The estimation of the order of an ARMA process using third-order statistics

The paper proposes a new approach to find an autoregressive moving average (ARMA) model order. The basic idea is to extend the previous approach proposed by Liang et al. to third order statistics (TOS). The algorithm uses data matrices rather than calculating cumulants of the observed signal. Hence, we avoid the non-stationary effects, which is due to finite-length observations. The system is driven by a zero-mean independent and identically distributed (i.i.d.) non-Gaussian process. The input signal is unobservable. The observed sequence is corrupted by a zero-mean additive Gaussian noise. Examples are given to demonstrate the performance of the proposed algorithm.     

含ARMA噪声系统模型的参数辨识方法*

实际问题中，大量的动态系统控制问题可归结为含ＭＡ，ＡＲＭＡ噪声系统模型的参数辨识问题。本文提出ＲＭＡ，ＲＡＲＭＡ两种系统模型参数辨识的一种新方法，主要手段是构造和研究特殊的辅助线性模型。理论分析和实际计算表明，本文方法较传统表度有明显提高。     

Identification of quadratic systems using higher order cumulants and neural networks: Application to model the delay of video-packets transmission

This work concerns the development of two approaches for the identification of diagonal parameters of quadratic systems from only the output observation. The systems considered are excited by an unobservable independent identically distributed (i.i.d), stationary zero mean, non-Gaussian process and corrupted by an additive Gaussian noise. The proposed approaches exploit higher order cumulants (HOC) (fourth order cumulants) and are the extension of the algorithms developed in the linear version 1D, which uses a non-Gaussian signal input. For test and validity purpose, these approaches are compared to recursive least square (RLS), least mean square (LMS) and neural network identification algorithms using non-linear model in noisy environment. To demonstrate the applicability of the theoretical methods on real processes, we applied the developed approaches to search for models able to describe the delay of the video-packets transmission over IP networks from video server. The simulation results show the correctness and the efficiency of the developed approaches.     

Development of Box–Jenkins type time series models by combining conventional and orthonormal basis filter approaches

A unified scheme for developing Box–Jenkins (BJ) type models from input–output plant data by combining orthonormal basis filter (OBF) model and conventional time series models, and the procedure for the corresponding multi-step-ahead prediction are presented. The models have a deterministic part that has an OBF structure and an explicit stochastic part which has either an AR or an ARMA structure. The proposed models combine all the advantages of an OBF model over conventional linear models together with an explicit noise model. The parameters of the OBF–AR model are easily estimated by linear least square method. The OBF–ARMA model structure leads to a pseudo-linear regression where the parameters can be easily estimated using either a two-step linear least square method or an extended least square method. Models for MIMO systems are easily developed using multiple MISO models. The advantages of the proposed models over BJ models are: parameters can be easily and accurately determined without involving nonlinear optimization; a prior knowledge of time delays is not required; and the identification and prediction schemes can be easily extended to MIMO systems. The proposed methods are illustrated with two SISO simulation case studies and one MIMO, real plant pilot-scale distillation column.     

Strongly consistent coefficient estimate for errors-in-variables models

For the single-input-single-output (SISO) errors-in-variables system it is assumed that the system input is an ARMA process and that the driven noise of the system input and the observation noise are jointly Gaussian. The two-dimensional observation made on system input and output is represented as a two-dimensional (2D) ARMA system of minimum phase driven by a sequence of 2D i.i.d. Gaussian random vectors (innovation representation). It is shown that the resulting ARMA system is identifiable, i.e., its coefficients are uniquely defined under reasonable conditions. Recursive algorithms are proposed for estimating coefficients of the ARMA representation including those contained in the original SISO system. The estimates are proved to be convergent to the true values with probability one and the convergence rate is derived as well. The theoretical results are justified by numerical simulation.     

应用于地震数据去卷的自校正白噪声估值器

本文把地震数据去卷问题处理为估计带观测噪声的ARMA模型的白噪声问题,应用时间 序列分析方法提出了不同于Mendel的新的稳态最优白噪声估值器,文章基于两个ARMA新 息模型的在线辨识,进一步给出了自校正白噪声估值器.     

Noisy texture classification: A higher-order statistics approach

In this paper, a novel texture classification scheme using higher-order statistics (HOS) functions as discriminating features is proposed. It is well known that such statistical parameters are insensitive to additive Gaussian noise. In particular, third-order statistical parameters, i.e. third-order cumulants and bispectrum, are insensitive to any symmetrically distributed noise, and also exhibit the capability of better characterizing non-Gaussian signals. By exploiting these HOS properties, it is possible to devise a robust method for classifying textures affected by noise with different distributions and even with very low signal-to-noise ratios.     

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.     

Identification of linear systems with noisy input using input-output cumulants

An identification method is presented for estimating the parameters of a discrete-time linear dynamic system excited by non-gaussian input signals using the fourth-order cumulants of the input and output signals, both of which are contaminated by additive gaussian noise. Two types of estimators of the fourth-order cumulants of the input and output signals are proposed for this method. The first is conventional. The second, which is new allows us only to have a recursive algorithm for computing the parameter estimators. The parameter estimators obtained by this algorithm are shown to be strongly consistent under certain weak conditions. Simulation examples are included to demonstrate the effectiveness of the proposed method.     

Modeling of the grounding resistance variation using ARMA models

This study addresses the problem of modeling the variation of the grounding resistance during the year. An AutoRegressive Moving Average (ARMA) model is fitted (off-line) on the provided actual data using the Corrected Akaike Information Criterion (AICC). The developed model is shown to fit the data in a successful manner. Difficulties occur when the provided data includes noise or errors and also when an on line/adaptive modeling is required. In both cases, and under the assumption that the provided data can be represented by an ARMA model, simultaneous order and parameter estimation of ARMA models under the presence of noise is necessary. In this paper, a new method based on the multi-model partitioning theory which is also applicable to on line/adaptive operation, is used for the solution of the above mentioned problem. The simulations show that the proposed method succeeds in selecting the correct ARMA model order and estimates the parameters accurately in very few steps and even with a small sample size. For validation purposes the method introduced is compared with three other established order selection criteria presenting very good results. The proposed method can be extremely useful in the studies of electrical engineer designers, since the variation of the grounding resistance during the year affects significantly power systems performance and must be definitely considered.     

ARMA Modeling of Time Series

A method for efficiently generating a rational model of a wide-sense stationary time series is presented. In this method the autoregressive parameters associated with an ARMA model consisting of q zeros and p poles are optimally chosen with the selection being based on a finite set of time series observations. This selection is made so that a set of Yule-Walker equation approximations are ``best' satisfied. The resultant autoregressive parameter estimates have the desired statistical feature of being unbiased and consistent. This estimation method has been found to provide a modeling performance which typically equals or exceeds that of contemporary alternatives. Moreover, this method is amenable to a computationally efficient adaptive solution procedure. The autoregressive parameters characterizing the resultant ARMA model estimate can serve the role of decision variables in pattern classification schemes. For example, these parameters can be utilized in determining whether or not a member(s) of a given signal class is contained within a noise corrupted measurement signal. This approach has been found to be particularly effective in Doppler radar and array processing applications in which one is looking for the presence of spectral lines (i.e., sinusoids) in the measurement signal.     

Rank pattern-based selection of input/output ARMA model order

Rational transfer functions are standard models for radar targets and adaptive beamforming. Fitting these models essentially involves estimating the transfer function “poles and zeroes.” A key preliminary step in this estimation process is to determine the numbers of poles and zeros, or equivalently to determine the order of the corresponding ARMA model. A pattern-based method of order selection using matrix ranks is proposed for input/output (I/O) ARMA models, where ARMA model inputs and outputs are each observed in additive noise with known variances. This I/O ARMA model encompasses two distinct scenarios: observational studies in which all observations—those of both inputs and outputs—are erred, and controlled experiments in which outputs are observed with error while inputs are known without error. The proposed rank pattern method exploits the eigenvalue structure of the covariance matrices associated with the observed data and performs well for short data records at moderate SNRs.     

基于小波和高阶累积量的数字调制识别算法

利用基于高阶累积量的数字调制识别算法对数字调制信号进行分类识别时,六阶及六阶以上累积量的计算过于复杂,且多进制频移键控(MFSK)与8PSK信号各阶累积量的值均相等,直接计算无法识别。针对此问题,提出了一种基于小波和高阶累积量相结合的分类算法,先对MFSK与8PSK信号进行小波变换,再利用四阶累积量进行识别。实验证明,利用该算法所提取的特征参数能有效抑制高斯白噪声,除了识别2ASK/BPSK,4ASK,2FSK,4FSK,QPSK,8PSK信号外,还可识别16QAM,并且计算量小,易于实现。当信噪比大于等于3dB时,总体识别率达到96%。与已有算法相比,仿真结果证明了该算法的优越性。     

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     

Cumulant-based inverse filters for blind deconvolution

Blind deconvolution of linear time-invariant (LTI) systems has received wide attention in various fields such as data communication and image processing. Blind deconvolution is concerned with the estimation of a desired input signal from a given set of measurements. This paper presents a technique for reconstructing the desired input from only the available corrupted data. The estimator is given in terms of an autoregressive moving average (ARMA) innovation model. This technique is based on higher order statistics (HOS) of a non-Gaussian output sequence in the presence of additive Gaussian or non-Gaussian noise. The algorithm solves a set of overdetermined linear equations using third-order cumulants of the given non-Gaussian measurements in the presence of additive Gaussian or non-Gaussian noise. The inverse filter is a finite impulse response (FIR) filter. Simulation results are provided to show the effectiveness of this method and compare it with a recently developed algorithm based on maximizing the magnitude of the kurtosis of estimate of the input excitation.     

Identifiability of the AR parameters of an ARMA process usingcumulants

The problem of estimating the autoregressive (AR)-order and the AR parameters of a causal, stable, single input single output (SISO) autoregressive moving average (ARMA) (p,q) model, excited by an unobservable i.i.d. process, is addressed. The observed output is corrupted by additive colored Gaussian noise, whose power spectral density is unknown. The ARMA model may be mixed-phase, and have inherent all-pass factors and repeated poles. It is shown that consistent AR parameter estimates can be obtained via the normal equations based on (p+1) 1-D slices of the mth-order ( m>2) cumulant. It is shown via a counterexample that consistent AR estimates cannot, in general, be obtained from a subset of these p+1 slices. Necessary and sufficient conditions for the existence of a full-rank slice are also derived     

Closed-loop subspace-based identification algorithm using third-order cumulants

The problem of closed-loop system identification for coloured noise system without any knowledge of feedback controller is considered. We develop a solution to this problem in the framework of subspace identification based on high-order cumulants. The key of the developed algorithm is using the properties that the third-order cumulants are insensitive to any coloured Gaussian noises. By post-multiplying a suitable instrumental variable to the noise terms, the cross third-order cumulants are constructed that become zero when the noises are Gaussian distributed, and meanwhile the column rank of extended observability matrix is maintained. Thus, the standard subspace identification algorithms can be extended to closed-loop system corrupted by arbitrary coloured noises. A numerical simulation is presented to demonstrate the proposed algorithm.     

Identification of dynamic errors-in-variables models: Approaches based on two-dimensional ARMA modeling of the data

In this paper we propose a parametric and a non-parametric identification algorithm for dynamic errors-in-variables model. We show that the two-dimensional process composed of the input-output data admits a finite order ARMA representation. The non-parametric method uses the ARMA structure to compute a consistent estimate of the joint spectrum of the input and the output. A Frisch scheme is then employed to extract an estimate of the joint spectrum of the noise free input-output data, which in turn is used to estimate the transfer function of the system. The parametric method exploits the ARMA structure to give estimates of the system parameters. The performances of the algorithms are illustrated using the results obtained from a numerical simulation study.