Symbolic Regression

In the first problem we apply our system to a symbolic regression problem summarised in Table 1 [Ryan 98a]. 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 examined is X⁴ + X³ + X² + X with the input values in the range [-1..+1]. As the target function has no constant values, only the variable X, it was decided to use only one terminal operand, i.e. X. So as not to unduly bias the system towards the correct solution, the terminal operators set consisted of more than the binary addition and multiplication operators, which alone would be sufficient to reach the target function. To determine the fitness of an individual program 20 test points were applied in the range -1 to 1, and the fitness was taken as the sum of the error.

 

Table 1: Grammatical Evolution Tableau for Symbolic Regression

Objective :	Find a function of one independent
	variable and one dependent
	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 independent 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