广义系统Wiener 滤波和Kalman 滤波新方法*

应用时域上的现代时间序列分析方法,基于ＡＲＭＡ新息模型和白噪声估计理论,提出了广义系统的Ｗｉｅｎｅｒ状态估值器和急剧记Ｋａｌｍａｎ估值器。它们可统一处理最优滤波,平滑和预后问题。     

An innovations approach to least-squares estimation--Part VII: Some applications of vector autoregressive-moving average models

We use the innovations method to solve some linear estimation problems for stochastic processes described as the solution of high-order linear difference equations driven by colored noise. Such models are often called vector or multivariable auto-regressive-moving average (ARMA) models. We illustrate how the use of ARMA models can provide some simplifications and some new results in the problem of state estimation in colored noise.     

广义系统Wiener状态滤波新算法

应用时域上的现代时间序列分析方法，基于ARMA新息模型和白噪声估计理论，由一种新的非递推最优状态估值器的递推变形，提出了广义系统Wiener状态滤波的一种新算法，它可统一处理滤波、平滑和预报问题，且具有渐近稳定性。同某些算法相比，它避免了求解Riccati方程和Diophantine方程，且避免了计算伪逆，因而减小了计算负担。仿真例子说明了其有效性。     

On the estimation of two-dimensional moving average parameters

The authors address the estimation problem of moving average (MA) parameters of a 2D autoregressive moving average (ARMA) model. The problem is equivalent to solving a set of overdetermined 2D transcendental equations. Based on some extensions of the Newton-Raphson method, an iterative algorithm is proposed for estimating 2D MA parameters. The performance of the algorithm is demonstrated by a numerical example. For 2D sinusoids in white noise spatial series, some interesting features of 2D ARMA modeling are observed     

On the application of the Wald statistic to order estimation ofARMA models

Consistent criteria for order estimation of autoregressive moving-average (ARMA) processes based on the Wald statistic are presented. The new criteria require only the estimation of the model parameters at the largest order, unlike alternative methods in the literature that require the estimation of the model parameters at all possible orders     

Spectral domain estimation of autoregressive moving-average models using shuttle iterative methods

An efficient iterative algorithm for the estimation of autoregressive moving-average (ARMA) time series models is presented. The prediction error criterion is transformed into the spectral domain, and the autoregressive (AR) parameters are computed (inner loop) using the Newton iterative steps by fixing the AR parameters. Use of data compression in the form of signal power spectrum (fewer frequency points compared to large data size), application of linear solution methods, and the ease of closed form computation of gradients and Hessian matrix of the optimality criterion result in an efficient estimation algorithm for systems with varied spectral forms. The algorithm is implemented on a PDP 11/35 mini computer, and may be used for on-line monitoring and diagnostics of dynamic processes (nuclear power reactors, chemical industry processes), pattern recognition systems, and evaluation of sensor response time characteristics.     

Uniquely identifiable state-space and ARMA parametrizations for multivariable linear systems

Multivariable systems can be represented, in a uniquely identifiable way, either by canonical forms or by so-called overlapping forms. The advantage of the latter is that they do not require the a priori estimation of a set of structural invariants (e.g. Kronecker invariants). We show here how to define uniquely identifiable overlapping parametrizations for state-space and ARMA models. We show that these parametrizations are all related to a set of intrinsic invariants, which are obtained from the Markov parameters of the system. Different forms of overlapping ARMA parametrizations are derived and their properties discussed.     

具有随机多丢包的ARMA信号最优线性估计

研究网络环境下具有随机丢包的自回归滑动平均(ARMA)信号的估计问题,其中丢包现象通过一个满足Bernoulli分布的随机变量描述.通过ARMA模型与状态空间模型的转化,将具有丢包的ARMA信号估计问题转化为具有丢包的状态空间模型的状态和白噪声估计问题.利用射影理论分别给出线性最小方差最优线性状态估值器和白噪声估值器,进而获得ARMA信号估值器.仿真结果表明,当存在数据丢失时,所提出的算法与以往基于完整数据的最优估计算法相比具有最优性和有效性.     

Wiener状态去卷滤波器

应用现代时间序列分析方法,基于ARMA新息模型提出了一类带多重观测滞后和带滑动平均（MA）有色观测噪声系统的Wiener状态去卷滤波器,它具有渐近稳定性和ARMA递推形式,可统一处理滤波、平滑和预报问题,且可用于解决带ARMA有色观测噪声系统状态估计和信号Wiener滤波与反卷积问题,二个仿真例子说明了其有效性。     

A method for adaptive estimation of ARMA processes

When the noise process in adaptive identification of linear stochastic systems is correlated, and can be represented by a moving average model, extended least squares algorithms are commonly used, and converge under a strictly positive real (SPR) condition on the noise model. In this paper, we present an adaptive algorithm for the estimation of autoregressive moving average (ARMA) processes, and show that it is convergent without any SPR condition, and has a convergence rate of O({loglog t)/t}^1/2).     

State and parameter estimation for dynamic systems with colorednoise: an accuracy approach based on the minimum discrepancymeasure

An autoregressive moving average (ARMA) order-determination criterion based on a minimum discrepancy measure (MDM) is proposed to model colored measurement noise. Based on this criterion, an algorithm is presented to perform the state and parameter estimation. The criterion is consistent in the sense that the MDM decision rule will choose the correct order as the number of observations tends to infinity. Experimental results are included to illustrate the effectiveness of this technique     

带白色和有色观测噪声系统解耦Wiener状态估值器

基于经典稳态Kalman滤波理论, 对带白色和有色观测噪声系统提出了设计最优Wiener状态估值器的新方法. 通过稳态Kalman滤波器建立ARMA新息模型, 由稳态最优非递推Kalman状态估值器的递推变形引出Wiener状态估值器, 可统一处理滤波、预报和平滑问题, 它们具有状态解耦的ARMA递推形式, 且具有渐近稳定性和最优性, 仿真结果表明了算法的有效性.     

Stochastic approximation algorithms for identifying ARMA processes

Following the convergence proofs for stochastic approximation identification of pure autoregressive (AR) processes with dependent observations, as derived by Saridis and Stein, it is shown that the convergence for mixed autoregressive-moving-average (ARMA) cases can also be proved when none of the AR or the MA parameters or of the covariances are assumed known. Consequently, a generalized stochastic approximations identification procedure for ARMA processes is derived, which is extendable to any linear Kalrman filter models.     

MEMS陀螺随机漂移的状态空间模型分析及应用

利用Kalman滤波器对MEMS陀螺随机漂移进行估计和补偿,需要将随机漂移的自回归滑动平均(ARMA)模型转化为相应的状态空间模型,从而有必要对模型之间的转换问题进行深入研究。针对国内外有关文献提出的三种转化形式,从数学角度进行了证明,并结合MEMS陀螺的试验数据,分别采用这三种状态空间模型进行了随机漂移和角速率估计试...     

Computation of the exact Cramer–Rao lower bound for the parameters of a nonsymmetric half-plane 2-D ARMA model

The Cramer–Rao lower bound (CRLB) that gives the minimal achievable variance/standard deviation for any unbiased estimator offers a useful tool for an assessment of the consistency of parameter estimation techniques. In this paper, a closed-form expression for the computation of the exact CRLB on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model with a nonsymmetric half-plane (NSHP) region of support is developed. The proposed formulation is mainly based on a matrix representation of 2-D real-valued discrete and homogeneous random field characterized by the NSHP ARMA model. Assuming that the random field is Gaussian, the covariance matrix of the NSHP ARMA random field is first expressed in terms of the model parameters. Then, using this matrix structure, a closed-form expression of the exact Fisher information matrix required for the CRLB computation of the NSHP ARMA model parameters is developed. Finally, the main formulas derived for the NSHP ARMA model are rearranged for its autoregressive and moving average counterparts, separately. Numerical simulations are included to demonstrate the behavior of the derived CRLB formulas.     

Invariants and canonical forms for systems structural and parametric identification

The basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled.With reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set.Then the set of polynomial input-output models of the type P(z)y(t)=Q(z)u(t) and the equivalence relation induced by the premultiplication of P and Q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced.Since the two sets of canonical forms share common sets of complete independent invariants, very simple algebraical links between state space and input-output canonical forms can be deduced.The previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input-output sequences generated by a linear, discrete, time-invariant multivariable system.The results obtained in the identification of a real process are then reported.     

Analyzing relationships among ARMA processes based on non-Gaussianity of external influences

The analysis of a relationship among variables in data generating systems is one of the important problems in machine learning. In this paper, we propose an approach for estimating a graphical representation of variables in data generating processes, based on the non-Gaussianity of external influences and an autoregressive moving-average (ARMA) model. The presented model consists of two parts, i.e., a classical structural-equation model for instantaneous effects and an ARMA model for lagged effects in processes, and is estimated through the analysis using the non-Gaussianity on the residual processes. As well as the recently proposed non-Gaussianity based method named LiNGAM analysis, the estimation by the proposed method has identifiability and consistency. We also address the relation of the estimated structure by our method to the Granger causality. Finally, we demonstrate analyses on the data containing both of the instantaneous causality and the Granger (temporal) causality by using our proposed method where the datasets for the demonstration cover both artificial and real physical systems.     

Parameter estimation in multivariate stochastic difference equations

We will review the principal methods of estimation of parameters in multivariate autoregressive moving average equations which have additional observable input terms in them and present some new methods of estimation as well. We begin with the conditions for the estimability of the parameters. In addition to the usual method of system representation, the canonical form I, we will present two new representations of the system equation, the so-called canonical forms II and III which are convenient for parameter estimation. We will mention, in some detail, the various methods of estimation like the various least-squares methods, the maximum likelihood methods, etc., and discuss them regarding their relative accuracy of the estimate and the corresponding computational complexity. We will introduce a new class of estimates, the so-called limited information estimates which utilizes the canonical forms II and III. The accuracy of these estimates is close to that of maximum likelihood, but their computation time is only a fraction of the computation time for the usual maximum likelihood estimates. We will present a few numerical examples to illustrate the various methods.