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Chapter 14 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

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Chapter 14 Genetic Algorithms

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  1. Chapter 14 Genetic Algorithms

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

  3. 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

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

  5. 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.

  6. Fitness • Fitness is an important concept in genetic algorithms. • 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)

  7. 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.

  8. 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 points: • Uniform crossover involves using a probability to select which genes to use from chromosome 1, and which from chromosome 2.

  9. Mutation • A unary operator – applies to one chromosome. • Randomly selects some bits (genes) to be “flipped” • 1 => 0 and 0 =>1 • Mutation is usually applied with a low probability, such as 1 in 1000.

  10. Termination Criteria • A genetic algorithm is run over a number of generations until the termination criteria are reached. • 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.

  11. 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) = 1). • 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).

  12. Optimizing a mathematical function 2 • Randomly create 4 chromosomes Chrom Gene val f(x) f’(x) fitness ratio • C1 = 1001 = 9 = .41 = 70 = 46.3% • C2 = 0011 = 3 = .14 = 57 = 37.7% • C3 = 1010 =10 = -.54 =22 = 14.9% • C4 = 0101 = 5 = -.96 =2 = 1.34% • F’(x) = 50(sin(x) + 1) fitness value

  13. Optimizing a mathematical function 3 • Generate random numbers to determine chromosomes to mate • 0 to 46 c1, 46 to 83 c2, … • Random num 1 = 56, c2 is chosen • Random num 2 = 38, c1 is chosen • Combine c1 and c2, randomly select a crossover, bt 2nd and 3rd genes • C5 = 1011, C6 = 0001

  14. Optimizing a mathematical function 3 • Similarly C1 and C3 produce C7 and C8 • C1 to C4 are replaced by C5 to C8 Chrom Gene val f(x) f’(x) fitness ratio C5 1011 11 -1 0 0% C6 0001 1 .84 92.07 48.1% C7 1000 8 .99 99.47 51.9% C8 1011 11 -1 0 0%

  15. Comments • Typical populations 100 to 500 chrom • Each chrom may have 100’s of genes • Typically provide near optimal solutions to combinatorial problems • Are a local search method

  16. Schemata (1) • As with the rules used in classifier systems, a schema is a string consisting of 1’s, 0’s and *’s. E.g.: 1011*001*0 Matches the following four strings: 1011000100 1011000110 1011100100 1011100110 • a schema with n *’s will match a total of 2n chromosomes. • Each chromosome of r bits will match 2r different schemata. • 101 matches 101,10*,1*0,*01,**1,*0*,1**,***

  17. Schemata (2) • The defining length dL(S) of a schema, S, isthe distance between the first and last defined bits. For example, the defining length of each of the following schemata is 4: **10111* 1*0*1** • The order O(S)is number of defined bits in S. The following schemata both have order 4: **10*11* 1*0*1**1 • A schema with a high order is more specific than one with a lower order.

  18. 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)

  19. 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.

  20. 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.

  21. 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.

  22. 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:

  23. The Schema Theorem • Holland’s 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 work. • It does not provide a complete answer.

  24. 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 chromosomes. • This tells us that it is important to choose a correct representation.

  25. Deception • 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.

  26. 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 overspecified: ((1,0), (2,1), (3,1), (3,0), (4,0))

  27. 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 generation. • Overspecified bits are usually dealt with by working from left to right and using the first value specified for each bit.

  28. Messy Genetic Algorithms (3) • MGAs use splice and cut instead of crossover. • Splicing involves simply joining two chromosomes together: ((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 two: ((1,0), (3,0), (4,1)) ((6,1), (2,1), (3,1), (5,0), (7,0), (8,0))

  29. 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.

  30. Co-Evolution • 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.

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