The Problem Space

As proof of concept, we have applied our system to a symbolic regression problem used by Koza [Koza 92]. The system is given a set of input and output pairs, and must determine the function that maps one onto the other. The particular function Koza examined is X⁴ + X³ + X² + Xwith the input values in the range [-1..+1]. Table  4 contains a tableau which summarizes his experiments.

 

Table 4: A Koza-style tableau

Objective :	Find a function of one independant variable and
	one dependant variable, in symbolic form
	that fits a given sample of 20 (xi, yi)
	data points, where the target functions is the
	quartic polynomial X4 + X3 + X2 + X
GPTerminal Set:	X (the independant variable)
GPFunction Set	$+, -, *, \%, \sin, \cos, \exp, \log$
Fitness cases	The given sample of 20 data points in the
	interval [-1, +1]
Raw Fitness	The sum, taken over the 20 fitness cases,
	of the error
Standardised Fitness	Same as raw fitness
Hits	The number of fitness cases for which the error
	is less than 0.01
Wrapper	None
Parameters	M = 500, G = 51

The production rules for <expr> are given above. As this and subsequent rules are the only ones that require a choice, they are the ones that will be evolved.

We adopt a similar style to Koza of summarizing information, using a modified version of his tableau. Notice how our terminal operands and terminal operators are analgous to GPTerminals and GPfunctions respectively.

 

Table 5: Grammatical Evolution Tableau

Objective :	Find a function of one independant variable and
	one dependant variable, in symbolic form
	that fits a given sample of 20 (xi, yi)
	data points, where the target functions is the
	quartic polynomial X4 + X3 + X2 + X
Terminal Operands:	X (the independant variable)
Terminal Operators	The binary operators +, *, /, and -
	The unary operators Sin, Cos, Exp and Log
Fitness cases	The given sample of 20 data points in the
	interval [-1, +1]
Raw Fitness	The sum, taken over the 20 fitness cases,
	of the error
Standardised Fitness	Same as raw fitness
Hits	The number of fitness cases for which the error
	is less than 0.01
Wrapper	Standard productions to generate
	C functions
Parameters	M = 500, G = 51