Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 195023, 10 pages

http://dx.doi.org/10.1155/2015/195023

## Measurement Data Fitting Based on Moving Least Squares Method

State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, China

Received 1 November 2014; Accepted 5 January 2015

Academic Editor: Mohamed Abd El Aziz

Copyright © 2015 Huaiqing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the electromagnetic field measurement data postprocessing, this paper introduced the moving least squares (MLS) approximation method. The MLS combines the concept of moving window and compact support weighting functions. It can be regarded as a combination of weighted least squares and segmented least square. The MLS not only can acquire higher precision even with low order basis functions, but also has good stability due to its local approximation scheme. An attractive property of MLS is its flexible adjustment ability. Therefore, the data fitting can be easily adjusted by tuning weighting function’s parameters. Numerical examples and measurement data processing reveal its superior performance in curves fitting and surface construction. So the MLS is a promising method for measurement data processing.

#### 1. Introduction

The measurement data of electromagnetic (EM) field plays a key role in EM environment assessment. However, the measurement points are limited, and, in order to describe the EM field distribution more accurately, the postprocessing is necessary. The data acquiring for nonmeasurement point is essentially a typical function approximation or surface construction problem. Owing to the instruments errors, environmental interference, or terrain changes, the deviation emerges inevitably and the fitting method is preferred in the postprocessing.

Currently, the least squares (LS) method has been most widely used in data fitting. The commonly used basis functions are polynomials [1], rational functions [2], Gaussian, exponential, smoothing spline in curve fitting, the B-spline [3], the nonuniform rational B-splines (NURBS) [4], Bézier surfaces [5], and radial basis function [6] in surface construction. And, simultaneously, the deformations of LS as RLS (recursive least squares), TLS (total least squares), PLS (partial least squares), WLS (weighted least squares), GLS (generalized least squares), and SLS (segmented least squares) have been also put forward. However, all the above LS based methods are global approximation schemes which are not suitable for large amount of data, irregular or scattered distribution cases. So the moving least squares (MLS) method which is local approximation was proposed in measurement data processing.

The MLS approximation was introduced by Lancaster and Salkauskas for surface generation problems [7]. It has been used for surface construction with unorganized point clouds [8], regression in learning theory [9], and sensitivity analysis [10]. However, the major applications of MLS are to form a lot of meshless methods [11], as the diffuse element method (DEM) [12], the well-known element-free Galerkin method (EFGM) [13], and the meshless local Petrov-Galerkin method [14]. These kinds of methods have high computational precision and stability. The disadvantage of the MLS lies in the algebra equations system that is sometimes ill-conditioned, so Cheng and Peng [15] proposed the improved method. The error estimates and stability of MLS [16–19] and the variation as complex or Hermite [20, 21] were intensively discussed. On the whole, the researches of MLS approximation theory are much less than applications.

As a data fitting method, the MLS can be regarded as a combination of WLS and SLS because of its compact support weighting function. Moreover, the introduced moving window in MLS shows superior performance versus SLS. Firstly, the compact support weighting function indicates that only partial measurement data nearby the unknown measurement point are involved in calculating which indicates the MLS inherits localized treatment of SLS. Then, the segmentation is rigid in SLS and causes the problems of how to select the segment and fitting discontinuity. However, the moving window in MLS acts as a soft segment. The segment selection is avoided and the fitting continuity and smoothness are guaranteed. Finally, weighting function parameters provide a convenient adjustment option for MLS.

Hence, this paper proposed the MLS method for measurement data fitting. The structure of the paper is as follows. In Section 2, a brief description is given for MLS approximation. Then, the weighting function is discussed in Section 3. In Section 4, the numerical examples of curve fitting are carried out. And finally, the measurement data fitting for substation is implemented in Section 5. Conclusions are drawn in Section 6.

#### 2. Moving Least Squares Approximation

In MLS, an arbitrary function can be approximated by where , , are basis functions, is the number of terms in basis functions, and are the coefficients. The basis functions can be polynomials, Chebyshev polynomials, Legendre polynomials, trigonometric function, wavelet function, radial basis function, and so forth. For example, the basis functions of one-dimensional polynomials have the following forms: linear basis as , and quadratic basis as , . In the following, we consider one-dimensional curve fitting for demonstration.

The obvious difference between the traditional LS method and MLS is the coefficients. For MLS, the coefficients varied with , while they are constant in LS. In order to determine the coefficients, a function which is similar to WLS is defined as where , , are the given nodes. are the weighting functions with compact support which can be also represented as . The subscript “” means the center of is located in . The schematic diagram of weighted scheme in MLS is shown in Figure 1.