1 SOFT COMPUTING Evolutionary Computing. What is a GA? GAs are adaptive heuristic search algorithm based on the evolutionary ideas of natural selection

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SOFT COMPUTINGEvolutionary Computing

What is a GA?

GAs are adaptive heuristic search algorithm based on the evolutionary ideas of natural selection and genetics.

As such they represent an intelligent exploitation of a random search used to solve optimization problems.

Although randomized, GAs are by no means random, instead they exploit historical information to direct the search into the region of better performance within the search space.

What is a GA?

The basic techniques of the GAs are designed to simulate processes in natural systems necessary for evolution, specially those follow the principles first laid down by Charles Darwin of "survival of the fittest.".

Since in nature, competition among individuals for scanty resources results in the fittest individuals dominating over the weaker ones.

Evolutionary Algorithms

Genetic Programming

Evolution Strategies

Genetic Algorithms

EvolutionaryProgramming

Classifier Systems

• genetic representation of candidate solutions• genetic operators• selection scheme• problem domain

History of GAs

Genetic Algorithms were invented to mimic some of the processes observed in natural evolution. Many people, biologists included, are astonished that life at the level of complexity that we observe could have evolved in the relatively short time suggested by the fossil record.

The idea with GA is to use this power of evolution to solve optimization problems. The father of the original Genetic Algorithm was John Holland who invented it in the early 1970's.

Classes of Search Techniques

F inonacci N ew ton

D irect m ethods Indirect m ethods

C alcu lus-based techn iques

Evolu tionary s trategies

C entra l ized D is tr ibuted

Para l le l

S teady-s ta te G enera tiona l

Sequentia l

G enetic a lgorithm s

Evolutionary a lgorithm s S im u lated annealing

G uided random search techniques

D ynam ic program m ing

Enum erative techn iques

Search techniques

Tabu Search Hill Climbing

DFS, BFS

Genetic Programming

Early History of EAs 1954: Barricelli creates computer simulation of life – Artificial Life 1957: Box develops Evolutionary Operation (EVOP), a non-computerised

evolutionary process 1957: Fraser develops first Genetic Algorithm 1958: Friedberg creates a learning machine through evolving computer

programs 1960s, Rechenverg: evolution strategies

a method used to optimize real-valued parameters for devices 1960s, Fogel, Owens, and Walsh: evolutionary programming

to find finite-state machines 1960s, John Holland: Genetic Algorithms

to study the phenomenon of adaptation as it occurs in nature (not to solve specific problems)

1965: Rechenberg & Schwefel independently develop Evolution Strategies 1966: L. Fogel develops Evolutionary Programming as a means of creating

artificial intelligence 1967: Holland and his students extend GA ideas further

The Genetic Algorithm Directed search algorithms based on the mechanics of

biological evolution Developed by John Holland, University of Michigan (1970’s)

To understand the adaptive processes of natural systems To design artificial systems software that retains the

robustness of natural systems

The genetic algorithms, first proposed by Holland (1975), seek to mimic some of the natural evolution and selection.

The first step of Holland’s genetic algorithm is to represent a legal solution of a problem by a string of genes known as a chromosome.

Evolutionary Programming First developed by Lawrence Fogel in 1966 for use in

pattern learning Early experiments dealt with a number of Finite State

Automata FSA were developed that could recognise recurring

patterns and even primeness of numbers Later experiments dealt with gaming problems

(coevolution) More recently has been applied to training of neural

networks, function optimisation & path planning problems

Biological Terminology• gene

• functional entity that codes for a specific feature e.g. eye color• set of possible alleles

• allele• value of a gene e.g. blue, green, brown• codes for a specific variation of the gene/feature

• locus• position of a gene on the chromosome

• genome• set of all genes that define a species• the genome of a specific individual is called genotype• the genome of a living organism is composed of several chromosomes

• population• set of competing genomes/individuals

Genotype versus Phenotype

• genotype• blue print that contains the information to construct an organism e.g. human DNA• genetic operators such as mutation and recombination modify the genotype during reproduction• genotype of an individual is immutable

(no Lamarckian evolution)

• phenotype• physical make-up of an organism• selection operates on phenotypes (Darwin’s principle : “survival of the fittest”)

Courtesy of U.S. Department of Energy Human Genome Program , http://www.ornl.gov/hgmis

Genotype Operators

• recombination (crossover)• combines two parent genotypes into a new offspring• generates new variants by mixing existing genetic material• stochastic selection among parent genes

• mutation• random alteration of genes• maintain genetic diversity

• in genetic algorithms crossover is the major operator whereas mutation only plays a minor role

Crossover

• crossover applied to parent strings with probability pc : [0.6..1.0]• crossover site chosen randomly

• one-point crossoverparent Aparent B

1 1 0 1 0

1 0 0 0 1

offspring Aoffspring B

1 1 0 1 1

1 0 0 0 0

• two-point crossoverparent Aparent B

1 1 0 1 0

1 0 0 0 1

offspring Aoffspring B

1 1 0 0

1 0 0 1

0

1

Mutation

• mutation applied to allele/gene with probability Pm : [0.001..0.1]• role of mutation is to maintain genetic diversity

offspring: 1 1 0 0 0

Mutate fourth allele (bit flip)

1 1 0 0 01mutated offspring:

recombination

10001

01011

10011

01001

mutation

x

f

phenotype space

Structure of an Evolutionary Algorithm

00111 11001

10001

01011

population of genotypes

coding scheme

fitness

selection

11001

10001

01011

10001

10111

01001

10

01

001

011

10

01 001

01110011

01001

Pseudo Code of an Evolutionary Alg.

Create initial random population

Evaluate fitness of each individual

Termination criteria satisfied ?

Select parents according to fitness

Recombine parents to generate offspring

Mutate offspring

Replace population by new offspring

stopyes

no

Roulette Wheel Selection

• probability of reproduction pi = fi / Sk fk

10001

11010

01011

00101

10001

11010

01011

0010110001

11010

01011

00101

10001

11010

01011

00101

10001 11010

01011

00101

• selected parents : 01011, 11010, 10001, 10001

• selection is a stochastic process

• automatic generation of computer programs by means of natural evolution see Koza 1999• programs are represented by a parse tree (LISP expression)• tree nodes correspond to functions : - arithmetic functions {+,-,*,/} - logarithmic functions {sin,exp}• leaf nodes correspond to terminals : - input variables {X1, X2, X3} - constants {0.1, 0.2, 0.5 }

Genetic Programming

+

*

X3X2

X1

tree is parsed from left to right: (+ X1 (* X2 X3)) X1+(X2*X3)

Genetic Programming : Crossover

+

*

X3X2

X1

-

/

X1-

X2

X3X2

parent A parent B

+ *

X3X2X1

-

/

X1-X2

X3X2

offspring A offspring B

Areas EAs Have Been Used In

Design of electronic circuitsDesign of electronic circuits

Telecommunication network Telecommunication network designdesign

Artificial intelligenceArtificial intelligence

Study of atomic clustersStudy of atomic clusters

Study of neuronal behaviourStudy of neuronal behaviour

Neural network training & designNeural network training & design

Automatic controlAutomatic control

Artificial lifeArtificial life

SchedulingScheduling

Travelling Salesman ProblemTravelling Salesman ProblemGeneral function optimisation General function optimisation Bin Packing ProblemBin Packing ProblemPattern learningPattern learningGamingGamingSelf-adapting computer programsSelf-adapting computer programsClassificationClassificationTest-data generationTest-data generationMedical image analysisMedical image analysisStudy of earthquakesStudy of earthquakes

Goldberg (1989)

Goldberg D. E. (1989), Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading.

Michalewicz (1996)

Michalewicz, Z. (1996), Genetic Algorithms + Data Structures = Evolution Programs, Springer.

Vose (1999)

Vose M. D. (1999), The Vose M. D. (1999), The Simple Genetic Simple Genetic Algorithm : Algorithm : Foundations and Foundations and Theory (Complex Theory (Complex Adaptive Systems). Adaptive Systems). Bradford Books; Bradford Books;

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SOFT COMPUTINGFuzzy-Evolutionary Computing

Genetic Fuzzy Systems (GFS’s)

• genetic design of fuzzy systems

• automated tuning of the fuzzy knowledge base

• automated learning of the fuzzy knowledge base

• objective of tuning/learning process

• optimizing the performance of the fuzzy system:

e.g.: fuzzy modeling : minimizing quadratic error

between data set and the fuzzy system outputs

e.g : fuzzy control system: optimize the

behavior of the plant + fuzzy controller

Genetic Fuzzy System for Data Modeling

Evolutionaryalgorithm

Evolutionaryalgorithm

Evaluationscheme

Evaluationscheme

Dataset : xi,yiDataset : xi,yiFuzzy SystemFuzzy System

Fuzzy system parameters

fitness

genotype

phenotype

Fuzzy SystemsKnowledge Base

Database :Definition of

fuzzy membership-function

a b c

Rule base:definition offuzzy rules

If X1 is A1 and … and Xn is An

then Y is B

Genetic Tuning Process

• tuning problems utilize an already existing rule base

• tuning aims to find a set of optimal parameters for

the database :

• points of membership-functions [a,b,c,d]

or

• scaling factors for input and output variables

Linear Scaling FunctionsChromosome for linear scaling:• for each input xi : two parameters ai,bi i=1..n• for the output y : two parameter a0,b0

Genetic Algorithms: • encode each parameter by k bit using Gray code total length = 2*(n+1)*k bit

Evolutionary Strategies:• each parameter ai or bi corresponds to one object variable xm m : 1… 2*(n+1)

a0

100101b0

011111a1

110101b2*(n+1)

100101. . .

a0x0,o

b0x1,1

a1x2,2

b2*(n+1) xm,m. . .

Descriptive Knowledge Base

• descriptive knowledge base

sm me lg

x

neg ze pos

y

• all rules share the same global membership functions :

R1 : if X is sm then Y is neg R2 : if X is me then Y is ze R3 : if X is lg then Y is pos

Approximate Knowledge Base

• each rule employs its own local membership function

R1 : if X is then Y is R1 : if X is then Y is

R1 : if X is then Y is

R1 : if X is then Y is

• tradeoff: more degrees of freedom and therefore better approximation but intuitive meaning of fuzzy sets gets lost

Tuning Membership Functions• encode each fuzzy set by characteristic parameters

x

(x)1

0 a b c d

Trapezoid: <a,b,c,d>

x

(x)1

0

Gaussian: N(m,s)

m

s

xx

(x)1

0 a b c

Triangular: <a,b,c>

Approximate Genetic Tuning Process

• a chromosome encodes the entire knowledge base, database and rulebaseRi : if x1 is Ai1 and … xn is Ain then y is Bi

encoded by the i-th segment Ci of the chromosomeusing triangular membership-functions (a,b,c)

(ai1, bi1, ci1, . . . , ain, bin, cin, ai, bi, ci, )Ci =

The chromosome is the concatenation of the individual segments corresponding to rules :C1 C2 C3 C4 . . . Ck

each parameter may be binary or real-coded

Descriptive Genetic Tuning Process• the rule base already exists• assume the i-th variable is composed of N i terms

(ai1, bi1, ci1, . . . , aiNi, biNi, ciNi )Ci =

A1 A2 A3

xiai1, bi1, ci1, ai2, bi2, ci2 ai3, bi3, ci3

The chromosome is the concatenation of the individual segments corresponding to variables :

C1 C2 C3 C4 . . . Ck

Descriptive Genetic Tuning• in the previous coding scheme fuzzy sets might change their order and optimization is subject to the constraints : aij < bij < cij

A1 A2 A3

x1 x2 x3

• encode the distance among the center points of triangular fuzzy sets and choose the border points such that i = 1

Fitness Function for Tuning

• minimize quadratic error among training data (xi,yi) and fuzzy system output f(xi) E = Sumi (yi-f(xi))2

Fitness = 1 / E (maximize fitness)

• minimize maximal error among training data (xi,yi) and fuzzy system output f(xi) E = maxi (yi-f(xi))2

Fitness = 1 / E (maximize fitness)

Genetic Learning Systems• genetic learning aim to :

• learn the fuzzy rule base or• learn the entire knowledge base

• three different approaches• Michigan approach : each chromosome represents a single rule• Pittsburgh approach : each chromosome represents an entire rule base / knowledge base• Iterative rule learning : each chromosome represents a single rule, but rules are injected one after the other into the knowledge base

Michigan Approach

11001 : R1: if x is A1 ….then Y is B100101 : R2: if x is A2 ….then Y is B210111 : R3: if x is A3 ….then Y is B311100 : R4: if x is A4 ….then Y is B401000 : R5: if x is A5 ….then Y is B511101 : R6: if x is A6 ….then Y is B6

Population:

Individual:

A1A3A4

A2A5A6

XB1

B3

B4

B2

B5

B6

Y

R4 : if x is small then Y is pos.F=-1.6

R1 : if x is large then Y is neg.F = 2.5

R3 : if x is small then Y is zeroF=-0.4

R2 : if x is med. then Y is zeroF=2.7

Cooperation vs. Competition Problem

• we need a fitness function that measures the accuracy of an individual rule as well as the quality of its cooperation with other rules

X

Y

small medium large

pos

zene

g

Fitness = number of correct classifications minus number of incorrect classifications

Michigan Approach• steady state selection:

• pick one individual at random• compare it with all individuals that cover the same input region• remove the “relatively” worst one from the population • pick two parents at random independent of their fitness and generate a new offspring

11001 : R1: if x is A1 ….then Y is B1









competitors:

removed from the population

4545

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That’s all.That’s all.

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1 SOFT COMPUTING Evolutionary Computing. What is a GA? GAs are adaptive heuristic search algorithm based on the evolutionary ideas of natural selection