关于直线拟合的注记

讨论了使残差的最大绝对值为最小准则和使残差的绝对值之和为最小的准则的直线拟合问题 ,采用线性规划可解决此问题 .最后讨论了多项式拟合的情况 .     

大数据条件下同时拟合多条空间直线和平面的方法研究

为利用总体最小二乘准则将三维空间数据中包含的多条空间直线和平面原型拟合出来,提出了在大数据条件下的一种估计空间直线和空间平面方程的快速计算方法.首先研究了在总体最小二乘准则下待拟合的空间直线和平面的特殊理论性质,为提出简明算法打下基础,其次,根据期望最大化(Expectation Maximization,EM)算法实现数据的自动分割,得到属于每条直线和每个平面的三维数据,最后,在数据被污染而含有较多噪声时,可先用DBSCAN算法清洗数据,再用EM算法自动分割数据,拟合出数据中包含的多条空间直线和多个空间平面原型.计算机仿真结果表明,在数据没有遭到污染和污染比例不太高大数据中,算法都可以快速准确地确定出多条直线和平面原型.     

关于直线拟合的注记

讨论了使残差的最大绝对值为最小准则和使残差的绝对值之和为最小的准则的直线拟合问题,采用线性规划可解决此问题,最后讨论了多项式拟合的情况。     

非线性回归方法的应用与比较

比较了非线性回归3种方法的数学原理:曲线直线化方法、非线性最小二乘方法、近似非线性法.说明了用方差分析确定回归模型的统计学意义、用决定系数R2描述曲线的拟合效果的理论依据.通过对同一问题用3种方法分析得出结论:非线性回归与近似非线性拟合方法决定系数相近(0.9966与0.9965),而曲线直线化决定系数为0.9738.因为近似非线性拟合方法无需选初值.建议应用近似非线性拟合方法.     

最小一乘线性回归(上)

1.引言 为使读者易于了解方法的要旨,而不为繁复的记号与数学推导分散掉注意力,本篇中先集中讨论只含一个自变量的线性回归的情形. 设有自变量X和因变量Y,X、Y都是一维,例如,X、Y分别是从同一个人身上量出的身高和体重.现在对X、 Y进行了。次观测,得数据 (x1,y1),(x2,y2),…,(xn,yn)(1)要作一条直线x= a bx,“最好地”代表(1)中那些点的趋势,或者说,与(1)中那些点“拟合”得最好,怎么叫拟合最好?这就要看你给出怎样的准则,最常见的准则是“最小二乘”准则,这准则要求这样决定a和b,使(1)中各点沿十轴到直线y=a bx的偏离yi-(a bx;)(i= 1,…     

基于矩阵奇异值分解的空间直线拟合

讨论空间直线拟合问题,先对已知数据进行中心化处理得到矩阵,对此矩阵进行奇异值分解,直线方程中的系数可由相应的列向量确定.该拟合过程不需迭代,简单易行,结果也比原文更接近真实值.     

基于3σ准则的分段拟合及其GARCH修正模型

在金融时间序列中,一组金融序列可被视为由不同时间段的分段函数拟合连接而成.利用3σ准则确定分段函数的临界点,并根据AIC准则及调整后R2对分段点进行验证,从而分段点把数据分割成两部分.对两序列分别用合适的函数进行拟合,并用ARMA-GARCH模型对残差序列进行修正.由上证综合指数数据的实证分析结果表明:3σ准则能很好地检索出临界点,同时建立的分段函数模型预测效果要优于ARMA与EGARCH模型,以及ARMA-GARCH模型的引入对模型的精确度有所提高.所介绍的方法简单易懂、便于操作、精度高,为金融投资者和学者提供参考价值.     

数据拟合函数的最小二乘积分法

数据拟合的方法很多,每种方法各有特点.本文探讨了积分准则下的数据拟合函数的方法,称为最小二乘积分法,并给出了两个常用拟合函数具体形式.     

数据拟合函数的加权最小二乘积分法

数据拟合的方法很多,每种方法各有特点.探讨了积分准则下的数据加权拟合函数的方法,称为加权最小二乘积分法,并给出了三个常用拟合函数具体形式.     

总体最小二乘准则下非线性EV模型的参数拟合方法

在非线性回归模型参数拟合问题中,当数据中的每个变量都存在不可忽略的误差时,在普通的最小二乘准则下拟合出的参数不是最优的.按照总体最小二乘准则,以观测点到拟合曲线或拟合曲面垂直距离平方和为目标函数,然后用最优化方法搜索出使目标函数值取最小值的参数和数据点估计,从而给出求最优模型参数的算法,最后,通过计算机仿真和与文献比较,验证了提出方法的正确性.     

Jump-preserving surface reconstruction from noisy data

A new local smoothing procedure is suggested for jump-preserving surface reconstruction from noisy data. In a neighborhood of a given point in the design space, a plane is fitted by local linear kernel smoothing, giving the conventional local linear kernel estimator of the surface at the point. The neighborhood is then divided into two parts by a line passing through the given point and perpendicular to the gradient direction of the fitted plane. In the two parts, two half planes are fitted, respectively, by local linear kernel smoothing, providing two one-sided estimators of the surface at the given point. Our surface reconstruction procedure then proceeds in the following two steps. First, the fitted surface is defined by one of the three estimators, i.e., the conventional estimator and the two one-sided estimators, depending on the weighted residual means of squares of the fitted planes. The fitted surface of this step preserves the jumps well, but it is a bit noisy, compared to the conventional local linear kernel estimator. Second, the estimated surface values at the original design points obtained in the first step are used as new data, and the above procedure is applied to this data in the same way except that one of the three estimators is selected based on their estimated variances. Theoretical justification and numerical examples show that the fitted surface of the second step preserves jumps well and also removes noise efficiently. Besides two window widths, this procedure does not introduce other parameters. Its surface estimator has an explicit formula. All these features make it convenient to use and simple to compute.     

A-acceptable exponentially fitted combinations of three Padé approximations

It is considered whether a linear combination of three A-acceptable Padé approximations to the exponential function remains A-acceptable when it is exponentially fitted to two distinct negative points. The results of Iserles [5] regarding linear combinations of two A-acceptable Padé approximations are generalized.     

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution

Starting from the question: What is the accident risk of an insured individual?, we consider that the customer has contracted policies in different insurance lines: motor and home. Three models based on the multivariate Sarmanov distribution are analyzed. Driven by a real data set that takes into account three types of accident risks, two for motor and one for home, three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals are considered and fitted to the data. To estimate the parameters of these three models, we discuss a method for approaching the maximum likelihood (ML) estimators. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM and with elliptical copula based models.     

带有不可忽略缺失数据的广义部分线性模型的贝叶斯分析

广义部分线性模型是广义线性模型和部分线性模型的推广,是一种应用广泛的半参数模型.本文讨论的是该模型在线性协变量和响应变量均存在非随机缺失数据情形下参数的Bayes估计和基于Bayes因子的模型选择问题,在分析过程中,采用了惩罚样条来估计模型中的非参数成分,并建立了Bayes层次模型;为了解决Gibbs抽样过程中因参数高度相关带来的混合性差以及因维数增加导致出现不稳定性的问题,引入了潜变量做为添加数据并应用了压缩Gibbs抽样方法,改进了收敛性;同时,为了避免计算多重积分,利用了M-H算法估计边缘密度函数后计算Bayes因子,为模型的选择比较提供了一种准则.最后,通过模拟和实例验证了所给方法的有效性.     

A note on the exponential fitting of blended,extended linear multistep methods

A class of exponentially fitted blended, extended linear multistep methods is investigated andA-stable formulae of order 3, 4 and 5 are derived.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

LPV identification of a turbocharged diesel engine

This work proposes a methodology of identifying linear parameter varying (LPV) models for nonlinear systems. First, linear local models in some operating points, by applying standard identifications procedures for linear systems in time domain, are obtained. Next, a LPV model with linear fractional dependence (LFR) with respect to measured variables is fitted with the condition of containing all the linear models identified in previous step (differential inclusion). The fit is carried out using nonlinear least squares algorithms. Finally, this identification methodology will then be applied to a nonlinear turbocharged diesel engine.     

Limiting Distributions of Linear Programming Estimators

Smith (1994) proposes estimation in linear regression models with non-negative errors by maximizing the sum of fitted values subject to the constraint that the fitted values can be no larger than the corresponding response value. In this paper, we consider the limiting distribution of these estimators under very general conditions. Some extensions to local polynomial estimation are also considered.     

Exponentially fitted multistep methods by generalized Hermite-Birkhoff interpolation

Generalized Hermite-Birkhoff interpolation gives a unifying method for constructing exponentially fitted linear multistep methods explicitly. Fitting may be done at any set of given (real) parameters. The construction principle permits study of the convergence and stability properties of these methods. Moreover, it shows how the properties depend on the corresponding properties of classical linear multistep methods with constant coefficients. The theoretical results are illustrated by some examples.     

The High Order Exponentially Fitted Nonequidistant Extrapolation Methods for Stiff Systems

A class of exponentially fitted nonequidistant extrapolation methods based on an L-stable linear single-step formula are studied. Theorems about the extrapolation coefficients are given. These methods keep good numerical stability "quasi-L-stability", while raising the accuracy of the original formula.