Yule Walker Research Articles

A new skew integer valued time series process

In this paper, we introduce a stationary first-order integer-valued autoregressive process with geometric–Poisson marginals. The new process allows negative values for the series. Several properties of the process are established. The unknown parameters of the model are estimated using the Yule–Walker method and the asymptotic properties of the estimator are considered. Some numerical results of the estimators are presented with a brief discussion. Possible application of the process is discussed through a real data example.

Discrete Fourier transform‐based parametric modal identification from ambient data of the power system frequency

This study proposes a discrete Fourier transform (DFT)-based parametric identification method by using a difference sequence between two sets of frequency data. An autocorrelation function of the frequency difference sequence can be represented as a linear combination of exponentially decaying sinusoidal functions, whose natural frequencies and damping ratios are equal to those of interarea modes. These modal parameters are calculated from the Laplace transform coefficients of the autocorrelation function. The coefficients are estimated by curve-fitting the DFT values of the autocorrelation function. The proposed method is compared with the modified extended Yule Walker method through simulations on signal-to-noise ratio. Finally, the feasibility of the proposed method is shown by identifying real power systems from frequency data of the frequency monitoring network system.

Bivariate zero truncated Poisson INAR(1) process

In this paper, we propose a new stationary bivariate first order integer-valued autoregressive (BINAR(1)) process with zero truncated Poisson marginal distribution. Some properties about this process are considered, such as probability generating function, autocorrelations, expectations and covariance matrix under conditional and unconditional situation. We also establish the strict stationarity and ergodicity of the process. Estimators of unknown parameters are derived by using Yule–Walker, conditional least squares and maximum likelihood methods. The performance of the proposed estimation procedures are evaluated through Monte Carlo simulations. An application to a real data example is also provided.

Improved gene prediction by principal component analysis based autoregressive Yule–Walker method

Spectral analysis using Fourier techniques is popular with gene prediction because of its simplicity. Model-based autoregressive (AR) spectral estimation gives better resolution even for small DNA segments but selection of appropriate model order is a critical issue. In this article a technique has been proposed where Yule–Walker autoregressive (YW-AR) process is combined with principal component analysis (PCA) for reduction in dimensionality. The spectral peaks of DNA signal are used to detect protein-coding regions based on the 1/3 frequency component. Here optimal model order selection is no more critical as noise is removed by PCA prior to power spectral density (PSD) estimation. Eigenvalue-ratio is used to find the threshold between signal and noise subspaces for data reduction. Superiority of proposed method over fast Fourier Transform (FFT) method and autoregressive method combined with wavelet packet transform (WPT) is established with the help of receiver operating characteristics (ROC) and discrimination measure (DM) respectively.

Random environment integer‐valued autoregressive process

An r states random environment integer‐valued autoregressive process of order 1, RrINAR(1), is introduced. Also, a random environment process is separately defined as a selection mechanism of differently parameterized geometric distributions, thus ensuring the non‐stationary nature of the RrNGINAR(1) model based on the negative binomial thinning. The distributional and correlation properties of this model are discussed, and the k‐step‐ahead conditional expectation and variance are derived. Yule–Walker estimators of model parameters are presented and their strong consistency is proved. The RrNGINAR(1) model motivation is justified on simulated samples and by its application to specific real‐life counting data.

Consensus-based decentralized real-time identification of large-scale systems

In this paper an approach is proposed to decentralized multi-agent identification of large-scale systems represented by linear discrete-time stochastic MIMO models. It is assumed that each agent: (a) has access only to a subset of noisy input–output variables; (b) communicates local data processing results to its neighborhood. The proposed algorithm consists of two stages. The first stage is a consensus-based stochastic approximation algorithm for estimating input–output correlation functions, while at the second stage each agent utilizes a stochastic approximation algorithm with expanding truncations derived from the modified Yule–Walker equations in order to generate all the system parameter estimates. It is proved that under nonrestrictive assumptions concerning the system properties and the multi-agent network topology the estimates of the correlation functions converge almost surely to their true values and those of the system parameters to a solution of the modified Yule–Walker equations, assuming intermittent observations and communication outages. Conditions are also given for the strong consistency of the parameter estimates. Simulation results provide an illustration of the algorithm properties.

Identifying an autoregressive process disturbed by a moving-average noise using inner–outer factorization

This paper deals with the identification of an autoregressive (AR) process disturbed by an additive moving-average (MA) noise. Our approach operates as follows: Firstly, the AR parameters are estimated by using the overdetermined high-order Yule–Walker equations. The variance of the AR process driving process can be deduced by means of an orthogonal projection between two types of estimates of AR process correlation vectors. Then, the correlation sequence of the MA noise is estimated. Secondly, the MA parameters are obtained by using inner–outer factorization. To study the relevance of the resulting method, we compare it with existing algorithms, and we analyze the identifiability limits. The identification approach is then combined with Kalman filtering for channel estimation in mobile communication systems.

Improving Localization Accuracy: Successive Measurements Error Modeling.

Vehicle self-localization is an essential requirement for many of the safety applications envisioned for vehicular networks. The mathematical models used in current vehicular localization schemes focus on modeling the localization error itself, and overlook the potential correlation between successive localization measurement errors. In this paper, we first investigate the existence of correlation between successive positioning measurements, and then incorporate this correlation into the modeling positioning error. We use the Yule Walker equations to determine the degree of correlation between a vehicle’s future position and its past positions, and then propose a p-order Gauss–Markov model to predict the future position of a vehicle from its past p positions. We investigate the existence of correlation for two datasets representing the mobility traces of two vehicles over a period of time. We prove the existence of correlation between successive measurements in the two datasets, and show that the time correlation between measurements can have a value up to four minutes. Through simulations, we validate the robustness of our model and show that it is possible to use the first-order Gauss–Markov model, which has the least complexity, and still maintain an accurate estimation of a vehicle’s future location over time using only its current position. Our model can assist in providing better modeling of positioning errors and can be used as a prediction tool to improve the performance of classical localization algorithms such as the Kalman filter.

MULTIVARIATE AR SYSTEMS AND MIXED FREQUENCY DATA: G-IDENTIFIABILITY AND ESTIMATION

This paper is concerned with the problem of identifiability of the parameters of a high frequency multivariate autoregressive model from mixed frequency time series data. We demonstrate identifiability for generic parameter values using the population second moments of the observations. In addition we display a constructive algorithm for the parameter values and establish the continuity of the mapping attaching the high frequency parameters to these population second moments. These structural results are obtained using two alternative tools viz. extended Yule Walker equations and blocking of the output process. The cases of stock and flow variables, as well as of general linear transformations of high frequency data, are treated. Finally, we briefly discuss how our constructive identifiability results can be used for parameter estimation based on the sample second moments.

Histopathology Image Analysis and Classification for Cancer Detection Using 2D Autoregressive Model

In traditional methods of cancer diagnosis using clinical pathology, pathologists inspects biopsy samples and make diagnostic inferences. These diagnostics are based on cell morphology and tissue distribution which represents randomness in growth and/or in placement. These methods are highly subjective and sometimes lead to incorrect diagnosis. But computer assisted diagnostics (CAD) enables objective judgment that is based on huge collected database. This paper presents the use of 2D autoregressive (AR) model in automated cancer diagnosis based on histopathology. This work emphasizes the contribution of 2D autoregressive models for analysis and classification of histopathological images. Autoregressive model parameter represents feature set of histopathological image of biopsy samples removed from patients. These features are further used for analysis, synthesis and classification. Yule walker Least Square (LS) method has used for parameter estimation. The test statistics for choice of proper order of model has also been suggested in paper. It has been inferred that the proper neighborhood for a given class sample images is unique and solely depends on the properties of samples under consideration. AR parameters provide features for classification of sample in three classes like benign tissue, healthy tissue and malignant tissue.

SCAD-penalized regression for varying-coefficient models with autoregressive errors

Varying-coefficient models are useful extension of classical linear models. This paper is concerned with the statistical inference of varying-coefficient regression models with autoregressive errors. By combining the estimated residuals, the smoothly clipped absolute deviation (SCAD) penalty and Yule–Walker equations, a novel method is proposed to fit the error structure, including determining the order of the autoregressive error and efficiently estimating the autoregressive parameters. With appropriate selection of the tuning parameters, we establish the consistency of this method and the oracle property of the resulting regularized estimators. Based on the fitted autoregressive error structure, we further propose a two-stage local linear estimation for the unknown coefficient functions of the mean model to improve efficiency. The procedure is based on a pre-whitening transformation of the dependent variable. The resultant estimator of the unknown coefficient functions is asymptotically efficient and has the same asymptotic distribution as it would be if the autoregressive error structure were known with certainty. Simulation studies demonstrate that our asymptotic theory is applicable for finite samples, and the analysis of two real data sets illustrates the usefulness of our developed methodologies.

A Poisson INAR(1) process with a seasonal structure

This paper introduces a non-negative integer-valued autoregressive (INAR) process with seasonal structure of first order, which is an extension of the standard INAR(1) model proposed by Al-Osh and Alzaid [First-order integer-valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275]. The main properties of the model are derived such as its stationarity and autocorrelation function (ACF), among others. The conditional least squares and conditional maximum likelihood estimators of the model parameters are studied and their asymptotic properties are established. Some detailed discussion is dedicated to the case where the marginal distribution of the process is Poisson. A Monte Carlo experiment is conducted to evaluate and compare the performances of these estimators for finite sample sizes. The standard Yule–Walker approach is also considered for comparison purposes. The empirical results indicate that, in general, the conditional maximum likelihood estimator presents much better performance in terms of bias and mean square error. The model is illustrated using a real data set.

First order non-negative integer valued autoregressive processes with power series innovations

In this paper, we introduce a first order non-negative integer valued autoregressive process with power series innovations based on the binomial thinning. This new model contains, as particular cases, several models such as the Poisson INAR(1) model (Al-Osh and Alzaid (J. Time Series Anal. 8 (1987) 261–275)), the geometric INAR(1) model (Jazi, Jones and Lai (J. Iran. Stat. Soc. (JIRSS) 11 (2012) 173–190)) and many others. The main properties of the model are derived, such as mean, variance and the autocorrelation function. Yule–Walker, conditional least squares and conditional maximum likelihood estimators of the model parameters are derived. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Special sub-models are studied in some detail. Applications to two real data sets are given to show the flexibility and potentiality of the new model.

A random-coefficient third-order autoregressive process

Many systems and physical processes require non-Gaussian probabilistic models to accurately capture their dynamic behaviour. In this paper, we present a random-coefficient mathematical form that can be used to simulate a third-order Laplace autoregressive (AR) process. The mathematical structure of the random-coefficient AR process has a Markovian property that makes it flexible and simple to implement. A detailed derivation of its parameters as well as its pseudo-code implementation is provided. Moreover, it is shown that the process has an autocorrelation property that satisfies Yule–Walker type of equations. Having such an autocorrelation property makes the developed AR process, particularly, convenient for deriving mathematical models for dynamic systems, as well as signals, whose parameters of interest are Laplace distributed.

A Real Valued Neural Network Based Autoregressive Energy Detector for Cognitive Radio Application.

A real valued neural network (RVNN) based energy detector (ED) is proposed and analyzed for cognitive radio (CR) application. This was developed using a known two-layered RVNN model to estimate the model coefficients of an autoregressive (AR) system. By using appropriate modules and a well-designed detector, the power spectral density (PSD) of the AR system transfer function was estimated and subsequent receiver operating characteristic (ROC) curves of the detector generated and analyzed. A high detection performance with low false alarm rate was observed for varying signal to noise ratio (SNR), sample number, and model order conditions. The proposed RVNN based ED was then compared to the simple periodogram (SP), Welch periodogram (WP), multitaper (MT), Yule-Walker (YW), Burg (BG), and covariance (CV) based ED techniques. The proposed detector showed better performance than the SP, WP, and MT while providing better false alarm performance than the YW, BG, and CV. Data provided here support the effectiveness of the proposed RVNN based ED for CR application.

Modelling and coherent forecasting of zero-inflated count time series

In this article, a new kind of stationary zero-inflated Pegram’s operator based integer-valued time series process of order p with Poisson marginal or ZIPPAR( p) process is constructed for modelling count time series consisting a large number of zeros compared to standard Poisson time series processes. Several properties like stationarity, ergodicity are examined. Estimates of the model parameters are studied using three methods of estimation, namely Yule–Walker, conditional least squares and maximum likelihood estimation. Also h-step ahead coherent forecasting distributions of the proposed process for p = 1, 2 are derived. Real data set is used to examine and illustrate the proposed process with some simulation studies.

Moderate Deviation Principles for Empirical Covariance in the Neighbourhood of the Unit Root

AbstractIn this paper, we consider the linear autoregressive model with varying coefficients θn∈[0,1). When θn tending to the unit root, the moderate deviation principle for empirical covariance is discussed, and as statistical applications, we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter θn.

Two dimensional autoregressive estimation from noisy observations as a quadratic eigenvalue problem

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.

Adaptive Algorithm for Multichannel Autoregressive Estimation in Spatially Correlated Noise

This paper addresses the problem of multichannel autoregressive (MAR) parameter estimation in the presence of spatially correlated noise by steepest descent (SD) method which combines low-order and high-order Yule-Walker (YW) equations. In addition, to yield an unbiased estimate of the MAR model parameters, we apply inverse filtering for noise covariance matrix estimation. In a simulation study, the performance of the proposed unbiased estimation algorithm is evaluated and compared with existing parameter estimation methods.

A New High-Resolution Spectral Approach to Noninvasively Evaluate Wall Deformations in Arteries

By locally measuring changes on arterial wall thickness as a function of pressure, the related Young modulus can be evaluated. This physical magnitude has shown to be an important predictive factor for cardiovascular diseases. For evaluating those changes, imaging segmentation or time correlations of ultrasonic echoes, coming from wall interfaces, are usually employed. In this paper, an alternative low-cost technique is proposed to locally evaluate variations on arterial walls, which are dynamically measured with an improved high-resolution calculation of power spectral densities in echo-traces of the wall interfaces, by using a parametric autoregressive processing. Certain wall deformations are finely detected by evaluating the echoes overtones peaks with power spectral estimations that implement Burg and Yule Walker algorithms. Results of this spectral approach are compared with a classical cross-correlation operator, in a tube phantom and “in vitro” carotid tissue. A circulating loop, mimicking heart periods and blood pressure changes, is employed to dynamically inspect each sample with a broadband ultrasonic probe, acquiring multiple A-Scans which are windowed to isolate echo-traces packets coming from distinct walls. Then the new technique and cross-correlation operator are applied to evaluate changing parietal deformations from the detection of displacements registered on the wall faces under periodic regime.