From Biological Systems to Computer Sciences and Back fileGenetic algorithms Genetics Genetic algorithm Theory of natural selection Searching Optimization Machine learning Population

From Biological Systems to

Computer Sciences and Back

Pedro Pablo González

e-mail: [email protected]

ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA

DEIS, SEDE DI CESENA, ITALIA

Course and Lab

Introduction to Genetic Algorithms

Pedro Pablo González

Generation t

Generation t+1

Selection operator

Crossover operator

Crossover operator

Mutation operator

+

1001101111001100

1001110011001001

Genetic algorithms

Genetics

Genetic algorithm

Theory of natural selection

Searching

Optimization

Machine learning

Population of chromosomes

Fitness function

Genetic operators

task

task

task

Genetic algorithms: an overview

• Genetics algorithms (GAs) are adaptive heuristic

search algorithms inspired by evolutionary ideas of

natural selection and genetic

• GAs were proposed and developed by John

Holland and its colleagues at the University of

Michigan in the 1960s and the 1970s.

• Initial Holland’s aims were to improve the

understanding of natural adaptation process, and

to design artificial systems having properties

similar to natural systems.

Genetic algorithms: an overview

• A function to optimize

• A population of chromosomes (bit strings)

• Selection method according to fitness

• Crossover to generate new chromosomes

• Random mutation to alter the value of bits of

the new chromosomes

A genetic algorithm is characterized by the following

elements:

Genetic algorithms: the method

• Work on the encoding of the function

parameters, rarely work on the parameters

themselves

• Work on a search space (a population of

points)

• Use the only values of the function to

optimize

• Use probabilistic transition rules

Genetic algorithms for solving

computational problems

• Searching in a search space

• Optimization

• Machine learning

• Genetic programming

Genetic terminology

Values of fitness function variables

itself

Phenotype

Bit string encoding of the values of

fitness function variables

Genotype

Bit position on the string Locus

Bit valueAllele

Bit or characteristic Gene

Bit stringChromosome

Genetic algorithmGenetic terminology

Elements of genetic algorithms

• A population of strings (chromosomes)

• A fitness function

• A set of genetic operators: selection,

crossover and mutation

The simplest form of genetic algorithm involves three

types of elements:


To use a genetic algorithm, the potential solutions to the

problem must be encoded in string structures, known as

chromosomes. Each bit (locus) in the chromosome can take

two possible values: 0 and 1. Each chromosome represents a

complete solution to the problem to optimize, and it can be

thought of as a point in the search space of potential solutions.

Population of strings (chromosomes)

Encoded

Potential

solution

128Chromosome

1 0 0 0 0 0 0 0

gene o bit


Example of a population of chromosomes

01110

01001

11100

01101

11101

00101

11011

10011

Chromosome

14

9

28

13

29

5

27

19

Value

8

7

6

5

4

3

2

1

Element

number

• X takes integer

values into the

interval [0,31]

• Chromosome

length is equal to

5


A fitness function assigns a score or fitness to each

chromosome in the current population. The fitness function

determines how acceptable each chromosome is for the

solution to the problem at hand.

Fitness function

Examples of fitness function

( )[ ] ( ) ( )[ ] 2

1

2

1

∑∑==

−==K

k

K

k

kykdkeE

),( XWd

( )Ae

Af−+

=1

1

• Weighted function

• Distance measure

• Sigmoid function

• Sum of the square

error

j

N

j

j XWnet ∑=

=1

A first example

Consider the following net function defined for real numbers,

X, W ∈ R:

A = -W1*X1 + W2*X2 - W3*X3 +W4*X4 + W5*X5 - 70

where:

W = (W1, W2, W3, W4, W5) is the weight vector, and each Wi

takes integer values into the interval [0,31]

X = (X1, X2, X3, X4, X5) is the input pattern and takes the fixed

values (2, 4, 1, 3, 2)

We would like to maximize the function A using the

genetic algorithm approach

A first example

First, we need to encode decision variables (weight vectors)

into binary strings. Here 25 bits are used to represent a

variable W = (W1, W2, W3, W4, W5). In the chromosome

population each chromosome is composed by five genes, and

each gene is represented by a binary string of five bits. The

mapping from a chromosome string to a real numbers for a

variable W is completed as follows:

Wchromosome = W1gene + W2gene + W3gene + W4gene +

W5gene

and

Wigene Widecodes

(binary) (integer)

A first example

The chromosome population and the fitness values of the chromosomes for

the net function:

A = -W1*X1 + W2*X2 - W3*X3 +W4*X4 + W5*X5 - 70

are shown in the table below

3414 19 13 15 1201110100110110101111011008

809 17 15 23 2301001100010111110111101117

-5928 13 24 7 2211100011011100000111101106

-3113 6 21 11 601101001101010101011001105

3329 23 4 25 2211101101110010011001101104

455 13 13 20 1300101011010110110100011013

-2227 11 21 15 1711011010111010101111100012

1619 14 7 11 2110011011100011101011101011

FitnessValoriStringaNo.


Genetic operators

• Selection

• Crossover

• Mutation


Selection operator

The selection of strings in the current population is done

considering theirs fitness function values. The selection

operator randomly selects the better chromosomes in the

current population, allowing them to pass on their genetic

material to the new population.

Selection operator

Generation t

f = 200

f = 240

f = 180

f = 160

f = 80

f = 60

f = 100

f = 70 f = 200

f = 200

f = 240

f = 240

f = 240

f = 180

f = 100

f = 160


Selection operator

• Roulette selection. A random selection method where the

selection probability of an individual is proportional to its

fitness.

• Tournament selection. The roulette selection is used N times

to produce a tournament subset of chromosome. The selected

chromosome will be the best chromosome in the subset.

• Top percent selection. This operator randomly selects a

chromosome from the top N percent of the population. The

value N is specified by the user.

Selection operator can be implemented in different ways:

A first example

The chromosome population and the fitness values of the chromosomes for the net

function: A = -W1*X1 + W2*X2 - W3*X3 +W4*X4 + W5*X5 - 70 are shown in the table

below

≈≈≈≈ 100%736Total

15.48

21.73

2.85

6.65

15.35

16.98

7.88

13.04

AFitness/

FitnessT

114

160

21

49

113

125

58

96

AFitness

3414 19 13 15 1201110100110110101111011008

809 17 15 23 2301001100010111110111101117

-5928 13 24 7 2211100011011100000111101106

-3113 6 21 11 601101001101010101011001105

3329 23 4 25 2211101101110010011001101104

455 13 13 20 1300101011010110110100011013

-2227 11 21 15 1711011010111010101111100012

1619 14 7 11 2110011011100011101011101011

FitnessValoriStringaNo.

A first example

Eight chromosomes are selected from the original population

using the roulette selection method:

58-2211011010111010101111100012

1133311101101110010011001101104

1608001001100010111110111101117

1608001001100010111110111101117

12545 00101011010110110100011013

114

58

96

AFitness

3401110100110110101111011008

-2211011010111010101111100012

1610011011100011101011101011

FitnessStringaNo.


Crossover operator

The crossover operator commonly combines two

chromosomes (parents) to produce two new chromosomes

(offsprings). This operator randomly selects a crossover point

in the chromosome (position in the string) and exchanges the

subsequences before and after that point between two

chromosomes to create the two offsprings.

The crossover operator can choose one or two crossover

points for determining the subsequence to exchange.

Crossover occurs according to a crossover probability, which

is defined by user.


Crossover operator

f = 200

f = 200

f = 240

f = 240

f = 240

f = 180

f = 100

f = 160

1

2

3

4

5

6

7

8

6

2 2

6 6’

2’

Crossover length: l = 8

Crossover site: k = 5

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

A first example

1608001001100010111110111101117

1133311101101110010011001101104

AFitnessFitnessSelected chromosomeNo.

With crossover probability pc, crossover the pair of selected strings at a

randomly crossover point to form two offsprings. The crossover used here

is one-cut-point method, which randomly selects one cut-point and

exchanges the right parts of two parents to generate offspring.

Crossover point = 10

4: 1110110111001001100110110

7: 0100110001011111011110111

01001100010010011001101107’

11101101110111110111101114’

AFitnessFitnessOffspringsNo.


Mutation operator

The mutation operator alters one or more gene values in the

chromosome (character in the string). This operator can be

applied at each gene in the chromosome with some

probability, usually very small. The application of the mutation

operator can result in entirely new chromosome being added

to the new population. This fact could help to genetic

algorithm to arrive at better solution than was previously

possible.

Functioning of a genetic algorithm

Crossover operator (step 1)

Crossover operator (step 2)

Selection operator

Generation t

Generation t+1

Mutation operator

+

1001101111001100

1001110011001001


The genetic algorithm creates a population of chromosomes and then

applies the selection, crossover and mutation operators to the strings in the

current population to generate a new population. A simple GA works as

shown in figure above and its algorithm is as follows:

1. Encoding of the problem in a binary string

2. Random generation of a population of n strings of

length n (where each string or chromosome

represents a candidate solution to the problem)

3. Calculate the value of fitness function f(x) for each

string x in the population

4. Repeat the following steps until n offsprings have

been created:


� Selection of the pair of parents strings from the current

population according to their share in the population

global fitness

� With crossover probability pc, crossover the pair of

selected strings at a randomly crossover point to form

two offsprings

� With mutation probability pm, mutate the two offspring

� Copy the two news strings in the population

5. Replace the old population with the new population.

6. Go to 3.

Other genetic operators and techniques

Operator/techniqueGA phase

Two-point crossover

Parameterized uniform crossover

Crossover

Elitism

Steady-state selection

Selection

Inversion Encoding

The premature convergence problem

The scaling problem

In order to reduce the premature convergence problem, it’s

possible to transform the fitness values. Here we present

three main methods:

• Exponential. This method consists in taking the squared

roots of the fitness plus one. In this way, the influence of

the strongest subjects is reduced.

• Linear transformation. This method applies a linear

transformation to each fitness (i.e. f’ = a*f + b).

• Windowing. This method reduces the fitness of each

individual by the fitness of the worse individual in the

population.

Parameter setting for a genetic algorithm

• Population size

• Crossover rate

• Mutation rate

GA schemas

• Schema. Genetic algorithms work in the space of

schemata. A schema is a set of bit strings that can be

described by a template made up of 1, 0 and *, the *

representing wild cards. A schema includes all the

strings with certain common string elements.

• Schema order. The order of a schema o(H) is the

number of fixed positions (1 or 0).

• Schema length. Schema length δ(H) is defined as the

distance between the first fixed position and the last

fixed position of the schema.

Genetic algorithm applications

• Evolving neural networks

• Evolving cellular automata

• Protein structure prediction

• Scheduling applications

• Automated design

• Learning robot behavior

• Electronic circuit design

• Game theory

• Image restoration

• Configuration applications

Genetic Algorithm tools

http://www.aridolan.com/ga/gaa/

gaa.html

It is a general purpose

genetic algorithm toolkit

where the user can

define and run his own

optimization problems

Genetic

Algorithms Toolkit

http://www.rennard.org/alife/engli

sh/gavgb.html

Demonstration applet of

the functioning of a

Genetic Algorithm

Genetic algorithm

viewer

httpDescriptionTool

From Biological Systems to Computer Sciences and Back fileGenetic algorithms Genetics Genetic algorithm Theory of natural selection Searching Optimization Machine learning Population

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