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# EE4047_Part3 - Part 3 Theory and Hypothesis City University...

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1 City University of Hong Kong City University of Hong Kong Part 3: Theory and Hypothesis Mathematical Models for GA box4 Macroscopic models square6 Focus on the properties of a large set of individuals box4 Microscopic models square6 Focus on the properties of a single string

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2 Hyperplanes Assume that we have a problem, where the solution can be encoded in 3 bits (X 1 ,X 2 ,X 3 ) X 2 X 1 X 3 000 100 010 110 101 001 011 111 **1 Terminology Schema: binary string may contain “ * ” (don’t care or called wild symbol) Each schema corresponds to a hyperplane in the search space Each chromosome (binary encoded) corresponds to a corner in the hypercube and is a member of 2 L different hyperplanes (*** is included.) Total number of hyperplanes is 3 L
3 Hyperplanes in 3-dimensional cubic Corners Lines Planes Cubic 0 0 0 * 0 0 * * 0 * * * 0 0 1 * 0 1 * * 1 0 1 0 * 1 0 * 1 * 0 1 1 * 1 1 * 0 * 1 0 0 0 * 0 1 * * 1 0 1 0 * 1 0 * * 1 1 0 1 * 0 1 1 1 1 * 1 0 0 * 0 1 * 1 0 * 1 1 * ----------------------------------------------------------------------- 8 12 6 1 total=27 Example Consider chromosome with 10 bits One wild symbol Two wild symbols [ * 1 1 1 0 0 1 0 0 1] [ * 1 * 1 1 0 0 1 0 0] It matches 2 strings It matches 4 strings [ 0 1 1 1 0 0 1 0 0 1]; [ 0 1 0 1 1 0 0 1 0 0]; [ 1 1 1 1 0 0 1 0 0 1] [ 0 1 1 1 1 0 0 1 0 0]; [ 1 1 0 1 1 0 0 1 0 0]; [ 1 1 1 1 1 0 0 1 0 0] Every schema matches exactly 2 r strings where r = the number of wild symbols For L= length of string, each string matches 2 L schemata . The TOTAL number of schemata is 3 L

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4 Order of Schema The order of the schema, o(S) , is the number of “0”s or “1”s in a chromosome S1=[ * * * 0 0 1 * 1 1 0 ] o(S1)=6 S2=[ * * * * 0 0 * * 0 *] o(S2)=3 S3=[ 1 1 1 0 1 * * 0 0 1 ] o(S3)=8 ) ( S o Defining Length The Defining Length of the schema S is the distance between the first and last fixed gene positions. It defines the compactness of information contained in a schema. S 1 =[ * * * 0 0 1 * 1 1 0 ] =10-4=6 S 2 =[ * * * * 0 0 * * 0 *] =9-5=4 S 3 =[ 1 1 1 0 1 * * 0 0 1 ] =10-1=9 A schema with a single fixed position has a defining length of zero ) ( S δ ( ) S δ ( ) 3 S δ ( ) 1 S δ ( ) 2 S δ
5 City University of Hong Kong City University of Hong Kong Holland’s Model Effect of Selection Denote Average Fitness of the schema, f (S,t) Assuming that there are ζ chromosomes

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EE4047_Part3 - Part 3 Theory and Hypothesis City University...

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