Robust and accurate ARX and ARMA model order estimation of non-Gaussian processes (original) (raw)
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Electrical and Electronics Engineering, 2009
In statistical signal processing, parametric modeling of non-Gaussian processes experiencing noise interference is a very important research area. The autoregressive moving average (ARMA) model is the most general and important tool of modeling system. This paper develops an algorithm for the selection of the proper ARMA model orders. The proposed technique is based on forming a third order cumulant matrix from the observed data sequence. The observed sequence is modeled as the output of an ARMA system that is excited by an unobservable input, and is corrupted by zero-mean Gaussian additive noise of unknown variance. Examples are given to demonstrate the performance of the proposed algorithm.
Cumulant-based order determination of non-Gaussian ARMA models
1990
Higher than second-order cumulants can be used for order determination of non-Gaussian ARMA processes. The two methods developed assume knowledge of upper bounds on the ARMA orders. The first method performs a linear dependency search among the columns of a higher order statistics matrix, via the Gram-Schmidt orthogonalization procedure. In the second method, the order of the AR part is found as the rank of the matrix formed by the higher order statistics sequence. For numerically robust rank determination the singular value decomposition approach is adopted. Furthermore, using the argument principle and samples of the polyspectral phase the relative degree of the ARMA model is obtained from which the order of the MA part can he determined. Statistical analysis is included for determining the correct MA order with high probability, when estimates of third-order cumulants are only available. Simulations verify the performance of our methods, and compare autocorrelation with cumulant-based order determination approaches.
ARMA parameter estimation using only output cumulants
1990
Several algorithms are developed to estimate the parameters of a causal nonminimum phase ARMA(p, q ) system which is excited by an unobservable independent identically distributed (i.i.d.) non-Gaussian process; the output is contaminated by additive colored Gaussian noise of unknown power spectral density. First we present a fiindamental result pertaining to the identifiability of AR parameters, based on the Yule-Walker type equations drawn from a (specific) set of ( p + 1 ) 1-D slices of the kth ( k > 2 ) order output cumulant. Next, we develop several MA parameter estimation algorithms: one method uses q 1-D slices of the output cumulant; a second method uses only two 1-D cumulant slices. These methods do not involve computation of the residual (i.e., AR compensated) time series or polynomial factorization. Multidimensional versions of the various algorithms are also presented. A simulation study demonstrating the effectiveness of our algorithms is included.
Signal Detection in Correlated Non-Gaussian Noise Using Higher-Order Statistics
Circuits, Systems, and Signal Processing, 2017
The authors of this paper study the synthesis of new models and methods for signal detection in additive correlated non-Gaussian noise. A new moment quality criterion decision making is proposed based on a random process description using moments and a formation of polynomial decision rules. Taking into account parameters of non-Gaussian distribution of random variables (such as the moments of third and higher orders and joint cumulants), it is shown that nonlinear processing of samples can increase the signal processing efficiency. A synthesis of effective methods and algorithms of data processing in non-Gaussian noise is also presented in this work.
Linear modeling of multidimensional non-gaussian processes using cumulants
Multidimensional Systems and Signal Processing, 1990
Extending the notion of second-order correlations, we define the cumulants of stationary non-Gaussian random fields, and demonstrate their potential for modeling and reconstruction of multidimensional signals and systems. Cumulants and their Fourier transforms called polyspeetra preserve complete amplitude and phase information of a multidimensional linear process, even when it is corrupted by additive colored Gaussian noise of unknown covafiance function. Relying on this property, phase reconstruction algorithms are developed using polyspectra, which can be computed via a 2-D FFT-based algorithm. Additionally, consistent ARMA parameter estimators are derived for identification of linear space-invariam multidimensional models which are driven by unobservable, i.i.d., non-Gaussian random fields. Contrary to autocorrelation based multidimensional modeling approaches, when cumulants are employed, the ARMA model is allowed to be non-minimum phase, asymmetric non-causal or non-separable.
Signal Processing, 2005
This paper presents a new corner location method to model order selection of an autoregressive moving average (ARMA) model. The criterion is determined in terms of the minimum eigenvalue of the third-order cumulant matrix derived from the observed data sequence. The observed sequence is modeled as the output of an ARMA system that is excited by an unobservable input, and is corrupted by zero-mean Gaussian additive noise. The system is driven by a zero-mean independent and identically distributed (i.i.d.) non-Gaussian sequence. The method is an extension to recent results based on third-order cumulant (TOC) by Al-Smadi and Wilkes. Simulations verify the performance of the proposed method even when the observed signal is heavily corrupted by additive noise. The proposed estimator, via computer simulation, is found to outperform the TOC estimator of Al-Smadi and Wilkes.
Higher-order statistics based blind estimation of non-Gaussian bidimensional moving average models
Signal Processing, 2006
In this paper, four batches least squares linear approaches are developed for non-minimum phase bidimensional non-Gaussian moving average (MA) models identification. A relationship between autocorrelation and cumulant sequences is established. One of the proposed methods is cumulant based. The others exploit both autocorrelation and mth-order cumulants (m42). Three of these proposed methods are obtained by transforming Brillinger-Rosenblatt's non-linear equation into linear one using the Tugnait's closed-form solution. We also generalize the 2-D version of Giannakis-Mendel's method to mth-order cumulant. The simulation results show that one of the three autocorrelation and cumulants based methods gives the best estimates in free-noise environments, but in a Gaussian noisy case, the cumulant-based one is more adequate when large data are available. We also show the usefulness of the relationship to improve the estimates of the autocorrelation-based method in colored noise environment.
Conference on Decision and Control, 2007
In this paper, the problem of identifying stochastic linear continuous-time systems from noisy input/output data is addressed. The input of the system is assumed to be non-Gaussian, whereas the noises contaminating the data are assumed to be Gaussian. The fourth-order cumulants of the input/output data are then (asymptotically) insensitive to the noises, that can be colored and/or mutually correlated. Two estimators based on this noise-cancellation property are proposed. The performance of the proposed algorithms are assessed through a numerical simulation.
Fast adaptive algorithms for AR parameters estimation using higher order statistics
IEEE Transactions on Signal Processing, 1996
With the rapid availability of vast and inexpensive computation power, models which are non-Gaussian even nonstationary are being investigated at increasing intensity. Statistical tools used in such investigations usually involve higher order statistics (HOS). The classical instrumental variable (IV) principle has been widely used to develop adaptive algorithms for the estimation of ARMA processes. Despite, the great number of IV methods developed in the literature, the cumulant-based procedures for pure autoregressive (AR) processes are almost nonexistent, except lattice versions of IV algorithms. This paper deals with the derivation and the properties of fast transversal algorithms. Hence, by establishing a relationship between classical (IV) methods and cumulant-based AR estimation problems, new fast adaptive algorithms, (fast transversal recursive instrumental variable-FTRIV) and (generalized least mean squares-GLMS), are proposed for the estimation of AR processes. The algorithms are seen to have better performance in terms of convergence speed and misadjustment even in low SNR. The extra compntational complexity is negligible. The performance of the algorithms, as well as some illustrative tracking comparisons with the existing adaptive ones in the literature, are verified via simulations. The conditions of convergence are investigated for the GLMS.
IEEE Transactions on Signal Processing, 2000
This paper addresses the problem of detecting the presence of colored multiplicative noise, when the information process can be modeled as a parametric ARMA process. For the case of zero-mean multiplicative noise, a cumulant based suboptimal detector is studied. This detector tests the nullity of a specific cumulant slice. A second detector is developed when the multiplicative noise is nonzero mean. This detector consists of filtering the data by an estimated AR filter. Cumulants of the residual data are then shown to be well suited to the detection problem. Theoretical expressions for the asymptotic probability of detection are given. Simulation-derived finite-sample ROC curves are shown for different sets of model parameters.