Curve And Surface Fitting With Improved Contraction-Expansion Algorithm

Regression analysis is a statistical method which quantifies the relationship between variables. Linear regression analysis has a longer applied history than most other statistical methods. For most applying assumptions could be satisfied, its statistical inference is valid on the whole. The results from regression analyses are reliable on most cases. Besides, the procedure of linear regression analysis is relative simple and can run on most statistical software. Most users can easily handle the method, so it has been widely used in many fields. However, the relationship between variable are rarely linear, most accurate functions between causal traits and effect ones are nonlinear, linear relation is just a special case of nonlinear relationship. Compared with extensive application of linear regression, nonlinear equations were rarely used to describe relationship between concerning variables in most researching fields. The application of nonlinear regression analyses are far more lagged behind other statistical methods. The reason is that curve and surface fitting, also referring as optimal parameters estimating of nonlinear equations, is much more complicated than linear regression. Lacking proper algorithm and software, the optimal fitting is usually difficult to reach. There has been two existing curve and surface fitting algorithms: namely, analytic and direct methods.The thesis introduced some common methods of above two classes and analyzed the advantage and disadvantage of them. The analytic method is characterized by solving the partial derivatives equations of objective function Q. The advantage of this kind of methods is that they can use partial derivation functions to guide the searching direction and the searching efficiency is usually high so that it can achieve optimized objective function with less time provided proper given initials. However, the analytic method, with precondition of the partial derivation function, was hard to obtain all partial derivative functions for the complexity of some nonlinear functions. Furthermore, appropriate initial values were difficulty to set in prior, otherwise it could easily be trapped in local optimal, and the global optimizing nonlinear parameters could not be easily obtained.The direct method is characterized by direct iterative estimation of parameters of nonlinear function. The advantage of this method is not needed to obtain the partial derivatives and suitable for all kinds nonlinear equations. However, low searching efficiency and poor global reaching ability especially for those big parameter number situations are the major disadvantages.The contraction- expansion algorithm, a direct method put forward by Shiliang Gu et al. in 1998, characterized by searching for the optimal parameter vector in the uniform distributed points in p dimensional space, had good effect in curve and surface fitting and could achieve the global optimal fitting in a wide range of initial values. It could also solve some complex constrained NLP. Similar to those direct methods, it also had the low efficient problem.In order to promote extensive application of nonlinear regression, this thesis focused on the method of curve and surface fitting and made some improvement on C-E algorithm by step length and central point adjustment. The feedback system, using scatter-spread of spring points to adjust search step length, was more sensitive than previous one. The most important improvement is to combine C-E algorithm with analytic method, Levenberg-Marquardt method. The new method could utilize C-E algorithm to jump off local minima and adapt the numerical derivatives to guide searching direction, so it could realize global optimization of objective function for most practical curve and surface fitting problems. The new C-E algorithm, combining the advantages and overcoming the disadvantages of two kinds of method, greatly enhanced ability to reach the global optimal estimation of nonlinear parameters and established a solid foundation for nonlinear regression analysis. The new Matlab program was compiled based on the improved algorithm, and various kinds of models and examples were used to demonstrate its function.