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Curve and Surface Fitting with Improved Contraction-Expansion Algorithm

Author: WanLinSheng
Tutor: GuShiLiang
School: Yangzhou University
Course: Crop Genetics and Breeding
Keywords: Nonlinear equations CE ALGORITHM Parameter Estimation Optimization Numerical differentiation Program
CLC: O213
Type: Master's thesis
Year: 2007
Downloads: 214
Quote: 2
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Abstract


Regression analysis is to describe the relationship between variables in the statistical analysis of quantitative methods, compared to other statistical methods, there are more long application history, because most of the assumptions of its application can be met, a solid foundation of statistical inference, so conclusions more credible. Linear (mono-or poly) Calculating regression process is relatively simple, there are many software available, and easy to grasp for most users, and thus the linear regression analysis of the great degree of application. However, the vast majority of agronomy, biology, economics, or in many other areas of the quantitative relationship between variables and non-linear, non-linear relationship is linear only in a special case. Compared to the widely used linear regression analysis to nonlinear equations depicts quantitative relationships between variables nonlinear analysis, application progress more slowly. The reason is mainly the curve and surface fitting that the optimal parameter estimation of nonlinear equations is very complex, due to lack of universally applicable algorithm, generally difficult to obtain the optimal parameter estimation. Throughout the existing curve, surface fitting ie parameter estimation algorithm, the main analytical method and the direct method two categories. In this paper, these two categories of methods appears some of the more common methods are summarized introduction, analysis of the advantages and limitations of various methods. Analytical method is based on solution of the partial derivative equation Q is characterized, and its biggest advantage is the ability to take advantage of the partial derivative function guide the direction of search optimization, search more efficient, more suited to the initial value in the conditions, you can quickly achieve the objective function optimization. But these methods to nonlinear equations based on the partial derivative of the function, and some nonlinear equations are complex, difficult to obtain all of its partial derivatives; fitting equation for most purposes and must provide appropriate initial value, otherwise it is easy fall into the trap of local optimum, the estimated parameters are not optimal. Direct method is based on the argument point directly into the objective function is estimated to be the main feature of iterations, and its biggest advantage is eliminating the trouble of providing partial derivatives, applicable to all types of non-linear relationship, versatility is strong. But its drawback is that search efficiency is relatively low, many parameters and nonlinear relationships in complex, global optimization of the capacity is low. Gu Shiliang and other many years ago proposed a further point to the main features of the direct method - shrink Zhang algorithm, curve and surface fitting has a good effect on the vast majority of the initial value problem can achieve a wide range of global best fit, in addition to the algorithm is more complex for most constrained and unconstrained nonlinear programming problem globally optimal solution also had an outstanding performance, but the algorithm is also a direct fit prevailing law efficiency is not high. In order to more effectively achieve all kinds of non-linear regression analysis, nonlinear regression promoting the wide application of this paper, the optimal curve and surface fitting method to focus on research, the better the performance of the original shrink Zhang made a step size algorithm , center adjustment several improvements, the most important improvement is that this algorithm, combined with numerical differentiation techniques and analytical method improved Gauss - Newton method, to establish a new curve and surface fitting algorithm - Improved shrink Zhang algorithms. New algorithms mainly use various types of compression algorithms Chang nonlinear model's versatility and ability to escape from local optima traps and achieve global optimal objective function; using the combination of modified Gauss - Newton method, accelerated iterative approximation regression statistics, including Curves and Surfaces statistical regression equation for each partial derivative of the function will use the numerical differentiation technique, not only eliminates the trouble of providing partial derivatives of the function, but also to approximate the partial derivatives of the function guide the optimization process, improve the efficiency of nonlinear statistical estimation. To combine these two key technologies to build the new algorithm can give full play to the advantages of the steps, but they can overcome the defects when used alone, greatly improve the implementation capacity curve and surface fitting, nonlinear regression analysis to establish a solid foundation. This new algorithm for MATLAB as a platform for the preparation of a software algorithm to a variety of analog and examples validate its effectiveness. Simulation and validation examples show that the new algorithm has higher efficiency and the only accurate fitting fitting results.

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CLC: > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Application of statistical mathematics
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