Fast Algorithms Of Moving Least Squares And Its Application

In practical engineering, there is often a large number of experimental data requires to be post-processed, fitting is one kind of important processing method. Usually, data fitting uses least square method, but it is a global method, it cannot take account of local needs and when dealing large amounts of data, there is a difficultytoset the model and the instability problem in calculation that caused by this. And moving least square method can solve the smooth fitting and localizationproblem better, it has great advantages in terms of data fitting, with the strong scatter adaptability, have local fitting or interpolation characteristics and has the advantages of high precision, therefore, data processing based on moving least squares method in curve and surface fitting has become the research hotspot in recent years.This paper introduces principle of moving least squares and introduces itto electromagnetic measurement data fitting, so as to analyze and evaluate the performance of the MLS fitting; due to the calculation amount of moving least squares method in the engineering application is too large, thus further study of the fast algorithm is needed, including the angle from the orthogonal basis function of processing and from equivalent FIR filtering, the main conclusions are as follows:This paper deduces the moving least squares fitting formula, with all kinds of meaning and calculation form are given in the course; introduces the definition of compactly supported weight function and its significance to moving least square method and points out that it is characteristic of compact support weight function makes the moving least square method with moving window characteristic, which has a great advantage in comparison with other least squares fitting method or other types; finally, compares the influence of weight function on the fitting results based on the three to seven spline weight function fitting, in addition introduces the principle ofselecting thefitting pointsof moving least squares method in the fitting process and how to judge whether a node is selected as the influencing node.Based on moving least squares method's advantages highlight performance in the data fitting, this paper will introduce the MLSto the field of electromagnetic measurement data processing, and substantive data lines and measured data fitting to deal with that corresponds to curve and surface fitting, through the comparison of least squares fitting and the results show that using moving least squares method in electromagnetic measurement data post-processing is better, the fitting results are more smooth, the error is lesser, the advantage is obvious; in the selection of compact support weighted function, finally by comparing the determination is using Gauss weight function.Considering that the relative large amounts of moving least square method, the paper studies its fast algorithm. First of all, through the basis function orthogonality, the inversematrix is transformed into a diagonal matrix, although the process of basic functions orthogonality increases the amount of calculation, but the inverse operation can significantly reduce the amount of calculation, MATLAB simulation examples show that the orthogonal process can achieve rapid calculation results, but when dealing a large amount of data the results is limited and still need further study; secondly, the essence of moving least square method of fitting is that fit the weight function support domain nodes by using the weighted least squares method, which is similar to moving average process, so it can be seen as data filtering, sothe angle of filter fast algorithm is proposed, the strict theoretical proof is given to identify the relationship between moving least squares and FIR filter, and the moving least squares fitting calculation is transformed into corresponding FIR filter coefficients, then the fast calculation can be reached through convolution or fast Fourier transform, and the feasibility of this method is verified by an example.Based on the fact that the moving least squares method and FIR filter is equivalent, moving least squares fitting can be seen as a FIR filter, specifically for the low pass, odd point linear phase, non-causal FIR, further discussion of its filtering performance is needed, mainly in considering of the affect of the weight function and shape parameters of support domain to the filtering performance, analyzes how the above parameters' changes influence the corresponding impulse response function and amplitude frequency characteristics of the filter, and draws the following conclusions: with the support domain increasing, the width of theimpulse response function increases whilethe amplitude decreases, at the same time, the cutoff frequency of the corresponding low-pass filter decreases; with the shape parameter increasing, the width of the impulse response function decreaseswhilethe amplitudeincreases,at the same time, the cutoff frequency of the corresponding low-pass filter increases;Finally, the paper summarizes the research work, and looks forward to the next step should be carried out to improve the work and the need to make an improvement.