1. 13 Basics of Genetic Algorithms and some possibilities Peter Spijker Technische Universiteit Eindhoven Department of Biomedical Engineering Division of Biomedical Imaging and Modeling California Institute of Technology Materials Process and Simulation Center Biochemistry & Molecular Biophysics November 25, 2003 12

2. 13 Presentation Overview Purpose of presentation General introduction to Genetic Algorithms (GA’s) Biological background Origin of species Natural selection Genetic Algorithm Search space Basic algorithm Coding Methods Examples Possibilities

3. 13 Purpose of presentation Optimising parameters of force fields is a difficult and time consuming task Use of optimising methods might be of use Methods: - steepest descent - simulated annealing (Monte Carlo) - genetic algorithms Brief introduction to genetic algorithms in lecture style

4. 13 General Introduction to GA’s Genetic algorithms (GA’s) are a technique to solve problems which need optimization GA’s are a subclass of Evolutionary Computing GA’s are based on Darwin’s theory of evolution History of GA’s Evolutionary computing evolved in the 1960’s. GA’s were created by John Holland in the mid-70’s.

5. 13 Biological Background (1) – The cell Every animal cell is a complex of many small “factories” working together The center of this all is the cell nucleus The nucleus contains the genetic information

6. 13 Biological Background (2) – Chromosomes Genetic information is stored in the chromosomes Each chromosome is build of DNA Chromosomes in humans form pairs There are 23 pairs The chromosome is divided in parts: genes Genes code for properties The posibilities of the genesfor one property is called: allele Every gene has an unique position on the chromosome: locus

7. 13 Biological Background (3) – Genetics The entire combination of genes is called genotype A genotype develops to a phenotype Alleles can be either dominant or recessive Dominant alleles will always express from the genotype to the fenotype Recessive alleles can survive in the population for many generations, without being expressed.

8. 13 Biological Background (4) – Reproduction Reproduction of genetical information Mitosis Meiosis Mitosis is copying the same genetic information to new offspring: there is no exchange of information Mitosis is the normal way of growing of multicell structures, like organs.

9. 13 Biological Background (5) – Reproduction Meiosis is the basis of sexual reproduction After meiotic division 2 gametes appear in the process In reproduction two gametes conjugate to a zygote wich will become the new individual Hence genetic information is shared between the parents in order to create new offspring

10. 13 Biological Background (6) – Reproduction During reproduction “errors” occur Due to these “errors” genetic variation exists Most important “errors” are: Recombination (cross-over) Mutation

11. 13 Biological Background (7) – Natural selection The origin of species: “Preservation of favourable variations and rejection of unfavourable variations.” There are more individuals born than can survive, so there is a continuous struggle for life. Individuals with an advantage have a greater chance for survive: survival of the fittest.

12. 13 Biological Background (8) – Natural selection Important aspects in natural selection are: adaptation to the environment isolation of populations in different groups which cannot mutually mate If small changes in the genotypes of individuals are expressed easily, especially in small populations, we speak of genetic drift Mathematical expresses as fitness: success in life

13. 13 Presentation Overview Purpose of presentation General introduction to Genetic Algorithms (GA’s) Biological background Origin of species Natural selection Genetic Algorithm Search space Basic algorithm Coding Methods Examples Possibilities

14. 13 Genetic Algorithm (1) – Search space Most often one is looking for the best solution in a specific subset of solutions This subset is called the search space (or state space) Every point in the search space is a possible solution Therefore every point has a fitness value, depending on the problem definition GA’s are used to search the search space for the best solution, e.g. a minimum Difficulties are the local minima and the starting point of the search

15. 13 Genetic Algorithm (2) – Basic algorithm Starting with a subset of n randomly chosen solutions from the search space (i.e. chromosomes). This is the population This population is used to produce a next generation of individuals by reproduction Individuals with a higher fitness have more chance to reproduce (i.e. natural selection)

16. 13 Genetic Algorithm (3) – Basic algorithm Outline of the basic algorithm 0 START : Create random population of n chromosomes 1 FITNESS : Evaluate fitness f(x) of each chromosome in the population 2 NEW POPULATION 0 SELECTION : Based on f(x) 1 RECOMBINATION : Cross-over chromosomes 2 MUTATION : Mutate chromosomes 3 ACCEPTATION : Reject or accept new one 3 REPLACE : Replace old with new population: the new generation 4 TEST : Test problem criterium 5 LOOP : Continue step 1 – 4 until criterium is satisfied

17. 13 Genetic Algorithm (4) – Coding Normal cells are diploid (containing 2 complete sets of chromosomes) On the contrary gametes are haploid Formalizing diploid reproduction is much more difficult than haploid Diploid populations have an extra dimension compared to haploid populations For simplicity therefore only haploid genetic algorithms

18. 13 Genetic Algorithm (5) – Coding Chromosomes are encoded by bitstrings Every bitstring therefore is a solution but not necisseraly the best solution The way bitstrings can code differs from problem to problem Either: sequence of on/off or the number 9 1 0 0 1

19. 13 Genetic Algorithm (6) – Coding Recombination (cross-over) can when using bitstrings schematically be represented: Using a specific cross-over point 1 0 0 1 1 0 1 0 1 0 1 1 1 0 X 1 0 0 1 1 1 0 0 1 0 1 1 0 1

20. 13 Genetic Algorithm (7) – Coding Mutation prevents the algorithm to be trapped in a local minimum In the bitstring approach mutation is simpy the flipping of one of the bits 1 0 0 1 1 0 1 1 1 0 1 1 0 1

21. 13 Genetic Algorithm (8) – Coding Both recombination and mutation depend a lot on the exact definition of the problem and the choice of representing the chromosomes (e.g. no bitstrings) Different encodings can be used: Binary encoding Permutation encoding Value encoding Tree encoding Focus in this presentation stays with binary encoding

22. 13 Example Minimum of Function (1) First example shows how to find the minimum of a function Minimum f(x) at x = 809 1100101001

23. 13 Example Minimum of Function (2) IndividualBest individual Mean fitness Best fitness Generations

24. 13 Example Minimum of Function (3) Interactive show of this algorithm with Matlab Using the function: genalg2() Variables: Population size Bitstringlength Mutation chance Recombination chance Starting population adaption

25. 13 Genetic Algorithm (9) – Remarks It is clear from the example that the convergence speed of the algorithm depends on many factors: Population size Mutation probability Recombination probability Elitism Selection methods Random selection of parents Roulette wheel selection of parents Strong point GA’s: mutation prevents from falling in a local minimum, recombination initiates a fast first convergence

26. 13 Example Checkboard (1) We are given an n by n checkboard in which every field can have a different colour from a set of four colours. Goal is to achieve a checkboard in a way that there are no neighbours with the same colour (not diagonal)

27. 13 Example Checkboard (2) Chromosomes represent the way the checkboard is coloured. Chromosomes are not represented by bitstrings but by bitmatrices The bits in the bitmatrix can have one of the four values 0, 1, 2 or 3, depending on the colour Crossing-over involves matrix manipulation instead of point wise operating. Crossing-over can be combining the parential matrices in a horizontal, vertical, triangular or square way Mutation remains bitwise changing bits in either one of the other numbers

28. 13 Example Checkboard (3) Fitnesscurve for the checkboard example This problem can be seen as a graph with n nodes and (n-1) edges, so the fitness f(x) is easily defined as: f(x) = 2 · (n-1) ·n

29. 13 Example Checkboard (4) Fitnesscurves for different cross-over rules

30. 13 Example Checkboard (5) Interactive show of this algorithm with Matlab Using the functions: main() checkers() bestindividual() mutate() recombine() select() showbestindividual()

31. 13 Possibilities Using the genetic algorithm to optimise parameters for a force field Parameters are real numbers, so adaptations of these algorithms is required Value incoding vs. bitstring encoding Difficulties: Definition fitness function Integration algorithm with software

32. 13 Further Questions ?