Evolving Intervening Variables for Response Surface Approximations

Genetic Programming (GP) is a powerful string processing technique based on the Darwinian paradigm of natural selection. Although initially conceived with the more general aim of automatically producing computer code for complex tasks, it can also be used to evolve symbolic expressions, provided that we have a fitness criterion that measures the quality of an expression. In this paper we present a GP approach for generating functions in closed analytic form that map the input space of a complex function approximation problem into one where the output is more amenable to linear regression. In other words, intervening variables are evolved in each dimension, such that the final approximation model has good generalization properties and at the same time, due to its linearity, can easily be incorporated into further calculations. We employ least squares and cross-validation error measures to derive the fitness function that drives the evolutionary process. Results are presented for a one-dimensional test problem to illustrate some of the proposed ideas – this is followed by a more thorough empirical study, including multi-dimensional approximations and an engineering design problem.