Polynomial Multivariate Approximation with Genetic Algorithms

Abstract

We discuss an algorithm which allows us to find the algebraic expression of a dependent variable as a function of an arbitrary number of independent ones where data is arbitrary, i.e. it may have arisen from experimental data. The possibility of such approximation is proved starting from the Universal Approximation Theorem (UAT). As opposed to the neural network (NN) approach to which it is frequently associated, the relationship between the independent variables is explicit, thus resolving the “black box” characteristics of NNs. It implies the use of a nonlinear function (called the activation function) such as the logistic 1/(1+e^− x). Thus, any function is expressible as a combination of a set of logistics. We show that a close polynomial approximation of logistic is possible by using only a constant and monomials of odd degree. Hence, an upper bound (D) on the degree of the polynomial may be found. Furthermore, we may calculate the form of the model resulting from D. We discuss how to determine the best such set by using a genetic algorithm leading to the best L∞- L2 approximation. It allows us to find the best approximation polynomial given a selected fixed number of coefficients. It then finds the best combination of coefficients and their values. We present some experimental results.