Multiple Adaptive Least Square Curve Fitting Algorithm And Applications

In the practical research and test projects,we often need to measure data from a serie of error bands[(Xi, Yi); I=1,2,...,m] and get the function relationship between variables X and Y, the most common case is that X and Y is linear relation, which is likely to get satisfactory linear relationship, but in most cases, for the nonlinear relationship is difficult to obtain analytical functions simply and easily. Generally,using the basic characteristics of the experimental data, with a specific regression or fitting to seek an approximate analytic function formula, such as common polynomial function using " fitting function curve" cannot be guaranteed through all of the samples, but commendably "close" them which fully reflects the inherent numerical relationship between the known data, and bring great convenience for the upper analysis of data, such as data Calculus properties of data and so on. Through the use of a set of simple,suitable and linearly independent basis functions (power function,exponential function, logarithmic function, special function and so on) to approximate the observed data, can effectively find the fitting function minimum f (x) with the overall experienced minimum-error, the least square method is the most commonly used method to solve the problem of curve. The method is through making least squares approximation to get a set of linear equations, solving linear equations can obtain fitted curve.The lease square method is established by Legendre,which first appeared in the appendix of "new method of the comet’s orbit calculation" by he published in1805,he considered balance error on the whole,and described the ideal of the lease square method,the specific practices of the method and its advantages.Gauss advanced the least square method more deeply than Legendre,and he elicited the method from the error function and elaborated the theory foundation of the method.For the development of the least square method,now there are many methods.such as moving least square method,piecewise least square method,orthogonal square method and so on.For the experimental observation data with multiple salient local features,the conventional method is the application of a set of basis functions in the global domain (such as polynomial) to fit, this method is fast and easy to achieve, but it will get the fitting function with a poor performance.The calculation result of example shows that based on the indirect adjustment principle is deduced in detail and correlation model,the fitting effect is better than the ordinary lease squares by using the orthogonal least square method. Piecewise curve fitting method is the local least squares fitting in each segment, however,the fitting function is not necessarily continuous in the piecewise point,and not ideal in the fitting effect near the interval boundary.Some people put forward the global continuous fitting method of the polynomial basis functions at the same time,but it only confined to two piecewise interval.The moving least square method establishing the fitting function does not use the tradition polynomial or its function,however,it is composed of a coefficient vector and matrix functions.The coefficient vector of general least square method is a constant while moving least square method’s is not constant,but a function argument X.On the basis of previous work,1implement an adaptive multiple iterative lease squares fitting method.The method is a global fitting iteration method,and before the data fitting we should engage in which include data translation, number field conversion, data translation once more, data expansion, the fitted data for inverse reconstruction and a series of operations. The reconstructed data is used to fit the measured data by different degree of polynomial expressions, each iteration correct data which skew great, while other fitting better data does not change, so that it keeps the character of all the data’s error offset size.After several iterations, until the interative error is very small, the iteration is terminated,comparing the degree of different fitting, polynomial correction and terminational error determines the most stable for the best fitting results. Through the numerical example and the measured data of the test, the algorithm is stable,which can return to the measured data better, especially it has better results for the measured data with large error.The method has been applied to process the original data of many research projects,and its effect is better than the commonly used.Adaptive multiple least square fitting algorithm in search of the best degree of a polynomial has a large amount of calculation, because the polynomials are calculated for each number, and it is the advanced iterative calculation, time-consuming, so seeking the best degree of a polynomial is the direction of further study.