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The aim of this paper is to develop a new recursive identification algorithm for autoregressive (AR) models corrupted by additive white noise. The proposed approach relies on a set of both low-order and high-order Yule–Walker equations and on a modified version of the overdetermined recursive instrumental variable method, leading to the estimation of both the AR coefficients and the additive noise variance. The main motivation behind our proposition is introducing model identification procedures suitable for implementation on edge-computing platforms and programmable logic controllers (PLCs), which are known to have limited capabilities and resources when dealing with complex mathematical computations (i.e., matrix inversion). Indeed, our development is focused on condition monitoring systems, with particular attention paid to their integration onboard industrial machinery. The performance of the recursive approach is tested using both numerical simulations and a laboratory case study. The obtained results are very promising.

This paper presents an overview of the main methods used to identify autoregressive models with additive noises. The classification of identification methods is given. For each group of methods, advantages and disadvantages are indicated. The article presents the simulation results of a large number of the described methods and gives recommendations on choosing the best methods. [ABSTRACT FROM AUTHOR]

In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule–Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example. [ABSTRACT FROM AUTHOR]

The celebrated Ljung-Box residual analysis is a widely used method in time series for the parameter estimation and the goodness of fit test for theARMAtime series models. The question is whether the autocorrelation function of the fitted ARMA(p, q) model for an observed time series, at different lags, in the Ljung-Box estimation method, is close to the correlogram of observed series. The answer indeed is not affirmative. In this article, firstly, we present a new procedure in solving the Yule-Walker equations for the exact computation of the autocorrelation functions of ARMA(p, q) models. Secondly, we provided a goodness of fit procedure using the limiting distribution of the the sample correlation function. Thirdly, we establish a new parameters estimation method based on examining the model autocorrelation function against the series autocorrelation coefficients. The effectiveness of the procedure is brought into sight using simulated data. [ABSTRACT FROM AUTHOR]

In a real life process evolving over time, the relationship between its relevant variables may change. Therefore, it is advantageous to have different inference models for each state of the process. Asymmetric hidden Markov models fulfil this dynamical requirement and provide a framework where the trend of the process can be expressed as a latent variable. In this paper, we modify these recent asymmetric hidden Markov models to have an asymmetric autoregressive component in the case of continuous variables, allowing the model to choose the order of autoregression that maximizes its penalized likelihood for a given training set. Additionally, we show how inference, hidden states decoding and parameter learning must be adapted to fit the proposed model. Finally, we run experiments with synthetic and real data to show the capabilities of this new model. [ABSTRACT FROM AUTHOR]

Vector‐valued‐60 extensions of univariate generalized binary auto‐regressive (gbAR) processes are proposed that enable the joint modeling of serial and cross‐sectional‐50 dependence of multi‐variate binary data. The resulting class of generalized binary vector auto‐regressive (gbVAR) models is parsimonious, nicely interpretable and allows also to model negative dependence. We provide stationarity conditions and derive moving‐average‐type representations that allow to prove geometric mixing properties. Furthermore, we derive general stochastic properties of gbVAR processes, including formulae for transition probabilities. In particular, classical Yule–Walker equations hold that facilitate parameter estimation in gbVAR models. In simulations, we investigate the estimation performance, and for illustration, we apply gbVAR models to particulate matter (PM10, 'fine dust') alarm data observed at six monitoring stations in Stuttgart, Germany. [ABSTRACT FROM AUTHOR]

In this paper, we consider the vector autoregressive time series with multidimensional α -stable noise where 1 < α < 2 . This model takes into account the short-term dependence in the multidimensional data and possible large observations (outliers). However, for the processes based on the α -stable distribution, the classical dependence measures known as the covariance function or the correlation function are not defined due to the diverging second moment. And therefore, the classical estimation method related to the covariance-based Yule–Walker (Y-W) equations should not be used in this case. Here, we propose to generalize the Y-W equations by replacing the covariance function with the covariation function which is a properly defined dependence measure for the symmetric α -stable distribution. Based on the covariation-based Yule–Walker equations, we propose a new estimation method and we demonstrate its efficiency by conducting a simulation study on the trajectories of two- and three-dimensional α -stable VAR(1) time series. Moreover, we compare the new covariation-based technique with the classical Yule–Walker method based on the covariance function. We emphasize that the estimation technique introduced in this paper extends the generalized Yule–Walker method introduced by Gallagher (Stat Prob Lett 53:381–390, 2001) for one-dimensional α -stable autoregressive models. The theoretical results are illustrated by the real data analysis of a bivariate data set describing the daily prices of KGHM and copper. [ABSTRACT FROM AUTHOR]

This paper presents an overview of the main methods used to identify autoregressive models with additive noises. The classification of identification methods is given. For each group of methods, advantages and disadvantages are indicated. The article presents the simulation results of a large number of the described methods and gives recommendations on choosing the best methods.

In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule–Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example.

The periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the "pure" model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model's parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise. [ABSTRACT FROM AUTHOR]

In this paper, a new method is proposed for estimating the frequency of a complex sinusoidal signal in multiplicative and additive noise. The proposed method is based on the linear prediction property of the signals' autocorrelation function. This frequency estimation method is a closed-form algorithm which can directly estimate the frequency without peak searching. The performance of the proposed frequency estimator is investigated by computer simulations. Simulation results demonstrate that the performance of the proposed technique is reasonable and is close to Cramer–Rao bound. [ABSTRACT FROM AUTHOR]

In this study, a block‐based estimation of noise due to blurriness distortion is proposed based on auto‐regressive (AR) modelling. In the proposed method; a de‐correlated, low‐energy version of the blurred image is auto regressively modelled. To this end, AR parameters are estimated using the Yule–Walker equations. As these equations include auto‐correlation function (ACF) coefficients, ACF estimation is also required. The Yule–Walker equations are solved making use of Durbin–Levinson algorithm. Finally, noise energy is mathematically defined and computed for each block. Since blurriness is a signal‐dependent image distortion, estimating and describing its characteristics via a noise like that of the AR model input, is significant. In fact, extracting features of such 'noise' can lead to the design and development of a new method of image quality metrics. Inspired by the 'stem cells' concept in medical science that is convertible to other cell types, the AR model input is called 'stem noise'. To visualise contribution of the 'Stem Noise' in the reconstruction of blurriness image distortion, a map called stem noise energy map is created. It is shown that the characteristics of the estimated noise energy are well correlated with the human subjective scores. [ABSTRACT FROM AUTHOR]

The present study addresses the problem of two-dimensional autoregressive estimation in the presence of additive white noise. The estimation method is based on combining the low-order and high-order Yule-Walker equations. The noise-compensated YW equations are solved using an iterative algorithm. The proposed method is also applied to joint frequency and direction of arrival estimation in uniform linear arrays. Using simulation study, the performance of the proposed algorithm is evaluated and compared with other methods. [ABSTRACT FROM AUTHOR]

The classical Least-Squares (LS) adjustment has been widely used in processing and analysing observations from Global Satellite Navigation System (GNSS). However, in detecting temporal correlations of GNSS observations, which can be described by means of autoregressive (AR) process, the LS method may not provide reliable estimates of process coefficients, since the Yule-Walker (YW) equations refer to structured Errors-In-Variables (EIV) equations. In this contribution, we proposed a Total Least-Squares (TLS) solution with the singular cofactor matrix to solve the YW equations. The proposed TLS solution is obtained based on the fact that random errors belong to column space of its cofactor matrix. In addition the proposed solution does not need any substitution of the squared true parameter vector as done by the current publications. Finally, we simulate the AR process to prove that our solution is more reliable than the existing methods. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of parameters estimation for stable periodic autoregressive (PAR) time series. Considered models generalize popular and widely accepted autoregressive (AR) time series. By examining measures of dependence for α -stable processes, first we introduce new empirical estimator of autocovariation for α -stable sequences. Based on this approach we generalize Yule–Walker method for estimation of parameter for PAR time series. Thus we fill a gap in estimation methods for non-Gaussian models. We test proposed procedure and show its consistency. Moreover, we use our approach to model real empirical data thus showing usefulness of heavy tailed models in statistical modelling. [ABSTRACT FROM AUTHOR]

In this paper we propose a noisy Multivariate Autoregressive (MVAR) process for modeling and estimating the correlated fading channels. The method can estimate joint MVAR processes and model parameters from noisy received signal. The proposed method is based on serial connection of two algorithms. In the first algorithm, we estimate MVAR model parameters. For this purpose, a combination of Yule-Walker equations is used. This combination is considered as a generalized eigenvalue problem; an estimate for receiver noise variance is obtained by solving this eigenvalue problem then Least-Squares method is used to estimate the MVAR model parameters. In the second algorithm, the fading process is estimated by using Kalman filter. Simulation results show that the proposed method has good performance compared with other existing methods. [ABSTRACT FROM AUTHOR]

We consider the problem of two dimensional (2-D) autoregressive (AR) parameter estimation in the presence of observation noise. The proposed method is based on Yule-Walker Equations. We express the Yule Walker equations as a quadratic eigenvalue problem then by solving these equations, the parameters of the signal and noise are estimated. We also apply the proposed method to (2-D) spectrum estimation of the sinusoidal signals in noise. The performance of the proposed method is evaluated by computer simulation examples. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes nonlinear autoregressive (AR) models for time series, within the framework of kernel machines. Two models are investigated. In the first proposed model, the AR model is defined on the mapped samples in the feature space. In order to predict a future sample, this formulation requires to solve a pre-image problem to get back to the input space. We derive an iterative technique to provide a fine-tuned solution to this problem. The second model bypasses the pre-image problem, by defining the AR model with an hybrid model, as a tradeoff considering the computational time and the precision, by comparing it to the iterative, fine-tuned, model. By considering the stationarity assumption, we derive the corresponding Yule–Walker equations for each model, and show the ease of solving these problems. The relevance of the proposed models is studied on several time series, and compared with other well-known models in terms of accuracy and computational complexity. [Copyright &y& Elsevier]

Abstract: The work in this paper continues and improves on the research that has been carried out on the generation of synthetic, correlated wind speed variates that are defined by marginal Weibull distributions and auto- and cross-correlations. Previously, in order to establish the matrix of auto-regressive coefficients for a small number of wind variates, a simple recursive method had been used. However, attempts to apply the model, using simple numerical approaches, to an increasing number of wind turbines or locations, proved useless. Therefore, a direct solution of the Yule–Walker equations is proposed here. These equations, originally in matrix form and of size N 2 for N wind variates, are put in symbolic notation and then broken down for numerical solution using the traditional Newton''s method. The complete procedure is explained through a symbolic example and illustrated with two very different numerical cases: with the first, wind in very close locations; with the second, wind in dispersed locations of a region. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problem of parameter estimation of autoregressive (AR) signals from observations corrupted with colored AR(1) noise. The proposed method is based on Yule-Walker equations. We express these equations as a quadratic eigenvalue problem and then the parameters of the signal and noise are estimated by solving this eigenvalue problem. We also apply the proposed method to the problem of sinusoidal frequency estimation in colored noise. The performance of the proposed algorithm is evaluated by computer simulation examples. [Copyright &y& Elsevier]

Abstract: In this paper we present a new method for estimating the parameters of an autoregressive (AR) signal from observations corrupted with white noise. The least-squares (LS) estimate of the AR parameters is biased when the observation noise is added to the AR signal. This bias is related to observation noise variance. The proposed method uses inverse filtering technique and Yule–Walker equations for estimating observation noise variance to yield unbiased LS estimate of the AR parameters. The performance of the proposed unbiased algorithm is illustrated by simulation results and they show that the performance of the proposed method is better than the other estimation methods. [Copyright &y& Elsevier]

Many methods have been proposed to assess small signal stability for either analysis or control purposes. In this paper, a new offline method is proposed to detect electromechanical modes and their associated damping levels in power systems. The new method estimates oscillatory performance by fitting an orthogonal polynomial expansion to data and extrapolating its spectrum to identify low frequency modes. Damping of individual modes is then performed using a sliding window technique previously developed with the use of a linear prediction algorithm. Performance of the new technique is assessed using test signals and measurements recorded from staged field tests along with noise probing measurements. Accuracy of the new method is measured against a least squares Prony algorithm and the Yule-Walker autoregressive technique using identical data sets. Estimation results reveal that the new method is reasonably accurate in the detection of modes and their damping levels when compared to other methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new method for estimation of the parameters of a noisy autoregressive (AR) signal using observations corrupted with colored noise. This method is an improved least-squares (ILS) based method that combines low-order and high-order Yule–Walker equations. The performance of the proposed method is illustrated using computer simulations. [Copyright &y& Elsevier]

We introduce the vector time-frequency autoregressive (VTFAR) model for a parsimonious parametric description of nonstationary vector random processes. The VTFAR model generalizes the recently proposed scalar TFAR model to the multivariate case. It is physically meaningful because nonstationarity and spectral correlation are represented in terms of frequency shifts, and it is parsimonious for the practically relevant class of underspread vector processes (i.e., nonstationary vector processes with rapidly decaying correlation in time and frequency). For vector processes with decaying correlation across the signals, we introduce a variant of the VTFAR model with banded parameter matrices. Furthermore, we present a VTFAR parameter estimator that is based on a-system of linear equations with two-level block-Toeplitz structure, and we develop an efficient order-recursive algorithm for solving these equations. We also present information criteria for estimating the VTFAR model order and the matrix bandwidth of the banded VTFAR model. The performance of the proposed VTFAR parameter and order estimators is assessed through numerical simulations. Finally, an application to nonstationary multivariate spectral analysis is presented. [ABSTRACT FROM AUTHOR]

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data. [ABSTRACT FROM AUTHOR]

We investigate a class of ARMA-type models for stationary binary time series developed in [M. Kanter, Autoregression for discrete processes mod 2, J. Appl. Probabil. 12 (1975), pp. 371-375, E. McKenzie, Extending the correlation structure of exponential autoregressive-moving-average processes, J. Appl. Prob. 18 (1981), pp. 181-189.], which we shall refer to as BinARMA models. This sparsely parameterized model family is even able to deal with negative autocorrelations, which occur in language modelling, for instance. While the autocorrelation structure of the BinAR(p) models has been studied before in [M. Kanter, Autoregression for discrete processes mod 2, J. Appl. Probabil. 12 (1975), pp. 371-375], we shall present new results on the autocorrelation structure of general BinARMA models. These results simplify in the BinMA(q) case, while the known results concerning BinAR(p) models are included as a special case. A real-data example indicates possible fields of application of these models. [ABSTRACT FROM AUTHOR]

We show that there exists a causal 2-D linear process in the nonsymmetric half-plane having the same autocorrelations as a noncausal 2-D linear process in the whole-plane; this property is called the autocorrelation equivalence relation, and can be used for practical fitting and modeling of 2-D processes. Some causal 2-D autoregressive (AR) models with various regions of support are considered such as half-cross, half-diamond, quarter-plane square, half-square, half-hexagon, half-octagon, and half-circle. Considerations of parsimony in 2-D model fitting are then focused not only on the number of parameters in our model, but also most importantly on the optimal shape of the region of support. Their 2-D Yule-Walker equations are derived, and a computationally efficient order-recursive algorithm is proposed to solve them. The autocorrelation equivalence relation and the order-recursive algorithm are utilized to specify a noncausal 2-D AR process as well as its spectrum from a given realization of a random field. [ABSTRACT FROM AUTHOR]

Abstract: The maximum entropy spectral estimator is widely used because of its high spectral resolution, but it lacks an easy procedure for evaluating the statistical significance of the spectral estimates. We implemented the non-parametric computer intensive permutation test in order to evaluate the statistical significance of the maximum entropy spectral estimates. There is the possibility of choosing between an underlying red or white noise in the permutation procedure. Two case studies, with a long and a short time series, illustrate the performance of the method. [Copyright &y& Elsevier]

Here we obtain difference equations for the higher order moments and cumulants of a time series{ X t} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method– the Whittle criterion– to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient. [ABSTRACT FROM AUTHOR]

Recently, as a result of the growing interest in modelling stationary processes with discrete marginal distributions, several models for integer value time series have been proposed in the literature. One of these models is the INteger-AutoRegressive (INAR) model. Here we consider the higher-order moments and cumulants of the INAR(1) process and show that they satisfy a set of Yule–Walker type difference equations. We also obtain the spectral and bispectral density functions, thus characterizing the INAR(1) process in the frequency domain. We use a frequency domain approach, namely the Whittle criterion, to estimate the parameters of the model. The estimation theory and associated asymptotic theory of this estimation method are illustrated numerically. [ABSTRACT FROM AUTHOR]

Many time series in the applied sciences display a time-varying second order structure. In this article, we address the problem of how to forecast these nonstationary time series by means of non-decimated wavelets. Using the class of Locally Stationary Wavelet processes, we introduce a new predictor based on wavelets and derive the prediction equations as a generalisation of the Yule-Walker equations. We propose an automatic computational procedure for choosing the parameters of the forecasting algorithm. Finally, we apply the prediction algorithm to a meteorological time series. [ABSTRACT FROM AUTHOR]

This paper considers the modelling and analysis of nonstationary vector ARMA processes from the theoretical point of view. Some extensions to the existing work (see, for example [1,2]) incorporating the nonstationarity of the underlying process are given using models with nonstationary innovations. It is shown that this class of models provides a very strong framework for many practical situations. The estimation procedures are discussed. Various prediction results are also given. Some examples are added to illustrate the theory. [Copyright &y& Elsevier]

Summary: In this paper it is shown that the famous Levinson-Durbin algorithm is just a recursive algorithm presented previously by Szegö. [ABSTRACT FROM AUTHOR]

The ordinary least squares method is the most commonly used estimation procedure in statistics but estimates of the input and output parameters through this method for transfer function models are not necessarily consistent. An iterative regression procedure is proposed to produce consistent estimates. Consistent moment estimates are also given. On the basis of these consistent estimates a method of model specification is proposed. An example is given to illustrate the procedure [ABSTRACT FROM AUTHOR]

Different methods for calculating partial autocorrelation coefficients can produce different estimates, and these differences can be non-trivial. It has long been known that the Yule-Walker equations are particularly susceptible to numerical error, yet it is the most widely used method in statistical and econometric software. Two other methods, conditional maximum likelihood and Burg's algorithm are known to be more reliable, yet are infrequently used. All three methods are applied to several datasets. A forecasting example shows that a model identified by Yule-Walker can produce inferior forecasts. [ABSTRACT FROM AUTHOR]

• Four new methods for estimating the parameters of an autoregressive (AR) process based on observations corrupted by white noise are presented. • The first three methods are iterative, while the last one is non-iterative. • The performance of the proposed methods is evaluated in terms of various numerical examples, which demonstrate their superiority in terms of accuracy and robustness against the observation noise. This paper presents four new methods for estimating the parameters of an autoregressive (AR) process based on observations corrupted by white noise. The first three methods are iterative, while the last one is non-iterative. One method is designed to achieve an unbiased estimation of the AR parameters by undermining the destructive impact of observation noise in terms of utilizing the null space of the AR parameter vector. Another one uses both low- and high-order Yule-Walker equations to construct a constrained least squares optimization problem, in which the variance of observation noise is estimated by alternating between two equations. One more method exploits an approximation which leads to reducing the problem of estimating the AR parameters with arbitrary order p to estimating just two parameters, while the last one estimates the variance of the observation noise using the minimum eigenvalue of the enlarged autocorrelation matrix. The performance of the proposed methods is evaluated in terms of various numerical examples, which demonstrate their superiority in terms of accuracy and robustness against the observation noise compared to state-of-the-art existing methods in most simulation examples. It makes the proposed methods a good fit for practical analysis of data contaminated by observation noise, when AR modeling is applicable, and gives a range of choices of methods for different data analysis situations. [ABSTRACT FROM AUTHOR]

• A low-complexity maximum-likelihood-like AR parameter estimation method. • The suggested method is especially suitable for short-data record applications such as radar signal processing applications. • The first stage of the suggested method (weighted forward-backward prediction) performs better than classical linear prediction based approaches, i.e. the auto-correlation, covariance, Burgs, Forward-Backward prediction methods. • The second stage of the suggested method further improves the likelihood value attained by the first stage estimate. A two-stage method for the parameter estimation of Gaussian autoregressive models is proposed. The proposed first stage is an improved version of the conventional forward-backward prediction method and can be interpreted as its weighted version with the weights derived from the arithmetic mean of the log-likelihood functions for different conditioning cases. The weighted version is observed to perform better than the conventional forward-backward prediction method and other linear prediction based methods (correlation method, covariance method, Burg's method etc.) in terms of attained likelihood value. The proposed second stage uses the estimate of the first stage as the initial condition and approximates the highly non-linear log-likelihood function with a quadratic function around the initial estimate. The optimization of the quadratic cost function yields the optimal perturbation vector that locally maximizes the likelihood in the vicinity of the initial condition. The proposed method is compared with other methods and it has been observed that the likelihood value attained at the end of two-stages is almost identical to the value attained by higher complexity numerical-search based optimization tools in a wide range of experiments. The maximum likelihood-like performance at a significantly lower implementation cost makes the proposed method especially valuable for the applications with short data-records and limited computational resources. [ABSTRACT FROM AUTHOR]

The serial dependence of categorical data is commonly described using Markovian models. Such models are very flexible, but they can suffer from a huge number of parameters if the state space or the model order becomes large. To address the problem of a large number of model parameters, the class of (new) discrete autoregressive moving-average (NDARMA) models has been proposed as a parsimonious alternative to Markov models. However, NDARMA models do not allow any negative model parameters, which might be a severe drawback in practical applications. In particular, this model class cannot capture any negative serial correlation. For the special case of binary data, we propose an extension of the NDARMA model class that allows for negative model parameters, and, hence, autocorrelations leading to the considerably larger and more flexible model class of generalized binary ARMA (gbARMA) processes. We provide stationary conditions, give the stationary solution, and derive stochastic properties of gbARMA processes. For the purely autoregressive case, classical Yule–Walker equations hold that facilitate parameter estimation of gbAR models. Yule–Walker type equations are also derived for gbARMA processes. [ABSTRACT FROM AUTHOR]