Automatic knot placement by a genetic algorithm for data fitting with a spline

In order to obtain a good spline model from many measurement data, frequently we have to deal with bets as variables. Then the problem to be solved becomes a continuous nonlinear and multivariate optimization problem with many local optima. Therefore, it is difficult to obtain a global optimum. We propose a new method to convert the original problem into a discrete combinatorial optimization problem and solve the converted problem by a genetic algorithm. We construct individuals by considering candidates of the locations of knots as genes, and convert the continuous problem into a discrete problem. We search for the best model among the candidate models by using H. Akaike's (1974) Information Criterion (AIC). Our method can determine appropriate number and locations of knots automatically and simultaneously. We don't need any subjective parameters such as error tolerance or a smoothing factor, and good initial location of knots for iterative search. Numerical examples are given to show the effectiveness of our method.

[1]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[2]  Gábor Renner,et al.  Spline interpolation with genetic algorithms , 1997, Proceedings of 1997 International Conference on Shape Modeling and Applications.

[3]  P. Dierckx.  An algorithm for least-squares fitting of cubic spline surfaces to functions on a rectilinear mesh over a rectangle , 1977 .

[4]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[5]  T. Lyche,et al.  A Data-Reduction Strategy for Splines with Applications to the Approximation of Functions and Data , 1988 .

[6]  Lalit M. Patnaik,et al.  Adaptive probabilities of crossover and mutation in genetic algorithms , 1994, IEEE Trans. Syst. Man Cybern..

[7]  Paul Dierckx,et al.  Curve and surface fitting with splines , 1994, Monographs on numerical analysis.

[8]  Ralph R. Martin,et al.  Reverse engineering of geometric models - an introduction , 1997, Comput. Aided Des..

[9]  H. M. Anthony.  The use of local polynomial approximations in a knot-placement strategy for least-squares spline fitting , 1989 .

[10]  D. Jupp.  Approximation to Data by Splines with Free Knots , 1978 .

[11]  J. G. Hayes,et al.  Numerical Approximations to Functions and Data. , 1971 .

[12]  Nina F. Thornhill,et al.  Fitting Spline Functions to Noisy Data Using a Genetic Algorithm , 1993, International Conference on Genetic Algorithms.

[13]  John R. Rice,et al.  Numerical methods, software, and analysis , 1983 .

[14]  H. Akaike.  A new look at the statistical model identification , 1974 .