Polynomial Genetic Programming for Response Surface Modeling

The 2nd order polynomial is commonly used for tting a response surface but the low order polynomial is not suf cient if the response surface is highly nonlinear. Based on genetic programming (GP), this paper presents a method with which high order smooth polynomials, which can model nonlinear response surfaces, can be built. Since, in many cases small samples are used to t the response surface, it is inevitable that the high order polynomial shows serious overtting behaviors. Moreover, the high order polynomial shows infamous wiggling, unwanted oscillations, and large peaks. To suppress such problematic behaviors, the paper introduces a novel method, called the directional derivative based smoothing (DDBS) that is very effective for smoothing out a high order polynomial. The role of GP is to nd appropriate terms of a polynomial through applying genetic operators to GP trees that represent polynomials.The GP tree is transformed into the standard form of a polynomial using the translation algorithm. To quickly estimate the coef cients of the polynomial, the ordinary least square (OLS) method that incorporates the DDBS and extended data set method is devised. Also, by using the classical Lagrange multiplier method, the modi ed OLS method enabling interpolation is presented. Four illustrative numerical examples are given to demonstrate the performance of GP with DDBS.