应用SAS解非线性回归问题

.应用SAS/STAT估计非线性回归模型中的参数.首先,通过变量代换,把可以线性化的非线性回归模型化为线性回归模型,并用普通最小二乘法、主成分分析法和偏最小二乘法求模型中的参数和回归模型.其次,通过改良的高斯—牛顿迭代法来估计Logistic模型和Compertz模型中的参数.     

线性子空间上求解AX=B的最小二乘问题的迭代算法

应用共轭梯度方法和线性投影算子,给出迭代算法求解了线性矩阵方程AX=B在任意线性子空间上的最小二乘解问题.在不考虑舍入误差的情况下,可以证明,所给迭代算法经过有限步迭代可得到矩阵方程AX=B的最小二乘解、极小范数最小二乘解及其最佳逼近.文中的数值例子证实了该算法的有效性.     

捕食模型高精度参数的估计

研究了捕食者模型在多种观测值条件下的非线性微分方程组参数拟合问题.首先利用龙格-库塔法进行微分方程数值计算,通过首次积分项变形建立线性回归方程,进行最小二乘拟合;其次,考虑到实验数据包含随机误差的扰动,引进正规方程组对模型进行误差分析;最后针对时间变量也出现误差,采用拉依达准则筛选,然后提出了一种较为简单的参数分段动态估计算法.     

生态系统的参数确定问题

通过不同观测数据研究捕食者—被捕食者生态系统参数确定问题.研究了四种情形1.观察数据无误差,并已知一个参数值.这种情况下,参数可由其相轨线和最小二乘法精确确定.2.观察数据无误差,但所有参数未知.此时仅靠相轨线的研究,无论给出多少组精确数据,都无法精确确定这些参数.通过原非线性模型的数值计算和网格搜索法,至少需要4组数据,同样得到了精度较高的参数值.3.当观测数据有误差时,根据解的周期性,引入标准周期的概念,在一个标准周期里讨论参数的确定问题,并利用标准周期内的捕食者与被捕食者的数量均值与系统的平衡点的关系对参数进行修正,然后使用网格法进行搜索,进一步提高了参数的精度.4.当观测时间也有误差时,先选取相对最优的随机正态数对观测时刻进行修正,然后再利用3.的方法估计参数.     

混合系数线性模型参数的广义Liu估计

本文研究连续测量数据情况下的混合系数线性模型的参数估计问题.利用压缩估计方法给出了该模型的一类新的有偏估计-广义Liu估计,并在均方误差意义下,证明此类估计分别优于最小二乘估计、Liu估计.最后讨论参数的选取问题.     

半参数回归模型的一类压缩估计

研究了半参数回归模型的参数估计问题,利用压缩估计方法给出了模型的一类有偏估计,并与最小二乘估计、岭估计、几乎无偏岭估计进行了比较.在均方误差意义下,新的压缩估计明显优于最小二乘估计.最后讨论了有偏参数选取的问题.     

纵向数据半参数回归模型的估计方法

本文考虑纵向数据半参数回归模型,通过考虑纵向数据的协方差结构,基于Profile最小二乘法和局部线性拟合的方法建立了模型中参数分量、回归函数和误差方差的估计量,来提高估计的有效性,在适当条件下给出了这些估计量的相合性.并通过模拟研究将该方法与最小二乘局部线性拟合估计方法进行了比较,表明了Profile最小二乘局部线性拟合方法在有限样本情况下具有良好的性质.     

矩阵方程AX＋XB=C的双对称解及其最佳逼近

提出一种求解线性矩阵方程AX＋XB=C双对称解的迭代法.该算法能够自动地判断解的情况,并在方程相容时得到方程的双对称解,在方程不相容时得到方程的最小二乘双对称解.对任意的初始矩阵,在没有舍入误差的情况下,经过有限步迭代得到问题的一个双对称解.若取特殊的初始矩阵,则可以得到问题的极小范数双对称解,从而巧妙地解决了对给定矩...     

半参数回归模型的迭代法

在补偿最小二乘法则下,采用迭代法考虑半参数回归模型Li=Ai^TX＋s（ti）＋△i（i=1,2,……n,）得到参数及非参数的估计;接着从理论上证明了该法的可行性,并给出了误差上界及确定迭代的最大次数;最后用模拟的算例说明该法的有效性．     

约束下线性混合模型参数估计的渐近解及其收敛性

在线性不等式约束下讨论了具有相同参数的两个线性混合模型的参数估计问题,给出了一种迭代算法,得到了这类模型中参数的最小二乘估计序列及其渐近解.在此基础上,利用多元多项式方程组解的个数定理和不动点定理,证明了此估计序列是依概率1收敛的.     

复共线联立方程模型参数的修正间接岭估计

在结构方程恰好被识别时,研究了外生变量设计矩阵X复共线时联立方程模型的参数估计问题,提出了参数的一种修正间接岭估计方法,并证明了这种参数估计的良好统计性质,最后给出了在修正间接岭估计均方误差最小意义下岭参数的一种选择方法.     

确定Lotka-Volterra生态系统模型高精度参数的研究

研究确定Lotka-Volterra生态系统模型的高精度参数估计问题.利用周期性,先对测量数据进行预处理;然后用三种不同的方法构造了误差函数,进行非线性最小二乘法参数估计;再用计算机仿真对其进行验证.结果表明该方法能够有效地解决高精度参数估计中消除测量数据误差的问题.     

Pareto分布形状参数的E-Bayes估计和多层Bayes估计及其应用

给出了参数的E-Bayes估计的定义,对Pareto分布在尺度参数已知时,在平方损失下给出了形状参数的E-Bayes估计和多层Bayes估计,并且用Monte Carlo方法给出了模拟算例.最后,结合高尔夫球手收入数据的实际问题进行了计算,结果表明本文提出的方法可行且便于应用.     

A random model approach for the LASSO

The least absolute selection and shrinkage operator (LASSO) is a method of estimation for linear models similar to ridge regression. It shrinks the effect estimates, potentially shrinking some to be identically zero. The amount of shrinkage is governed by a single parameter. Using a random model formulation of the LASSO, this parameter can be specified as the ratio of dispersion parameters. These parameters are estimated using an approximation to the marginal likelihood of the observed data. The observed score equations from the approximation are biased and hence are adjusted by subtracting an empirical estimate of the expected value. After estimation, the model effects can be tested (via simulation) as the distribution of the observed data given that all model effects are zero is known. Two related simulation studies are presented that show that dispersion parameter estimation results in effect estimates that are competitive with other estimation methods (including other LASSO methods).     

带缺失数据的偏正态众数回归模型的参数估计

针对现实生活中大量数据存在偏斜的情况,构建偏正态数据下的众数回归模型.又加之数据的缺失常有发生,采用插补方法处理缺失数据集,为比较插补效果,考虑对响应变量随机缺失情形进行统计推断研究.利用高斯牛顿迭代法给出众数回归模型参数的极大似然估计,比较该模型在均值插补,回归插补,众数插补三种插补条件下的插补效果.随机模拟和实例分...     

变系数部分线性模型的加权随机约束s-K估计

研究随机约束条件下半参数变系数部分线性模型的参数估计问题,当回归模型线性部分变量存在多重共线性时,基于Profile最小二乘方法、s-K估计和加权混合估计构造参数向量的加权随机约束s-K估计,并在均方误差矩阵准则下给出新估计量优于s-K估计和加权混合估计的充要条件,最后通过蒙特卡洛数值模拟验证所提出估计量的有限样本性质.     

A parameter estimation scheme for a class of sequential hybrid systems

It may happen that the equations governing the response of dynamical systems have some parameters whose values may not be known a priori and have to be obtained using parameter estimation schemes. In this article, we present a parameter estimation scheme for a class of sequential hybrid systems. By hybrid systems, we refer to those systems whose response is described by different governing equations corresponding to various regimes/modes of operation along with some criteria to switch between the same. In a sequential hybrid system, the different modes are arranged in a specific sequence and the system can switch from a given mode to either the previous mode or the following mode in this sequence. Here, we consider those systems whose governing equations consist of ordinary differential equations and algebraic equations. The conditions for switching between the various modes (referred to as transition conditions) are in the form of linear inequalities involving the system output. We shall first consider the case where the transition conditions are known completely. We present a parameter update scheme along with sufficient conditions that will guarantee bounded parameter estimation errors. Then, we shall consider the case where the transition conditions are not known in the sense that some parameters in these conditions are not known. We present a parameter estimation scheme for this case. We illustrate the performance of the parameter estimation scheme in both cases with some examples.     

Parameter estimation of delay differential equations: An integration-free LS-SVM approach

This paper introduces an estimation method based on Least Squares Support Vector Machines (LS-SVMs) for approximating time-varying as well as constant parameters in deterministic parameter-affine delay differential equations (DDEs). The proposed method reduces the parameter estimation problem to an algebraic optimization problem. Thus, as opposed to conventional approaches, it avoids iterative simulation of the given dynamical system and therefore a significant speedup can be achieved in the parameter estimation procedure. The solution obtained by the proposed approach can be further utilized for initialization of the conventional nonconvex optimization methods for parameter estimation of DDEs. Approximate LS-SVM based models for the state and its derivative are first estimated from the observed data. These estimates are then used for estimation of the unknown parameters of the model. Numerical results are presented and discussed for demonstrating the applicability of the proposed method.     

Consistent least squares fitting of ellipsoids

Summary. A parameter estimation problem for ellipsoid fitting in the presence of measurement errors is considered. The ordinary least squares estimator is inconsistent, and due to the nonlinearity of the model, the orthogonal regression estimator is inconsistent as well, i.e., these estimators do not converge to the true value of the parameters, as the sample size tends to infinity. A consistent estimator is proposed, based on a proper correction of the ordinary least squares estimator. The correction is explicitly given in terms of the true value of the noise variance.Mathematics Subject Classification (2000): 65D15, 65D10, 15A63Revised version received August 15, 2003     

A truncated estimation method with guaranteed accuracy

This paper presents a truncated estimation method of ratio type functionals by dependent sample of finite size. This method makes it possible to obtain estimators with guaranteed accuracy in the sense of the $L_m$ -norm, $m\ge 2$ . As an illustration, the parametric and non-parametric estimation problems on a time interval of a fixed length are considered. In particular, parameters of linear (autoregressive) and non-linear discrete-time processes are estimated. Moreover, the parameter estimation problem of non-Gaussian Ornstein-Uhlenbeck process by discrete-time observations and the estimation problem of a multivariate logarithmic derivative of a noise density of an autoregressive process with guaranteed accuracy are solved. In addition to non-asymptotic properties, the limit behavior of presented estimators is investigated. It is shown that all the truncated estimators have asymptotic properties of basic estimators. In particular, the asymptotic efficiency in the mean square sense of the truncated estimator of the dynamic parameter of a stable autoregressive process is established.