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Genetic Algorithms Lecture 5 Addendum to the slides of Jason Noble

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Evolutionary Algorithms NNII NI NI NINI NN NI II

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Evolutionary Algorithms Binary Genetic Algorithms Variable Encoding and Decoding Generating a Population Generating Offspring: Crossover Mutation Selection Real chromosomes diploid GA almost always (haploid) exception will be explicitly mentioned

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Evolutionary Algorithms Example (Haupt & Haupt, 2004) Suppose we have a digital map of a mountain range We want to find the highest peak on the map Automatically! Difficult problem for conventional optimization techniques

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Evolutionary Algorithms

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Variable Encoding VariableValueDecimalBinary Latitude (min) 40 o,1510000000 Latitude (max) 40 o,161281111111 Longitude (min) 105 0 3610000000 Longitude (max) 105 0 37301281111111 Chromosome =[11000110011001] X y

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Evolutionary Algorithms Generate an Initial Population 0010111100011012359 1110010110010011872 0011001000110013477 0010111100100012363 1100111111101111631 0100010111101112097 1110110000000112588 0100110111001111860 Assign each chromosome a fitness (in this case simply the height of the point)

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Evolutionary Algorithms Selection (at 50% selection rate) 0010111100011012359 0011001000110013477 0010111100100012363 1110110000000112588 Selection: Choose a selection rate Choose a threshold (all above survive)

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Evolutionary Algorithms Assume random mating 0010111100011012359 0011001000110013477 0010111100100012363 1110110000000112588 Selection: Choose a selection rate Choose a threshold (all above survive)

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Evolutionary Algorithms Assume random mating 0010111100011012359 0011001000110013477 0010111100100012363 1110110000000112588 Selection: Choose a selection rate Choose a threshold (all above survive)

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Evolutionary Algorithms Crossover 00101111000110 00110010001100 00101111001000 11101100000001

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Evolutionary Algorithms Crossover 00101111000110 00110010001100 00101111001000 11101100000001 Crossover point

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Evolutionary Algorithms Crossover 00101111000110 00110010001100 00101111001000 11101100000001 Crossover point

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Evolutionary Algorithms Crossover 00101111000110 00110010001100 00101111001000 11101100000001 Crossover point

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Evolutionary Algorithms Crossover 0010111100011000101100000001 00110010001100 00101111001000 1110110000000111101111000110 Crossover point

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Evolutionary Algorithms Crossover 0010111100011000101100000001 00110010001100 00101111001000 1110110000000111101111000110 Crossover point

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Evolutionary Algorithms Crossover 0010111100011000101100000001 0011001000110000110111001000 0010111100100000101010001100 1110110000000111101111000110 Crossover point

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Evolutionary Algorithms New generation (almost) 0010111100011000101100000001 0011001000110000110111001000 0010111100100000101010001100 1110110000000111101111000110 Crossover point

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Evolutionary Algorithms Mutation 0010111100011000101100000001 0011001010110000110111001000 0010111100100000101010101100 1110110000000111101111000110

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Evolutionary Algorithms Over the generations ….

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Evolutionary Algorithms Summary: Canonical Genetic Algorithm S1: Set t= 0 S2: Initialize chromosome population P(t) Usually by random init. S3: Evaluate P(t) by a fitness measure S4: while (termination not satisfied) do: begin S4.1 Select for recombination chromosomes from P(t). Let P 1 be the set of selected chromosomes. Choose individuals from P 1 to enter the mating pool (MP) S4.2 Recombine the chromosomes in MP forming P 2. Mutate chromosomes in P 2 forming P 3 S4.3 Select for replacement from P 3 and P(t) forming P(t+1) t = t + 1 end

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Evolutionary Algorithms Generating offspring Weighted random pairing (roulette wheel weighting) Rank weighting Cost weighting Equivalently allow a number of copies to mate (weighted)

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Evolutionary Algorithms Generating offspring Roulette wheel selection How many copies take part in mating? Rank weighting Cost weighting http://www.edc.ncl.ac.uk/highlight/rhjanuary2007g02.php/

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Evolutionary Algorithms Roulette wheel weighting (1) 0011001000110013477 1110110000000112588 0010111100011012359 0010111100100012363 0.4 0.30.7 0.20.9 0.11.0 Rank weighting: assign distributions by rank, make sure that total probability is 1.0 pnpn ΣpnΣpn

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Evolutionary Algorithms Roulette wheel weighting (2) 0011001000110013477 1110110000000112588 0010111100100012363 0010111100011012359 0.265 0.2480.513 0.2430.756 0.2431.0 Fitness based weighting: assign distributions by fitness, make sure that total probability is 1.0 pnpn ΣpnΣpn

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Evolutionary Algorithms Roulette wheel weighting (3) 0011001000110013477-12097 = 1380 1110110000000112588-12097 = 491 0010111100100012363-12097 = 266 0010111100011012359-12097 = 262 0.575 0.2050.780 0.1110.891 0.1091.0 Cost based weighting: assign distributions by cost, make sure that total probability is 1.0 pnpn ΣpnΣpn

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Evolutionary Algorithms Pairing offspring Randomly Rank based (1-2) (3-4) Prevents overly quick convergence Tournament: Pick two chromosomes Set parameter k Generate random nr r If (r < k) pick fitter (eg. k = 0.75) Elitism: always pick (a couple of) the best

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Evolutionary Algorithms Real valued chromosomes In case of example: (x, y), where x and y are the spatial coordinates Crossover Discrete recombination Intermediate recombination Line recombination

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Evolutionary Algorithms Real valued chromosomes Crossover Line recombination (122, 45, 77) and (3,4,3) Generate random positions: (1,2,2) and (1,1,2) Results in: (122, 4, 3) and (122, 45, 3)

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Evolutionary Algorithms Real valued chromosomes Intermediate recombination Offspring = parent 1 + α(parent 2 – parent 1) Allow area to be slightly larger than hypercube defined by parents Generate α for each position Line recombination Use a single value for α

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Evolutionary Algorithms Genetic Programming (Koza, 1992,1994,…) Example (Mitchell, 1998): Keplers law P 2 = cA 3, P orbital Period, A average distance from sun Programme for Mars: Program Orb // Mars A = 1.52; P = SQRT(A*A*A); PRINT P END

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Evolutionary Algorithms Genetic Programming Lisp version (defun orbital_period () ; Mars; (setf A 1.52) (sqrt (A A (* A A))))

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Evolutionary Algorithms Parse tree for LISP expression SQRT * A * AA

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Evolutionary Algorithms Kozas algorithm Trees consist of functions and terminals Choose a set of functions and terminals, e.g { +, -, *, /, }; {A} Generate random programmes (trees) which are syntactically correct Evaluate fitness Apply crossover (mutation)

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Evolutionary Algorithms Crossover / A/ // AAAA * A- * AAA / A/ // AAA * A- *A AA X A

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Overview Last two weeks we looked at evolutionary algorithms.

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