- author = "Y. S. Yeun and Y. S. Yang and W. S. Ruy and B. J. Kim",
- title = "Polynomial genetic programming for response surface modeling Part 1: a methodology",
- journal = "Structural and Multidisciplinary Optimization",
- year = "2005",
- volume = "29",
- number = "1",
- pages = "19--34",
- month = jan,
- keywords = "genetic algorithms, genetic programming, directional derivative-based smoothing, extended data-set method, high-order polynomial, interpolation, overfitting,response surface",
- ISSN = "1615-147X",
-
DOI = "
doi:10.1007/s00158-004-0460-6",
-
abstract = "The second-order polynomial is commonly used for
fitting a response surface but the low-order polynomial
is not sufficient 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 fit the response surface, it is
inevitable that the high-order polynomial shows serious
overfitting behaviors. Moreover, the high-order
polynomial shows infamous wiggling, unwanted
oscillations, and large peaks. To suppress such
problematic behaviors, this paper introduces a novel
method, called directional derivative-based smoothing
(DDBS) that is very effective for smoothing a
high-order polynomial. The role of GP is to find
appropriate terms of a polynomial through the
application of 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 estimate the coefficients of the
polynomial quickly 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 modified OLS method enabling interpolation is presented. Four illustrative numerical examples are given to demonstrate the performance of GP with DDBS.",
