基于最小二乘法的曲线拟合及其在Matlab中的应用

物理量之间的函数关系的确定在实际研究工作中有很重要的作用。目前我们用于曲线拟合的方法主要是三次多项式插值法,抛物线加权平均法,张力样条函数插值法等,但这些方法计算量大。本文结合最小二乘法的基本原理,利用最小二乘方法进行曲线拟合,计算过程简便。首先介绍了最小二乘法拟合的基本原理,然后介绍了用Matlab实现曲线拟合以得到函数关系的方法和步骤,最后举例详细介绍了该方法的应用。     

最小二乘曲线拟合在SKE测向中的应用

本文介绍了最小二乘曲线拟合方法,阐述了位置保持设备(以下简称SKE)及其测向原理。根据SKE定向天线信号的特点,给出了利用最小二乘曲线拟合实现SKE测向的方法。     

最佳逼近法在晶体频温曲线拟合和TCXO中的应用

在频率控制领域中,常用最小二乘法来进行数学逼近,但所需的是频率偏差最小而不是它的平方和最小。为此,在频率控制领域首次引入了最佳逼近算法,它能给出频率偏差的最小-最大解。该文不仅给出了最佳逼近算法用于晶体谐振器频率温度特性曲线拟合的方法和例子,并将该算法与最小二乘法作比较;从中看出该算法与最小二乘法比较具有优势。而且给出了它用于单电容温补晶振(TCXO)设计的方法和例子,并且在这个基础上制作了一个TCXO样品,经过补偿后该TCXO的频偏小于±1×1-0 6。结果表明:在频率控制领域最佳逼近算法比最小二乘法更有效。该算法对曲线拟合和工作温度范围在晶体谐振器两个零温度系数点之间的TCXO有实际意义。     

基于曲线拟合的测向误差分析方法

本文首先分析了短波测向误差,然后采用曲线拟合的最小二乘法得出了频率与测向误差的函数关系,并结合实验结果,证明了曲线拟合是分析测向天线阵环境变化对测向误差影响的有效方法。     

基于有约束最小二乘法的IQA算法曲线拟合

为了实现准确评估图像质量评价算法的目标,采纳VQEG报告中提到的性能指标,完成了将算法数据的曲线拟合过程转换为有约束最小二乘法形式的数学模型,然后编写Matlab程序求解。以PSNR、SSIM和MSSIM三种图像质量评价算法为例的评估结果表明,此法结果准确,求解效率高。     

CMOS MOCCII高频非理想特性研究

本文用高频小信号等效电路对MOCCII电路进行了分析,提出了vx/vy的双二次模型理论.采用最小二乘法中的最速下降法根据MOCCII的PSPICE仿真曲线对该模型进行最小二乘曲线拟合,得出双二次模型各参数.拟合后曲线与PSPICE仿真曲线在取样点误差的方均根值为10-7数量级:同时对Iz/Ix的单极点模型和Iz-/Ix的双极点模型均进行了最小二乘曲线拟合,得出了模型参数,拟合后曲线与PSPICE仿真曲线在取样点误差的方均根根值为10-5数量级.这些MOCCII(CCII)高频模型在高频滤波器等电路的设计中具有重要的意义.     

非线性曲线拟合程序及其在半导体物理中的应用

本文论述了曲线拟合的数学模型及其用途与存在的困难。提出联合套用阻尼最小二乘法和复合形法可具有放宽初值要求、收敛速度快等优点。对拟合结果检验的必要性和方法作了讨论,给出非线性拟合的详细通用程序框图。应用本程序对硅太阳电池光谱响应、磷化镓发光二极管光电流谱,硅N-N~+-P结构光伏谱、磷化镓瞬态光电容谱、掺氮或掺铋磷化镓光致发光谱等半导体物理问题作了曲线拟合,并借助拟合结果分析完善了物理模型。     

全自动晶圆磨边机平面度误差测量与评定

提出了在全自动晶圆磨边设备中对晶圆平面度误差的等角度单圆周测量方案,讲述了最小二乘法的算法原理;并根据最小二乘法的算法原理,基于Matlab软件编写平面度误差评定及其结果的可视化程序;运行程序对全自动磨边机测量的数据进行处理,程序运行结果表明,该算法运行稳定可靠,能够快速确定最小二乘法平面方程和评定平面度误差。     

基于激光诱导击穿光谱的燃煤热值定量分析

煤的热值是影响燃煤锅炉运行最重要的煤质特性之一。燃煤热值在线分析能够为实时优化调整锅炉运行提供重要依据。采用激光诱导击穿光谱技术对烟煤样品热值进行了定量分析。实验用Nd:YAG固体激光器1 064 nm激光脉冲在空气环境下烧蚀烟煤样品表面而形成等离子体。采用偏最小二乘法和基于主导因素的偏最小二乘法提高煤热值定量分析的准确性。对于基于主导因素的偏最小二乘法,利用碳双原子分子（C2）和碳氮双原子分子（CN）的谱线强度建立主导因素模型。采用偏最小二乘法建立的定标曲线拟合优度和预测均方根误差分别为0.94和1.46 MJ/kg;采用基于主导因素的偏最小二乘法建立的定标曲线拟合优度和预测均方根误差分别为0.99和1.18 MJ/kg。结果表明:激光诱导击穿光谱技术对燃煤热值在线分析具有很大的应用潜力。     

基于最小二乘曲线拟合的时间配准方法研究

针对单一手段探测低慢小目标能力不足的问题,提出了运用雷达/红外手段联合探测的方法.但在雷达红外数据融合之前,需要解决时间配准的问题.为解决这个问题,提出了一种基于最小二乘曲线拟合的时间配准改进方法,推导出了改进的最小二乘曲线拟合公式,分析了影响拟合精度的因素,通过仿真分析验证了该方法的有效性.     

空战模拟器导弹火控工作式推导研究

为了求解空战模拟器上导弹火控工作式,建立了空空导弹发射区的计算模型,根据拟合导弹火控工作式的要求,计算了空空导弹发射区离散边界点,并采用多元非线性回归方法进行数学拟合,进而求得空战模拟器上导弹火控工作式。     

Calculation of Fourier coefficients of a function given at a set of arbitrary points

A numerical method is presented for the calculation of Fourier coefficients of a function which is given at a discrete set of arbitrary points. The function is approximated by a sum of Cheby?shev polynomials. This is performed by Clenshaw's method of curve fitting, which is a least-squares method. The Cheby?shev coefficients are then used to construct linear combinations of Bessel functions, which are very good approximations of the Fourier coefficients.     

Gain measurements of Fabry-Pe/spl acute/rot semiconductor lasers using a nonlinear least-squares fitting method

A method for the measurement of the gain-reflectance product of Fabry-Pe/spl acute/rot (F-P) semiconductor lasers is proposed and compared to other techniques. The method is based on a nonlinear, least-squares fitting of the F-P modes to an Airy function. A separate fitting is performed over each mode, as measured with an optical spectrum analyzer (OSA), so that the gain-reflectance parameters are extracted. The influence of the OSAs response function is considered by convolution of the Airy function with the response function of the OSA. By comparing with the Hakki-Paoli method, the mode sum/min method, and the Fourier series expansion method, we find that the nonlinear fitting method is the least sensitive to noise. However, owing to a broadening of the F-P modes of the semiconductor laser, the mode sum/min method combined with a deconvolution technique gives the least underestimated gain above threshold.     

Least-square-exponential approximation

A method is presented for solving the problem of fitting decay-type experimental data by sums of exponentials. Both the exponents and the coefficients of the exponentials are considered as unknowns. The novel idea presented here is to consider the sum of exponentials as a solution of an integral equation. A step-by-step procedure is given for the solution of the problem in the least-squares sense. An advantage of the proposed method is that the data need not be equidistant.     

基于人工神经网络的智能化架空线路应变快速解调方法北大核心

为了提高智能化光纤复合架空线路态势感知的实时性,将人工神经网络方法应用于光纤沿线应变解调,确定了神经网络的结构。编程实现了基于洛伦兹模型的最小二乘谱拟合方法和神经网络方法,采用不同信噪比和布里渊频移的布里渊谱训练神经网络,将它们应用于某光纤复合架空线路沿线光纤应变的测量,从不同角度比较了两种方法的计算结果。计算结果表明,神经网络方法能有效获得光纤沿线的布里渊频移进而获得应变,具有与谱拟合方法相似的准确性,但应变解调时间仅约为谱拟合方法的1/20000。研究结果为提高智能光纤复合架空线路态势感知的实时性提供了参考。     

Stable Parametric Macromodeling Using a Recursive Implementation of the Vector Fitting Algorithm

A novel least-squares fitting technique is presented for the macromodeling of parameterized frequency responses. Such parametric macromodel can be used for the design, study, and optimization of microwave structures. A key benefit of the proposed method, is that the poles of the macromodel are guaranteed stable by construction. This can easily be enforced when using the presented macromodel representation.      

ARMA spectral estimation based on partial autocorrelations

This paper presents a new method for estimating the coefficients of autoregressive moving-average parameters of stationary time series. The method is based on computing the sample autocorrelations of the given time series and fitting an ARMA model so as to approximate the partial autocorrelations in a least-squares sense. When the given time series are characterized by spectral zeroes near the unit circle, they tend to have relatively long sequences of nonzero partial autocorrelations; hence the new method is especially effective in such cases. This paper contains a derivation of all necessary mathematical details, as well as several numerical examples illustrating the performance.     

Least-squares fitting of data by rational functions: Levy's method (Part 1) [By the Numbers]

We have, in various previous columns [1-5], looked at the "approximation" problem. There are two broad approaches to dealing with this problem: 1. Interpolation, which finds a suitable fitting function that passes through all the data points available, and 2. Fitting, according to some criterion for goodness, a suitable function to a set of data without restricting that function to coincide with the data points. In the second approach, both the criterion for goodness of fit and the class of fitting functions are important to the outcome of the approximation, and both are left to the person working with the data. One popular criterion is the minimization of the sum (or the integral) of the squared error between the chosen fitting function and the data. An alternative to this "least-squares" criterion is to minimize the maximum error between the fitting function and the data. The fitting function itself can be anything that "makes sense"--a polynomial, a trigonometric function, etc. For a wide variety of problems in science and engineering (e.g., the characterization of a linear, time-invariant system from frequency-response data), it might be particularly appropriate to choose a rational function as the fitting function. In this installment, we will provide a brief introduction to least-squares fitting of data by rational functions by presenting Levy's Method [6], a classical approach that is relatively straightforward in concept and easy to program, and which gives respectable results in a broad range of circumstances.     

Automatic Microwave Q Measurement for Determination of Small Attenuations

The Q's of a waveguide transmission cavity with and without insertion of a length of waveguide give the insertion loss. Least-squares fitting of transmission versus frequency data have been used to determine Q and resonant frequency f/sub r/ with computer-controlled equipment in an iterative process. Preliminary values of Q and f/sub r/ are used to calculate an improved set of test frequencies, and so on. A rapid least-squares method is described which minimizes truncation errors.     

A simple and efficient design of variable fractional delay FIR filters

In this paper, a simple and efficient approach for designing one-dimensional variable fractional delay finite impulse response digital filters is proposed. Two matrix equations, based respectively on the weighted least-squares function of the optimum fixed fractional delay filter and the filter coefficient polynomial fitting, are formulated in tandem to form the design algorithm, which only has the computation complexity comparable with that of designing fixed finite impulse response digital filters. A design example is also given to justify the effectiveness and advantages of the proposed design method.