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Genetic Algorithms


Chapter 14 Genetic Algorithms Chapter 14 Contents (1) Representation The Algorithm Fitness Crossover Mutation Termination Criteria Optimizing a mathematical function ... – PowerPoint PPT presentation

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Title: Genetic Algorithms

  • Chapter 14
  • Genetic Algorithms

Chapter 14 Contents (1)
  • Representation
  • The Algorithm
  • Fitness
  • Crossover
  • Mutation
  • Termination Criteria
  • Optimizing a mathematical function

Chapter 14 Contents (2)
  • Schemata
  • The Effect of Reproduction on Schemata
  • The Effect of Crossover and Mutation
  • The Schema Theorem
  • The Building Block Hypothesis
  • Deception
  • Messy Genetic Algorithms
  • Evolving Pictures
  • Co-Evolution

  • Genetic techniques can be applied with a range of
  • Usually, we have a population of chromosomes
  • Each chromosome consists of a number of genes.
  • Other representations are equally valid.

The Algorithm
  • The algorithm used is as follows
  • Generate a random population of chromosomes (the
    first generation).
  • If termination criteria are satisfied, stop.
    Otherwise, continue with step 3.
  • Determine the fitness of each chromosome.
  • Apply crossover and mutation to selected
    chromosomes from the current generation to
    generate a new population of chromosomes (the
    next generation).
  • Return to step 2.

  • Fitness is an important concept in genetic
  • The fitness of a chromosome determines how likely
    it is that it will reproduce.
  • Fitness is usually measured in terms of how well
    the chromosome solves some goal problem.
  • E.g., if the genetic algorithm is to be used to
    sort numbers, then the fitness of a chromosome
    will be determined by how close to a correct
    sorting it produces.
  • Fitness can also be subjective (aesthetic)

Crossover (1)
  • Crossover is applied as follows
  • Select a random crossover point.
  • Break each chromosome into two parts, splitting
    at the crossover point.
  • Recombine the broken chromosomes by combining the
    front of one with the back of the other, and vice
    versa, to produce two new chromosomes.

Crossover (2)
  • Usually, crossover is applied with one crossover
    point, but can be applied with more, such as in
    the following case which has two crossover
  • Uniform crossover involves using a probability to
    select which genes to use from chromosome 1, and
    which from chromosome 2.

  • A unary operator applies to one chromosome.
  • Randomly selects some bits (genes) to be
  • 1 gt 0 and 0 gt1
  • Mutation is usually applied with a low
    probability, such as 1 in 1000.

Termination Criteria
  • A genetic algorithm is run over a number of
    generations until the termination criteria are
  • Typical termination criteria are
  • Stop after a fixed number of generations.
  • Stop when a chromosome reaches a specified
    fitness level.
  • Stop when a chromosome succeeds in solving the
    problem, within a specified tolerance.
  • Human judgement can also be used in some more
    subjective cases.

Optimizing a mathematical function
  • A genetic algorithm can be used to find the
    highest value for f(x) sin (x).
  • Each chromosome consists of 4 bits, to represent
    the values of x from 0 to 15.
  • Fitness ranges from 0 (f(x) -1) to 100 (f(x)
  • By applying the genetic algorithm it takes just a
    few generations to find that the value x 8 gives
    the optimal solution for f(x).

Schemata (1)
  • As with the rules used in classifier systems, a
    schema is a string consisting of 1s, 0s and
    s. E.g.
  • 10110010
  • Matches the following four strings
  • 1011000100
  • 1011000110
  • 1011100100
  • 1011100110
  • a schema with n s will match a total of 2n
  • Each chromosome of r bits will match 2r different

Schemata (2)
  • The defining length dL(S) of a schema, S, is
    the distance between the first and last defined
    bits. For example, the defining length of each of
    the following schemata is 4
  • 10111
  • 101
  • The order O(S) is number of defined bits in S.
    The following schemata both have order 4
  • 1011
  • 1011
  • A schema with a high order is more specific than
    one with a lower order.

Schemata (3)
  • The fitness of a schema is defined as the average
    fitness of the chromosomes that match the schema.
  • The fitness of a schema, S, in generation i is
    written as follows
  • f(S, i)
  • The number of occurrences of S in the population
    at time i is
  • m(S, i)

The Effect of Reproduction on Schemata
  • The probability that a chromosome c will
    reproduce is proportional to its fitness, so the
    expected number of offspring of c is
  • a(i) is the average fitness of the chromosomes in
    the population at time i
  • If c matches schema S, we can rewrite as
  • c1 to cn are the chromosomes in the population at
    time i which match the schema S.

The Effect of Reproduction on Schemata
  • Since
  • We can now write
  • This tells us that a schema that is fit will have
    more chance of appearing in a subsequent
    generation than less fit chromosomes.

The Effect of Crossover
  • For a schema S to survive crossover, the
    crossover point must be outside the defining
    length of S.
  • Hence, the probability that S will survive
    crossover is
  • This tells us that a short schema is more likely
    to survive crossover than a longer schema.
  • In fact, crossover is not always applied, so the
    probability that crossover will be applied should
    also be taken into account.

The Effect of Mutation
  • The probability that mutation will be applied to
    a bit in a chromosome is pm
  • Hence, the probability that a schema S will
    survive mutation is
  • We can combine this with the effects of crossover
    and reproduction to give

The Schema Theorem
  • Hollands Schema Theorem, represented by the
    above formula, can be written as
  • Short, low order schemata which are fitter than
    the average fitness of the population will appear
    with exponentially increasing regularity in
    subsequent generations.
  • This helps to explain why genetic algorithms
  • It does not provide a complete answer.

The Building Block Hypothesis
  • Genetic algorithms manipulate short, low-order,
    high fitness schemata in order to find optimal
    solutions to problems.
  • These short, low-order, high fitness schemata are
    known as building blocks.
  • Hence genetic algorithms work well when small
    groups of genes represent useful features in the
  • This tells us that it is important to choose a
    correct representation.

  • Genetic algorithms can be deceived by fit
    building blocks that happen not to combine to
    give the optimal solutions.
  • Deception can be avoided by inversion this
    involves reversing the order of a randomly
    selected group of bits within a chromosome.

Messy Genetic Algorithms (1)
  • An alternative to standard genetic algorithms
    that avoid deception.
  • Each bit in the chromosome is represented as a
    (position, value) pair. For example
  • ((1,0), (2,1), (4,0))
  • In this case, the third bit is undefined, which
    is allowed with MGAs. A bit can also
  • ((1,0), (2,1), (3,1), (3,0), (4,0))

Messy Genetic Algorithms (2)
  • Underspecified bits are filled with bits taken
    from a template chromosome.
  • The template chromosome is usually the best
    performing chromosome from the previous
  • Overspecified bits are usually dealt with by
    working from left to right and using the first
    value specified for each bit.

Messy Genetic Algorithms (3)
  • MGAs use splice and cut instead of crossover.
  • Splicing involves simply joining two chromosomes
  • ((1,0), (3,0), (4,1), (6,1))
  • ((2,1), (3,1), (5,0), (7,0), (8,0))
  • ((1,0), (3,0), (4,1), (6,1), (2,1), (3,1),
    (5,0), (7,0), (8,0))
  • Cutting involves splitting a chromosome into
  • ((1,0), (3,0), (4,1))
  • ((6,1), (2,1), (3,1), (5,0), (7,0), (8,0))

Evolving Pictures
  • Dawkins used genetic algorithms with subjectives
    metrics for fitness to evolve pictures of
    insects, trees and other creatures.
  • The human selection of fitness can be used to
    produce amazing pictures that a person would
    otherwise not be able to produce.

  • In the real world, the presence of predators is
    responsible for many evolutionary developments.
  • Similarly, in many artificial life systems,
    introducing predators produces better results.
  • This process is known as co-evolution.
  • For example, Ramps, which were evolved to sort
    numbers parasites were introduced which
    produced sets of numbers that were harder to
    sort, and the ramps produced better results.