Parameter estimation for generalized Dirichlet distributions from the sample estimates of the first and the second moments of random variables

Generalized Dirichlet distributions have a more flexible covariance structure than Dirichlet distributions, and the computation for the moments of a generalized Dirichlet distribution is still tractable. For situations under which Dirichlet distributions are inappropriate for data analysis, generalized Dirichlet distributions will generally be an applicable alternative. When the expected values and the covariance matrix of random variables can be estimated from available data, this study introduces ways to estimate the parameters of a generalized Dirichlet distribution for analyzing compositional data. Under the assumption that the sample mean of every variable must be considered for parameter estimation, we present methods for choosing the statistics from a sample covariance matrix to construct a generalized Dirichlet distribution. Some rules for removing inappropriate statistics from a sample covariance matrix to speed up the estimation process are also established. An example for Taiwan’s car market is introduced to demonstrate the applicability of the parameter estimation methods.     

杂波协方差矩阵结构的融合估计方法

针对特定杂波背景下的最优或次优杂波协方差矩阵估计方法难以适应过渡杂波环境的问题,提出协方差矩阵结构的融合估计方法,通过调整参数涵盖现有的3种杂波协方差矩阵估计方法,并分析所提出方法对应的自适应归一化匹配滤波器的自适应特性.其次,确定了控制参数的经验公式,经验公式符合数值结果.最后,从估计精度、恒虚警率特性和检测性能3个方面对所提出方法和已有方法进行对比分析.仿真结果表明,在过渡杂波环境中,所提出方法的精度更高、检测效果更好,对实际杂波非高斯程度时空渐变性具有较强的适应能力.     

Estimation of the matrices of parameters and covariations of the perturbation vectors in multidimensional discrete-time dynamic systems under special structure of the unknown covariance matrices

For the multidimensional linear dynamic system obeying a difference equation with an unknown covariance matrix of the vector of random perturbations having dependent components, consideration was given to estimation of the matrix of system parameters and the covariance matrix represented by a linear combination of the given symmetrical matrices. The family of joint probability densities of the observation vector was factorized, and the sufficient statistics was determined. For the estimates of the maximum likelihood of the system parameter matrices and the estimates of the coefficients of expansion of the covariance matrix, equations were presented. Developed was a recurrent procedure for joint estimation of the system parameter matrices and the covariance matrix with arrival of observations.     

Analysis of the Kalman filter based estimation algorithm: an orthogonal decomposition approach

In this paper we shall provide new analysis on some fundamental properties of the Kalman filter based parameter estimation algorithms using an orthogonal decomposition approach based on the excited subspace. A theoretical analytical framework is established based on the decomposition of the covariance matrix, which appears to be very useful and effective in the analysis of a parameter estimation algorithm with the existence of an unexcited subspace. The sufficient and necessary condition for the boundedness of the covariance matrix in the Kalman filter is established. The idea of directional tracking is proposed to develop a new class of algorithms to overcome the windup problem. Based on the orthogonal decomposition approach two kinds of directional tracking algorithms are proposed. These algorithms utilize a time-varying covariance matrix and can keep stable even in the case of unsufficient and/or unbounded excitation.     

Identification of a Class of Linear Time—Varying Continuous—Time Systems

This paper deals with the identification of linear time—varying continuous—time systems whose parameters have bounded first-order time-derivative under square-integrable output noise. The proposed parameter updating algorithm is one of -modification type with Covariance Matrix. The Covariance Matrix updating rule is performed by using a modification with respect to standard previous estimation schemes used for time-invariant plants. Such a modification consists of including, apart from the standard ones in least-squares estimation with forgetting factor, two extra additive weighted terms involving the identity matrix as well as the second power of such a matrix with the appropriate signs. That parameter estimation strategy ensures that the norm of the covariance matrix is always finitely upper-bounded without using adaptation freezing techniques over a prescribed norm threshold. The main role played by the -modification relies on introducing a penalty for large values of the covariance inverse and the parametrical error. Furthermore, in the most general formalism setting, the knowledge of absolute upper-bounds on either the parameter vector or its time-derivative are not requested. The estimation scheme guarantees the asymptotic convergence of the prediction error to zero and that of the estimates to the true parameters under standard properties of persistent excitation of the input and controllability of both plant and filter realizations in the case when the plant is noise-free and either time-invariant after finite time asymptotically time-invariant. If the plant is slowly time-varying and the covariance matrix is updated with the standard least-squares rule, the variations of the integrals of the squares of the estimation vector and its time-derivative norms vary not faster than uniformly linearly with the interval lengths considered for integration. Furthermore, the overall changes are arbitrarily small over any finite time-interval provided that the time-derivative of the true parameter vector and the correction coefficients in the covariance updating rule are arbitrarily small.     

参数估计的Systolic算法

本文根据最小二乘原理在三角形Ｓｙｓｔｏｌｉｃ阵列上实现了单输入单输出系统的递推参数估计算法，首先利用矩阵的三角分解给出了待估参数及协方差阵的递推公式，然后利用正交平面旋转并结合三角形Ｓｙｓｔｏｌｉｃ阵列的特点给出了相应的Ｓｙｓｔｏｌｉｃ递推参数估计算法，最后还考虑了算法实现时的性能指标，其后是一些数值仿真结果，由于文中利用了正交平面旋转，因而所得算法是数值稳定的。     

A Gaussian pseudolikelihood approach for quantile regression with repeated measurements

To enhance the efficiency of regression parameter estimation by modeling the correlation structure of correlated binary error terms in quantile regression with repeated measurements, we propose a Gaussian pseudolikelihood approach for estimating correlation parameters and selecting the most appropriate working correlation matrix simultaneously. The induced smoothing method is applied to estimate the covariance of the regression parameter estimates, which can bypass density estimation of the errors. Extensive numerical studies indicate that the proposed method performs well in selecting an accurate correlation structure and improving regression parameter estimation efficiency. The proposed method is further illustrated by analyzing a dental dataset.     

Parameter estimation of the 2D‐GTD model and RCS reconstruction based on an improved 2D‐ESPRIT algorithm

The two‐dimensional estimating signal parameter via rotational invariance techniques (2D‐ESPRIT) algorithm is a classical method to estimate parameters of the two‐dimensional geometric theory of diffraction (2D‐GTD) model. While as signal‐to‐noise‐ratio (SNR) decreases, the parameter estimation performance of 2D‐ESPRIT algorithm is severely influenced. To solve this problem, a performance‐enhanced 2D‐ESPRIT algorithm is proposed in this article. The improved 2D‐ESPRIT algorithm combines the conjugate data with the original back‐scattered data and obtains a novel covariance matrix by squaring the original total covariance matrix. Simulation results indicate that the improved algorithm has a better noise robustness and a more stable parameter estimation performance than the classical ESPRIT algorithm and the classical TLS‐2D‐ESPRIT algorithm. To further validate the superiority of the improved 2D‐ESPRIT algorithm, reconstructed radar cross section (RCS) is presented in this article. Compared with the classical 2D‐ESPRIT algorithm, the proposed algorithm presents higher RCS fitting precision. Furthermore, the impacts of other factors on parameter estimation, such as matrix pencil parameters and paring parameters, are also studied in this article.     

Efficient semiparametric estimation via Cholesky decomposition for longitudinal data

Semiparametric methods for longitudinal data with dependence within subjects have recently received considerable attention. Existing approaches that focus on modeling the mean structure require a correct specification of the covariance structure as misspecified covariance structures may lead to inefficient or biased mean parameter estimates. Besides, computation and estimation problems arise when the repeated measurements are taken at irregular and possibly subject-specific time points, the dimension of the covariance matrix is large, and the positive definiteness of the covariance matrix is required. In this article, we propose a profile kernel approach based on semiparametric partially linear regression models for the mean and model covariance structures simultaneously, motivated by the modified Cholesky decomposition. We also study the large-sample properties of the parameter estimates. The proposed method is evaluated through simulation and applied to a real dataset. Both theoretical and empirical results indicate that properly taking into account the within-subject correlation among the responses using our method can substantially improve efficiency.     

Linear minimum variance estimation fusion

This paper shows that a general multisensor unbiased linearly weighted estimation fusion essentially is the linear minimum variance (LMV) estimation with linear equality constraint, and the general estimation fusion formula is developed by extending the Gauss-Markov estimation to the random parameter under estimation. First, we formulate the problem of distributed estimation fusion in the LMV setting. In this setting, the fused estimator is a weighted sum of local estimates with a matrix weight. We show that the set of weights is optimal if and only if it is a solution of a matrix quadratic optimization problem subject to a convex linear equality constraint. Second, we present a unique solution to the above optimization problem, which depends only on the covariance matrix Ck.Third, if a priori information, the expectation and covariance, of the estimated quantity is unknown, a necessary and sufficient condition for the above LMV fusion becoming the best unbiased LMV estimation with known prior informatio     

利用Voronoi协方差矩阵重建隐式曲面

目的 针对含少量离群点的噪声点云,提出了一种Voronoi协方差矩阵的曲面重建方法。方法 以隐函数梯度在Voronoi协方差矩阵形成的张量场内的投影最大化为目标,构建隐函数微分方程,采用离散外微分形式求解连续微分方程,从而将曲面重建问题转化为广义特征值求解问题。在点云空间离散化过程中,附加最短边约束条件,避免了局部空间过度剖分。并引入概率测度理论定义曲面窄带,提高了算法抵抗离群点能力,通过精细剖分曲面窄带,提高了曲面重建精度。结果 实验结果表明,该算法可以抵抗噪声点和离群点的影响,可以生成不同分辨率的曲面。通过调整拟合参数,可以区分曲面的不同部分。结论 提出了一种新的隐式曲面重建方法,无需点云法向、稳健性较强,生成的三角面纵横比好。     

Inference about clustering and parametric assumptions in covariance matrix estimation

Selecting an estimator for the covariance matrix of a regression’s parameter estimates is an important step in hypothesis testing. From less to more robust estimators, the choices available to researchers include Eicker/White heteroskedasticity-robust estimator, cluster-robust estimator, and multi-way cluster-robust estimator. The rationale for choosing a less robust covariance matrix estimator is that tests conducted using this estimator can have better power properties. This motivates tests that examine the appropriate level of robustness in covariance matrix estimation. In this paper, we propose a new robustness testing strategy, and show that it can dramatically improve inference about the proper level of robustness in covariance matrix estimation. In an empirically relevant example, namely the placebo treatment application of Bertrand, Duflo and Mullainathan (2004), the power of the proposed robustness testing strategy against the null hypothesis “no clustering” is 0.82 while the power of the existing robustness testing approach against the same null is only 0.04. We also show why the existing clustering test and other applications of the White (1980) robustness testing approach often perform poorly, which to our knowledge has not been well understood. The insight into why this existing testing approach performs poorly is also the basis for the proposed robustness testing strategy.     

The optimal filtering problem for a discrete-time distributed parameter system

In this paper, the optimal filtering problem for a discrete-time linear distributed parameter system is considered. Using the least squares estimation error criterion, the Wiener-Hopf equation for the discrete-time distributed parameter system is derived. Based on the Wiener-Hopf equation, the equations satisfied by the optimal filtering estimate and the minimum error covariance matrix function are derived by using the matrix inversion lemma for a distributed parameter system. Finally, we show that the approximation of the results obtained for a distributed parameter system by using the Fourier expansion method produces those of the Kalman filtering problem for the lumped parameter system.     

Adaptive divided difference filtering for simultaneous state and parameter estimation

A novel adaptive version of the divided difference filter (DDF) applicable to non-linear systems with a linear output equation is presented in this work. In order to make the filter robust to modeling errors, upper bounds on the state covariance matrix are derived. The parameters of this upper bound are then estimated using a combination of offline tuning and online optimization with a linear matrix inequality (LMI) constraint, which ensures that the predicted output error covariance is larger than the observed output error covariance. The resulting sub-optimal, high-gain filter is applied to the problem of joint state and parameter estimation. Simulation results demonstrate the superior performance of the proposed filter as compared to the standard DDF.     

MLMATERN: A computer program for maximum likelihood inference with the spatial Matérn covariance model

The Matérn covariance scheme is of great importance in many geostatistical applications where the smoothness or differentiability of the random field that models a natural phenomenon is of interest. In addition to the range and nugget parameters, the flexibility of the Matérn model is provided by the so-called smoothness parameter which controls the degree of smoothness of the random field. It has been the usual practice in geostatistics to fit theoretical semivariograms like the spherical or exponential, thus implicitly assuming the smoothness parameter to be known, without questioning if there is any theoretical or empirical basis to justify such assumption. On the other hand, if only a small number of sparse experimental data are available, it is more critical to ask if the smoothness parameter can be identified with statistical reliability. Maximum likelihood estimation of spatial covariance parameters of the Matérn model has been used to address the previous questions. We have developed a general algorithm for estimating the parameters of a Matérn covariance (or semivariogram) scheme, where the model may be isotropic or anisotropic, the nugget variance can be included in the model if desired, and the uncertainty of the estimates is provided in terms of variance–covariance matrix (or standard error-coefficient of correlation matrix) as well as likelihood profiles for each parameter in the covariance model. It is assumed that the empirical data are a realization of a Gaussian process. Our program allows the presence of a polynomial trend of order zero (constant global mean), one (linear trend) or two (quadratic trend). The restricted maximum likelihood method has also been implemented in the program as an alternative to the standard maximum likelihood. Simulation results are given in order to investigate the sampling distribution of the parameters for small samples. Furthermore, a case study is provided to show a real practical example where the smoothness parameter needs to be estimated.     

关于鞅超收敛定理与遗忘因子最小二乘算法的收敛性分析

鞅超收敛定理是研究随机时变系统辨识算法有界收敛性的一个有效数学工具,它是鞅收益是在随机时变系统中的推广。文「１」用它证明了遗忘因子最小二乘算法参数估计误差的有界收敛性,但是文「１」假设系统的理各态遍历的,且协方差阵是用它的数学期望代替的,所得到的结果是近似的。而本文精确地给出了协方差阵的上下界,改进了文「１」的结果。     

Parameter identification in a class of nonlinear systems

Two approaches are proposed for on-line identification of parameters in a class of nonlinear discrete-time systems. The system is modeled by state equations in which state and input variables enter nonlinearly in general polynomial form, while unknown parameters and random disturbances enter linearly. All states and inputs must be observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite, but unknown, covariance. One method of parameter estimation is based upon a least squares approach, the second is a related stochastic approximation algorithm (SAA). Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability 1. Examples illustrate convergence of the LSA which even in recursive form requires inversion of a matrix at each step. The SAA requires no matrix inversions, but experience with the algorithm indicates that convergence is slow relative to that of the LSA.     

基于低秩矩阵恢复的DOA稀疏重构方法

为提高非均匀噪声下波达方向（direction of arrival,DOA）角估计算法的估计精度和分辨率,基于低秩矩阵恢复理论,提出了一种二阶统计量域下的加权L1稀疏重构DOA估计算法。该算法基于低秩矩阵恢复方法,引入弹性正则化因子将接收信号协方差矩阵重构问题转换为可获得高效求解的半定规划（semidefinite programming,SDP）问题以重构无噪声协方差矩阵;而后在二阶统计量域下利用稀疏重构加权L1范数实现DOA参数估计。数值仿真表明,与传统MUSIC、L1-SVD及加权L1算法相比,所提算法能显著抑制非均匀噪声影响,具有较好的DOA参数估计性能,且在低信噪比条件下,所提算法具有较高的角度分辨力和估计精度。     

Active fault tolerant control using state-space self-tuning control approach

This paper presents a new fault tolerant control scheme for unknown multivariable stochastic systems by modifying the conventional state-space self-tuning control approach. For the detection of faults, a quantitative criterion is developed by comparing the innovation process errors occurring in the Kalman filter estimation algorithm, which, for faulty system recovery, a weighting matrix resetting technique is developed by adjusting and resetting the covariance matrices of the parameter estimate obtained in the Kalman filter estimation algorithm to improve the parameter estimation of the faulty systems. The proposed method can effectively cope with partially abrupt and/or gradual system faults and/or input failures with fault detection. The modified state-space self-tuning control scheme can be applied to the multivariable stochastic faulty system without requiring prior knowledge of system parameters and noise properties.     

New scheme for discrete observer design with estimation error covariance assignment

This article presents an innovative method for solving an estimation error covariance assignment problem to design an observer for a stochastic linear system. In the proposed method, the covariance assignment problem is converted to the problem of finding an extra noise-like input to the observer. Using appropriate matrix manipulation, the Riccati equation of the estimation error covariance assignment problem, is converted to a new deterministic linear state-space model. Also, the extra noise-like input to the observer is modelled as an input to the new deterministic linear state-space model. Therefore, all the conventional and well-defined control strategies could be applied and there is no need to solve a complicated Riccati equation. Moreover, using the proposed method, a multi-objective estimation error covariance tracking problem would be easily converted to the problem of controlling a standard deterministic linear state-space system. Based on the integral control method, which is applied to the new state-space model, formulations for the proposed covariance feedback law are presented. The control law results in a stable closed-loop covariance system and assigns a pre-specified covariance matrix to the estimation errors.