evolutionary computation and fuzzy systems

Electronic Data Systems, Inc.

Evolutionary Computationand

Fuzzy Systems

Yuhui Shi

Electronic Data Systems, Inc. Embedded Systems1401 E. Hoffer St.,

Kokomo, Indiana 46902


Outline of tutorial session

zBrief introduction of EC and FSzInteractions between EC and FSySupportingyCollaborating

zDiscussions and Conclusions


Evolutionary Computation

Evolutionary computation comprises four main areas:yGenetic algorithmsyEvolutionary programmingyEvolution strategiesyGenetic programming



EC paradigms differ from traditional search and optimization paradigms in that EC paradigms:yUse a population of points in their searches,yUse direct “fitness” information, instead of

function derivatives or other related knowledge, and,yUse probabilistic rather than deterministic

transition rules.



Population:yEach member is a point in the hyper-space

problem domain, and thus is a potential solutionyA new population is generated each epochyPopulation typically remains the same sizeyOperators such as crossover and mutation

significantly enhance parallel search capabilities



Fitness InformationyAuxiliary information such as derivatives used

to minimize sum-squared error in neural nets is not usedythe fitness value optimized is directly

proportional to the function value being optimizedyIf fitness is proportional to profit, for

example, then the fitness rises as the profit rises



Probabilistic Transition Rules:ySearch is not randomySearch is directed toward regions that are

likely to have higher fitness valuesyDifferent EC paradigms make different uses

of stochasticity



Encoding of ParametersyTypically encoded as binary stringsyAny finite alphabet can be usedyTypically, population member string is of

fixed length



Generic EC ProcedureyInitialize the populationyCalculate the fitness for each individualyReproduce selected individuals to form new

populationyPerform evolutionary operations (mutation,

etc.)yGo to step 2 until some condition is met



Generic EC Procedure, Cont’d.yInitialization often done by randomizing

initial population(can use promising values sometimes)yFitness value is proportional to value of

function being optimized (often scaled 0-1)ySelection for reproduction based on fitness

values (some paradigms such as PSO retain all population members)



Applying EC Tools:yOptimization and classificationyMostly optimization⌧non-differentiable⌧many local optima⌧many not know optimum⌧system may be dynamic, changing with time, or

even chaotic

yoptimization versus meliorization (perhaps try other approaches first)


Fuzzy Systems

Lotfi ZadehyThe single most significant developer and

champion of fuzzy logic theory and applicationsySignificant contributions to field of systems

theoryy“Fuzzy Sets” paper published in 1965yAwarded the IEEE Medal of Honor in 1995


Fuzzy Systems

Fuzzy System Theory and ParadigmsyVariation on 2-valued logic that makes

analysis and control of real (non-linear) systems possibleyCrisp “first order” logic is insufficient for

many applications because almost all human reasoning is imprecise


Fuzzy Systems

Fuzzy versus CrispyFuzzy logic comprises fuzzy sets and

approximate reasoningyA fuzzy “fact” is any assertion or piece of

information, and can have a “degree of truth”, usually a value between 0 and 1yFuzziness: “A type of imprecision which is

associated with … classes in which there is no sharp transition from membership to non-membership” -- Zadeh (1970)


Fuzzy Systems

Fuzziness is not probability:yProbability is used, for example, in weather

forecastingyProbability is a number between 0 and 1 that

is the certainty that an event will occuryThe event occurrence is usually 0 or 1 in

crisp logic, but fuzziness says that it happens to some degree


Fuzzy Systems

Fuzziness is not probability, Cont’d.yFuzziness is more than probability;

probability is a subset of fuzzinessyProbability is only valid for future/unknown

eventsyFuzzy set membership continues after the

event


Fuzzy Systems

Fuzzy Set Membership:yIn fuzzy logic, set membership occurs by

degreeyFuzzy set membership values are between 0

and 1yWe can now reason by degree, and apply

logical operations to fuzzy sets


Fuzzy Systems

Fuzzy Set Membership FunctionsyFuzzy sets have “shapes”: the membership

values plotted versus the variableyFuzzy membership function: the shape of the

fuzzy set over the range of the numeric variable⌧Can be any shape, including arbitrary or irregular⌧Is normalized to values between 0 and 1⌧Often uses triangular approximations to save

computation time


Fuzzy Systems

Linguistic VariablesyLinguistic variable: “a variable whose values

are words or sentences in a natural or artificial language.” -- ZadehyLinguistic variables translate ordinary

language into logical or numerical statementsyImprecision of linguistic variables makes

them useful for reasoning


Fuzzy Systems

Approximate ReasoningyFuzzy reasoning involves different processes

than binary logicyRelations and operators have similar names

(AND, OR, etc.) but have different meaningsyIn Aristotlian logic:⌧Law of Noncontradition: A∩A ≡null set⌧Law of Excluded Middle: A∪A ≡ universal set

yNeither law holds in fuzzy logic


Fuzzy Systems

Equality of Fuzzy SetsyIn traditional logic, sets containing the same

members are equal: {A,B,C}={A,B,C}yIn fuzzy logic, however, two sets are equal if

and only if all elements have identical membership values:{.1/A,.6/B,.8C}={.1/A,.6/B,.8/C}


Fuzzy Systems

Fuzzy ContainmentyIn traditional logic, A⊂ B if and only if all

elements in A are also in ByIn fuzzy logic, containment means that the

membership values for each element in a subset is less than or equal to the membership value of the corresponding element in the supersetyAdding a hedge can create a subset or

superset.


Fuzzy Systems

Fuzzy Complement:yIn traditional logic, the complement of a set

is all of the elements not in the setyIn fuzzy logic, the value of the complement

of a membership is (1-membership_value)


Fuzzy Systems

Fuzzy Intersection:yIn standard logic, the intersection of two sets

contains those element in both setsyIn fuzzy logic, the weakest element

determines the degree of membership in the intersection

(Law of Noncontradiction does not hold)


Fuzzy Systems

Fuzzy Union:yIn traditional logic, all elements in either (or

both) set(s) are includedyIn fuzzy logic, union is the maximum set

membership value


Fuzzy Systems

General Forms of Fuzzy Rules:yIf X1 is A1 and ... and Xn is An then Y is ByIf X1 is A1 and ... and Xn is An then Y = p0 +

p1 X1 + ... + pn Xn

yIf X1 is A1 and ... and Xn is An then Y = f(X1 , ... , Xn )yIf X1 is A1 and ... and Xn is An then Y is class

I with confident degree = CDi


Fuzzy Systems

Fuzzy Rules Fire in Parallel:yIn a fuzzy system, all rules are activated in

parallelyIn traditional AI systems, rules fire in series -

must sometimes “back out” and start again


Fuzzy Systems

Combining Fuzzy Sets:yThis involves the combination of antecedent

sets, i.e., the sets on the if side of an if-thenruleyThis step is sometimes called fuzzification


Fuzzy Systems

Defuzzification:ycombines set of if-then rules into a specific

values of a control (output) variable


Fuzzy Systems

Fuzzy Logic Summary:yDistinction between A and not-A has been

weakenedyRules can be executed in parallel - puts fuzzy

logic in harmony with neural nets and evolutionary algorithmsyRules in fuzzy logic allow for unanticipated

solutions -- solutions can emerge rather that being imposed


Fuzzy Systems

Fuzzy Logic is not Fuzzy at all !!!Fuzzy Logic is not Fuzzy at all !!!


Interactions between EC and FS

ySupportive Approaches⌧From the Fuzzy System Side⌧From Evolutionary Computation Side⌧Michigan vs. Pittsburgh Approaches

yCollaborative Approaches⌧Environment level adaptation⌧Population Level Adaptation⌧Individual Level Adaptation⌧Component Level Adaptation


From Fuzzy System Side

zA fuzzy System is comprised of:ymembership function of fuzzy setsyset of fuzzy rules

zThe Design of Fuzzy System can be Categorized:yFuzzy rule setyFuzzy membership functionsyBoth


From Fuzzy System Side

Why employ EC: the design of a fuzzy system can be formulated as a search problem in high-dimensional space which has the following characteristics:yinfinitely largeynon-differentiableycomplex and noisyymulti-modalydeceptive


From Fuzzy System Side--- rule set only

zEncoding all possible fuzzy ruleszEncoding a portion of fuzzy rules


From Fuzzy system Side-- decision table

Low Medium High

Low High Medium Medium

Medium High Medium Low

High Medium Medium Low


Encoding All Fuzzy Rules

zEncode all possible rules: 1-- on; 0 -- offzEquivalent to the determination of the

decision tableyOnly consequent parts are required to be

encoded⌧integer: {0,1,2,3,4,5} represent

{NB,NS,ZR,PS,PB, don’t care}⌧binary bits: 3 bits represent 7 fuzzy sets plus one

don’t care symbol


Encoding All Fuzzy Rules

Drawbacks for large systems:ycomputational efficiency associated with

fuzzy logic is lostyrobustness decrease with increasing the

number of rules


Partially Encoding

zMaximum number of rules is known or can be guessedzThe experts can provide enough

information, for example, only the consequent parts of required rules are required to be determined


From Fuzzy System Side-- Membership Function’s Shape only

zRule set provided by experts or clustering algorithmsyRepresentations depend on the membership

function used: triangular, Gaussian, etc.yBinary; integer; real value

zEach rule has its own definition of membership function for each variable --corresponds to each expert


From Fuzzy System Side--- Rule Set & MF’s Shape

zRule set and membership function are co-dependent: designed at the same timezCombine previous two together. Mostly

the length of the chromosome is fixedyfixed type of membership functionyfixed number of parameters for each MFyfixed number of fuzzy rules



How to represent membership functionsytriangular⌧adjacent MFs to be fully overlapped & one MF has

its center resting at the lower boundary of the input range⌧symmetric and fixed center points; only need to

encode the base length

yGaussianysymmetrical exponential MFsyetc.



How many rules to encodeyComplete or partialyFor uniform representation, rule

representation needs to be consistent with the membership function representationyCan also encode the number of rules in the

rule set


From Fuzzy System Side -- Rule Set & MF’s Shape & Range

Uniform distribution of the fuzzy partitions (fuzzy set) is usually not optimalytoo coarse: performance may be lowytoo fine: too many rules ⌧lack of training patterns

ysome parts of space require fine partition; while the others coarse partition



Fuzzy partitionyConcept of distributed fuzzy rules⌧use several different partitions⌧encode all the possible rules corresponding to all

partitions: 1-- on; 0 -- off⌧use EC to remove unnecessary rules⌧drawback: chromosome is too long, especially for

large problems



Fuzzy partition, Cont’d.yEmploy several different types of MFs⌧triangular, Gaussian, sigmoid, etc.⌧each fuzzy set can be associated with different

type of MF⌧determined through evolution⌧both the MF type and MF parameters are encoded

into the chromosome



ConclusionsyFour parts need to be considered for fuzzy

system design⌧the number of rules⌧determination of rule structures⌧selection of membership function’s type⌧tune the membership function’s parameters

yWhich combination of the four parts should be considered is problem dependent



Conclusions, Cont’d.yAll four parts can be considered separately in

sequence when necessaryyNo single method is optimal to all problems


From EC Side

zRepresentationzInitializationzEvaluationzEvolutionary operationszValues for the parameters that EC uses

(population size, probabilities of applying evolutionary operators, etc.)


From EC Side

Representation:yBinary representationyInteger representationyReal value representationyMixed representationyDifferent representation requires different

evolutionary operators


From EC Side

zBinary representationyfuzzy set: ⌧n bits represent n fuzzy sets

10001 represents this variable is very low or very high

⌧using 0 means don’t care; 1 to 7 means large negative to large positive; then 3 bits can be used to represent these 8 symbols

• if fuzzy sets plus 1 is not 2’s power, then biase comes in


From EC Side

zBinary representationyMembership function’s parameter⌧a string of bits to represent each parameter

yMembership function’s typesyUse 1 and 0 to represent whether this rule

exists in the whole rule set


From EC Side

zInteger representationyfuzzy set: 0 to n represent n fuzzy sets plus

one don’t care symbolyMembership function’s parameter: divide the

range of the parameter into n partsyMembership function’s types: 1 to n

represent n different function typesyNumber of the rules inside the rule set


From EC Side

zReal value representationyMembership function’s parametersyAdvantage:⌧precision⌧capacity to exploit the gradulity of the functions

with continuous variables

zMixed representation


From EC Side

zInitialization of the populationyincluding the previous knowledge into the

population initialization to speed up the automatic designyobtain some individuals by equally

partitioning the problem space into varying number of linguistic terms


From EC Side

zFitnessyProblem dependent⌧classification problems⌧control problems⌧fuzzy classifier systems

yGeneral information: for example, include the complexity of the system into the fitness to prefer a simple system


From EC Side

zClassification problemsynumber of correctly and/or wrongly classified

training patternsymean-square error (absolute difference

error): if you prefer your system to have a bigger tolerance yrelative difference error: if you prefer your

system to have similar accuracy for any target output value


From EC Side

zControl problemsyOff-line design⌧prepare the training data set

• balance between the number of samples and the computation time

⌧measure of convergence ⌧rise-time, setting time, overshoot⌧the complexity of the controller⌧etc.


From EC Side

zFuzzy classifier systemsyeach chromosome represents only a portion

of the systemywhole population represents a single systemythe whole population obtains a single payoff

through evaluation function and shared by all the chromosomes (rules) by some distribution methods, like credit assignment approach


Michigan vs. Pittsburgh

yPittsburgh approach⌧more useful for off-line environment⌧each individual represents a fuzzy system⌧rules cooperate while systems compete

yMichigan approach⌧most useful in an on-line, real-time environment⌧whole population represents a fuzzy system⌧rules cooperate and compete


Michigan vs. Pittsburgh

⌧Alternative: iterative rule learning approach• each individual represents a single rule• only the best individual is considered as the solution• procedure

– Use a EC to obtain a rule for the system– Incorporate this rule into the final set of rules– Penalize this rule to avoid obtaining the same rule

by eliminating from training set all those examples that are covered by the set of rules obtained previously

– If the set of rules obtained is suffice to represent the examples in the training set, the system ends up. Otherwise return to step 1


Collaborative Approaches

Use a fuzzy system to tune EC where the EC is employed to design another fuzzy systemyEnvironment level adaptationyPopulation level adaptationyIndividual level adaptationyComponent level adaptation



zEnvironment level adaptationydivide the design process into two stages⌧evolution stage: find satisfactory controller⌧refinement stage: refine the previous designed

controller to minimize the amount of time needed to reach the destination⌧different fitness functions for different stages⌧fuzzy transition between these two stages

ydivide the problem into different subspaces



zPopulation level adaptationytrapped in a local optimum⌧lack of diversity in the population⌧a disproportionate global/local search ability⌧key factors for balancing global/local search

• mutation rate• crossover rate• population size

⌧interaction of EC parameters and performance is complex and problem dependent



zPopulation level adaptationyutilize previous experience to build up a fuzzy

system to tune EC parametersyInputs to fuzzy system can be any

combination of EC performance measures or current parameter settingsyOutputs can be any of the EC parameters



zPopulation level adaptationymutation & crossover rate yrepresentationypopulation diversityyfuzzy stop criterionyAdvantage: better understanding of the

complex relationship between parameters and performance of EC


Population Level Adaptation

zMutation & crossover rate yinputs to fuzzy system⌧the variance average chromosome (VAC)⌧the average variance alleles (AVA)

yOutputs of fuzzy system⌧mutation rate change⌧crossover rate change

ye.g. VAC small, increase crossover rate slightly



zRepresentationyconsider the complexity of development from

genotype to a mature phenotype⌧several stages⌧influenced by the environment

yusing fuzzy decision values φ ∈ (0,1) instead of binary bitsyDifferent genetic operators (fuzzy genetic

operators)



zPopulation Diversityybalance between exploration (global search)

and exploitation (local search) abilityycrossover operator is a key point for

exploration/exploitation tradeoffyemploy several different crossover operators

while some facilitate exploration and the others facilitate exploitation; when to employ which operator based on some fuzzy rules



a bc d

exploration

exploitation

exploration



zFuzzy stop criterionycrisp goal: goal of finding the “optimum

solution”yfuzzy goal: goal of finding an approximate

solution that is close to the optimal solution⌧estimate the membership value of EC “optimal”

solution based on the previous EC iterations⌧stop if is above a user-defined acceptance level


Individual Level Adaptation

zDynamic mutation and crossover ratesyEach chromosome has its own crossover and

mutation probabilitiesyVaried according to its current fitness, the

best and mean fitness of the population


Individual Level Adaptation

zDynamic selection of crossover operators; yapplies two different crossover operators;

one with exploitation properties and the other with exploration propertiesyuses the linear ranking selection mechanismyThe parameters of the mechanism that

determines the selection pressure produced by the selection are adjusted using fuzzy systems


Component level Adaptation

zEach element in the chromosome has its own mutation probabilityyposition diversity measures as inputs to a

fuzzy systemyposition’s mutation rates as outputs


Discussions & Conclusions

Rule SetMembership functions(shape, range, etc.)FuzzificationDefuzzificationFuzzy operators

Representation(genotype)

Genetic operatorsselectioncrossovermutation

FitnessInitialization

phenotype

perform



zThe FSs and ECs are connected through the phenotypezEC get feedback from the evaluation of

the system it representszTwo sides are required to be considered

togetherzAll components inside EC and FS can play

a crucial role in a evolutionary FS



zFurther considerationsyevolutionary selection of inputs featuresyindirectly encoding scheme for large system⌧simple representation combined with a

developmental process

yEC for finding a near-optimal FS, then hill-climbing approach for finely tuning the obtained FS



zFurther considerations, Cont’d.yhand tuning is still necessary for some

problems⌧EC provides some insight and more

understanding of the system, and then provide some guideline for hand tuning

yThe one with the best fitness is not always the one has the best performance⌧modular evolutionary fuzzy system



zFurther considerations, Cont’d.ycollaboration among operators make ECs

work; try to understand⌧ the relationship between the operators ⌧what influence one operator can bring to another

operator⌧come out a better order for applying operators

yhow much evolution and/or fuzzification is enough

evolutionary computation and fuzzy systems

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