Walsh and Haar functions in genetic algorithms

Abstract

Theoretical analysis of fitness functions in genetic algorithms has included the use of Walsh functions [14]. They form a convenient basis for the expansion of fitness functions [3]. These orthogonal, rectangular functions have also been used to compute the average fitness values of schemata [5]. This work explores the use of Haar functions [7] for the same purposes. While 2` non-zero terms are required for the expansion of a given function as a linear combination of Walsh functions, at most `+ 1 non-zero terms are required with the Haar expansion, where ` is the size of each binary string in the solution space. Similarly, Haar coefficients require less computation than their Walsh counterparts. The total number of terms required for the expansion of the fitness function at a given point using Haar is of order 2`, substantially less than Walsh’s 22`. A comparison of Haar functions and Walsh functions with respect to fitness averages shows that the use of Haar functions will reduce computation time. Furthermore, the advantage of Haar over Walsh functions remains large (of order ` 2`) when fast transforms are used.