequency o a e w o o 0 1 0 0 5 10 15 20 25 generation Figure 3 4 Graphical output produced by the plotoperators option in the graphics parameter 37 Chapter 4 State The next sections describe aspects related to the state variables used by GPLAB These variables store information that reflect the current running conditions of the algorithm as well as the last batch of results produced Some variables also store historic information concerning the results produced useful for a posterior analysis including visualization see Sect 5 of the complete run Although not part of the state structure the current population of individuals pop will also be described as a state variable see Sect 2 2 Each subsection concerns one or more state variables Table 4 1 indicates the location of each state variable in this manual Table 4 1 Location of state variables in this manual State variable Section Page adaptwindow 4 4 41 arity 4 3 41 avgfitness 4 6 42 avgintronshistory 4 9 43 avglevelhistory 4 9 43 avgnodeshistory 4 9 43 avgtreefillhistory 4 9 43 bestfithistory 4 7 42 bestintronshistory 4 7 4
56. of the new tree is close to the specified size inicmaxlevel and only then terminals are chosen Unlike the standard procedure the size variation may not be able to create trees with the exact size specified but only close never exceeding e growinit this is the Grow method In the standard procedure each new node is randomly chosen between terminals and non terminals except nodes at the initial tree depth level which must be terminals Trees created with this method may be very unbalanced with some branches much longer than others and their depth may be anywhere from 1 to the value of the inicmaxlevel parameter 17 If using the size variation nodes are also chosen randomly but prior to reaching the size specified in inicmaxlevel care is taken on the choice on the internal nodes based on their arity so as to guarantee the inic maxlevel will not be exceeded by the respective arguments which now have to be terminals e rampedinit this is the Ramped Half and Half method In the standard procedure an equal number of individuals are initialized for each depth between 2 and the initial tree depth value For each depth level considered half of the individuals are initialized using the Full method and the other half using the Grow method The population of trees resulting from this initialization method is very diverse with balanced and unbalanced trees of several different depths In the size variation an equal
57. of most parameters Here is a sum mary of three different profiles of usage that may be given by different types of users the layman the regular user and the advanced researcher 2 3 1 The layman This user wants to try a genetic programming algorithm to achieve a solution to a standard problem without having to learn about available parameters or how to set them The available functions are vars b gplab g n vars b gplab g vars where g is the maximum number of generations to run the algorithm and n is the population size The first function initializes all the parameters of the algorithm with the default values and runs it for g generations with n individuals The user will be asked about the location of the data files to use see Sect 3 8 It returns vars all the variables of the algorithm and b the best individual found which is the 10 same as vars state bestsofar The second function continues a previously started run for another g generations and also needs vars as an input argu ment These two functions correspond to the operation modules GEN POP and GENERATION shown in Fig 2 1 GPLAB also provides some demonstration functions to illustrate its usage in different types of problems see Chap 6 for a complete list of functions e demo runs a symbolic regression problem the quartic polynomial with 50 individuals for 25 generations with automatic adaptation of operator probabilities and performing cross v
58. r the new population Table 3 4 lists the appropriate list of filters for each combination of the 3 depth size related parameters fixedlevel dynamiclevel depthnodes Once again the list of filters is chosen automatically by GPLAB in the be ginning of the run 3 6 Selection for reproduction sampling tournamentsize As shown in Fig 2 1 genetic operators need parent individuals to produce their children In GPLAB these parents are selected according to one of four sampling methods as indicated in the parameter variable sampling e roulette this method acts as if a roulette with random pointers is spun and each individual owns a portion of the roulette that corresponds to its expected number of children see Sect 3 7 e sus this method also relies on the roulette but the pointers are equally spaced 3 e tournament this method chooses each parent by randomly drawing a number of individuals from the population and selecting only the best of them 25 e lexictour this method implements lexicographic parsimony pressure 9 Like in tournament a random number of individuals are chosen from the population and the best of them is chosen The main difference is if two individuals are equally fit the shortest one the tree with less nodes is chosen as the best This technique has shown to effectively control bloat in different types of problems see 9 for details When either of
59. researcher It was tested on different MATLAB versions and computer platforms and it does not require any additional toolboxes This manual is accompanied by a zip file containing all the functions that form the toolbox released under GNU General Public Licence Both are freely available for download at http gplab sourceforge net Chapter 2 describes the operational structure of GPLAB Details on the available parameters and state variables are found in Chaps 3 and 4 respectively Chapter 5 shows the available offline graphical capabilities of GPLAB and Chapter 6 presents a summary of all toolbox functions divided in functional groups 1 1 Update from version 2 GPLAB is slowly growing and hopefully improving The changes are always biased towards my own work but I also try to incorporate different things that I have come to realize other users need Version 2 had already suffered some updates since its original release some minor bug fixes and an efficiency patch published on the GPLAB website These introduced modifications to sev eral functions plotpareto checkvarsparams stopcondition regfitness mutation crossover maketree provided a new function updatenodeids and collapsed two functions findnode swapnode into a single one swapnodes This new release does not differ much from the previous updated one Some bugs on the artificial ant problem had been spotted and another function mypower needed improvement This
60. ronnodes tree2str swapnodes updatenodeids 6 12 Data manipulation xy2inout anttrail saveall 51 6 13 Expected number of children e absolute e rank85 e rank89 e calcpopexpected 6 14 Sampling e roulette e sus e tournament e lexictour e sampling 6 15 Genetic operators e crossover e mutation 6 16 Fitness e calcfitness e regfitness e antfitness e calcpopfitness 6 17 Diversity measures e uniquegen e hamming 52 6 18 Automatic operator probability adaptation isoperator setinitialprobs automaticoperatorprobs moveadaptwindow addcredit updateoperatorprobs 6 19 Runtime graphical output These are called by gplab and should not be called by the user graphicsinit graphicsstart graphicscontinue graphicsgenerations 6 20 Offline graphical output desired_obtained accuracy_complexity plotpareto operator_evolution drawtree antsim 6 21 Utilitarian functions explode implode scale normalize shuffle 53 e orderby e intrand e countfind e findfirstindex e isvalid e ranking e fixdec e uniquenosort 6 22 Text input files These are used in pairs exp_ txt exponential and quartic_ txt quar tic polynomial zt z 2 x contain 21 equidistant points in the inter val 1 to 1 parity bit_ txt contain all cases santafetrail txt and santafepellets txt contain respectively the Santa Fe artificial ant trail and the number of food pellets in it e exp
61. suffering the action of the operators Tree crossover and tree mutation are the genetic operators provided by GPLAB implemented as follows Crossover In tree crossover random nodes are chosen from both parent trees and the respective branches are swapped creating two offspring There is no bias towards choosing internal or terminal nodes as the crossing sites Mutation In tree mutation a random node is chosen from the parent tree and substituted by a new random tree created with the terminals and functions available This new random tree is created with the Grow initialization method and obeys the size depth restrictions imposed on the trees created for the initial generation see Sect 3 1 Although these are the only genetic operators provided the addition of oth ers is straightforward thanks to the modular structure shown in Fig 2 1 A new genetic operator is simply a MATLAB function used as a plug and play device to module OPERATOR and the declaration of its existence to the algorithm is made similarly to the setting of functions and terminals see Sect 3 3 with one of the toolbox functions use help setoperators and help addoperators in the MATLAB prompt for usage params setoperators params operator1 2 2 operator2 2 1 params addoperators params operator1 2 2 operator2 2 1 22 The first function defines the set of genetic operators as containing opera tors operator1 and
62. t of solutions for which no other solution was found which both has a smaller tree and better fitness and the sizes and fitnesses of the current population vars pop This plot can easily be coupled as a runtime plot updated in every generation as it does not request a new figure to be drawn upon Use the command figure before calling this function to see the plot in a different window if necessary 5 3 Desired versus Obtained This plot is drawn by the function desired_obtained use help desired_ob tained in the MATLAB prompt for usage It draws lines representing the 44 function the algorithm was trying to approximate and several approximations obtained in different generations Fig 5 3 Appropriate for symbolic regression problems only 5 4 Operator Evolution This plot is drawn by the function operator_evolution use help opera tor_evolution in the MATLAB prompt for usage It draws lines representing the evolution of the operator s probabilities during the run Fig 5 4 It is not as detailed as the plotoperators drawn in runtime see Sect 6 19 5 5 Tree visualization This plot is drawn by the function drawtree use help drawtree in the MAT LAB prompt for usage It draws a GPLAB tree with the respective node labels Enlarge the figure if labels are overlapped Accuracy versus Complexity 405 fitness level nodes 35 30 fitness level nodes rm nN o ar
63. time being variable or fixed their initial values in the beginning of a run can be set either by the user or subject to an initial adaptation procedure closely related to the one previously described To specify the desired initial operator probabilities one should set the pa rameter variable initialprobstype to fixed and initialfixedprobs to a list of probability values following the same order as operatornames see Sect 3 4 If initialfixedprobs is left blank all the probabilities will be set to equal values To allow the initial adaptation procedure to run one should set initialprobstype to variable Additionally and because the initial adaptation procedure also needs initial probability values to start the adaptation one can set initialvarprobs with a list of probability values If left blank all the probabilities will be set to equal values The initial adaptation procedure creates an initial random population of individuals and runs the algorithm until adaptinterval new individuals have been created It then adapts the operator probabilities as described in Sect 3 12 repeats the process including the creation of a random population and averages both sets of adapted operator probabilities With the new operator probabilities set to the average values the whole process is repeated until the difference between old and new probabilities is no larger than smalldifference This parameter is initially set with the the maximum ch
64. tput arguments and set the parameter variable calcfitness with the name of the new function params calcfitness new_calcfitness_method The new function must accept as input arguments the string expression of the individual to measure vars pop i str the parameters vars params the data variable vars data the terminals vars state terminals and the varsvals string see Sect 4 5 containing all the fitness cases in a format ready for assignment it must output the fitness value of the individual the vector 28 of values returned by the individual on each fitness case and if necessary the updated state variable Please see functions regfitness and antfitness for prototypes The parameter lowerisbetter should be set accordingly Calculating fitness may be a time consuming task and during the evolu tionary process the same tree is certainly evaluated more than once To avoid this the parameter keepevalssize specifies how many evaluations are kept in memory for future use in case their results are needed again Evaluations used less often are the first to be discarded when making room for new ones If left empty keepevalssize will be automatically set to the population size The ideal balance between CPU time and memory is not easy to find and one must not forget that searching the memory for the results of previous evalua tions may also be a time consuming task Nevertheless it is almost essential to use this option in
65. uces Details on how to build new plug and play user functions can be found in Chap 3 please search for the particular subsection that applies and the way to integrate them into GPLAB is described in Sect 2 4 2 4 Plug and play Figure 2 1 shows that most modules of the GPLAB operational structure are based on a set of user functions that act as plug and play devices There are three important aspects related to these functions how to build them how to use them and how to integrate them in GPLAB 2 4 1 Building plug and play functions Building a new plug and play function is like building any other MATLAB function while following the rules pertaining input and output arguments Each module defines its own set of input and output arguments so the interested user should refer to the appropriate section in Chap 3 2 4 2 Using new plug and play functions To use a newly built plug and play function the user must declare its existence in the algorithm s parameters Once again each module is associated to different parameter variables and the user should refer to the appropriate section in Chap 3 but the general form of doing this is vars params lt specificvariable gt name_new_func where lt specificvariable gt is the parameter that refers to the module adopting the new function This may look equivalent to doing params setparams params lt specificvariable gt name_new_func 12 but this form of setting p
66. ynnodes the same as the previous one but considering size number of nodes instead of depth e heavydyndepth this filter measures the fitness of an individual and checks its depth If it is deeper than the dynamic maximum allowed depth if the individual is better than the best so far or if it is no deeper than the deepest of its parents the filter increases the dynamic depth if needed and accepts the individual otherwise rejects it If the individual is less deep than the dynamic maximum allowed depth if it is the better than the best so far the filter accepts it and lowers the dynamic depth and does nothing otherwise e heavydynnodes the same as the previous one but considering size number of nodes instead of depth 24 Table 3 4 List of filters for each combination of parameters Filters list fixedlevel dynamiclevel depthnodes Er 0 0 gt dyndepth 0 1 1 dynnodes 0 1 2 heavydyndepth 0 2 1 heavydynnodes 0 2 2 strictdepth 1 0 1 strictnodes 1 0 2 strictdepth dyndepth 1 1 1 strictnodes dynnodes 1 1 2 strictdepth heavydyndepth 1 2 1 strictnodes heavydynnodes 1 2 2 The above filters may reject an individual accept an individual or do nei ther After passing through all the filters the individuals that still haven t been rejected or accepted will finally be accepted as candidates fo

GPLAB A Genetic Programming Toolbox for MATLAB

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