Three-stage recursive least squares parameter estimation for controlled autoregressive autoregressive systems

A three-stage recursive least squares parameter estimation algorithm is derived for controlled autoregressive autoregressive (CARAR) systems. The basic idea is to decompose a CARAR system into three subsystems, one of which contains one parameter vector, and to identify the parameters of each subsystem one by one. Compared with the recursive generalized least squares algorithm, the dimensions of the involved covariance matrices in each subsystem become small and thus the proposed algorithm has a high computational efficiency. Finally, we verify the proposed algorithm with a simulation example.     

Normalized fractional adaptive methods for nonlinear control autoregressive systems

The trend of applying mathematical foundations of fractional calculus to solve problems arising in nonlinear sciences, is an emerging area of research with growing interest especially in communication, signal analysis and control. In the present study, normalized fractional adaptive strategies are exploited for automatic tuning of the step size parameter in nonlinear system identification based on Hammerstein model. The brilliance of the methodology is verified by mean of viable estimation of electrically stimulated muscle model used in rehabilitation of paralyzed muscles. The dominance of the schemes is established by comparing the results with standard counterparts in case of different noise levels and fractional order variations. The results of the statistical analyses for sufficient independent runs in terms of Nash-Sutcliffe efficiency, variance account for and mean square error metrics validated the consistent accuracy and reliability of the proposed methods. The proposed exploitation of fractional calculus concepts makes a firm branch of nonlinear investigation in arbitrary order gradient-based optimization schemes.     

Subset selection for vector autoregressive processes via adaptive Lasso

Subset selection is a critical component of vector autoregressive (VAR) modeling. This paper proposes simple and hybrid subset selection procedures for VAR models via the adaptive Lasso. By a proper choice of tuning parameters, one can identify the correct subset and obtain the asymptotic normality of the nonzero parameters with probability tending to one. Simulation results show that for small samples, a particular hybrid procedure has the best performance in terms of prediction mean squared errors, estimation errors and subset selection accuracy under various settings. The proposed method is also applied to modeling the IS-LM data for illustration.     

Efficient estimation and variable selection for infinite variance autoregressive models

In this paper, a self-weighted composite quantile regression estimation procedure is developed to estimate unknown parameter in an infinite variance autoregressive (IVAR) model. The proposed estimator is asymptotically normal and more efficient than a single quantile regression estimator. At the same time, the adaptive least absolute shrinkage and selection operator (LASSO) for variable selection are also suggested. We show that the adaptive LASSO based on the self-weighted composite quantile regression enjoys the oracle properties. Simulation studies and a real data example are conducted to examine the performance of the proposed approaches.     

Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model

We derive the asymptotics of the OLS estimator for a purely autoregressive spatial model. Only low-level conditions are used. As the sample size increases, the spatial matrix is assumed to approach a square-integrable function on the square (0,1)². The asymptotic distribution is a ratio of two infinite linear combinations of χ² variables. The formula involves eigenvalues of an integral operator associated with the function approached by the spatial matrices. Under the conditions imposed identification conditions for the maximum likelihood method and method of moments fail. A corrective two-step procedure using the OLS estimator is proposed.     

Prediction of multivariate time series by autoregressive model fitting

Suppose the stationary r-dimensional multivariate time series {y_t} is generated by an infinite autoregression. For lead times h≥1, the linear prediction of y_t+h based on y_t, y_t−1,… is considered using an autoregressive model of finite order k fitted to a realization of length T. Assuming that k → ∞ (at some rate) as T → ∞, the consistency and asymptotic normality of the estimated autoregressive coefficients are derived, and an asymptotic approximation to the mean square prediction error based on this autoregressive model fitting approach is obtained. The asymptotic effect of estimating autoregressive parameters is found to inflate the minimum mean square prediction error by a factor of (1 + kr/T).     

Quantile regression for linear models with autoregressive errors using EM algorithm

In this paper, we consider the quantile linear regression models with autoregressive errors. By incorporating the expectation–maximization algorithm into the considered model, the iterative weighted least square estimators for quantile regression parameters and autoregressive parameters are derived. Finally, the proposed procedure is illustrated by simulations and a real data example.     

Moderate deviation principle for autoregressive processes

A moderate deviation principle for autoregressive processes is established. As statistical applications we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter of an autoregressive process. The main assumption on the autoregressive process is the Gaussian integrability condition for the noise, which is weaker than the assumption of Logarithmic Sobolev Inequality in [H. Djellout, A. Guillin, L. Wu, Moderate deviations of empirical periodogram and nonlinear functionals of moving average processes, Ann. I. H. Poincaré-PR 42 (2006) 393–416].     

An innovative fractional order LMS algorithm for power signal parameter estimation

Parameter estimation is an important issue for the quality monitoring and reliability assessment of power systems. In this study, an innovative fractional order least mean square (I-FOLMS) adaptive algorithm is presented for an effective parameter estimation. The I-FOLMS algorithm exploits the fractional gradient in its recursive parameter update mechanism, because its performance can be tuned by means of the fractional order. High values of the fractional order are good for fast convergence, but lead to steady state mis-adjustments. While, low values provide a smooth steady state behavior, but require a compromise in the convergence rate. The effective performance of I-FOLMS is verified and validated through two numerical examples of power signals estimation for different levels of noise variance and values of the fractional orders.     

Covariance operator estimation of a functional autoregressive process with random coefficients

We deal with the covariance and cross covariance operators estimation of a Hilbert space valued autoregressive process with random coefficients. We establish bounds for empirical estimators in mean square error and almost sure convergence in Hilbert–Schmidt norm. Consistent estimators of the eigenvalues are also derived.     

Empirical likelihood-based subset selection for partially linear autoregressive models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.     

Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise

In this paper we study the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order p with regular stationary Gaussian noise. We prove the large sample asymptotic properties of the MLE under very mild conditions. We do simulations for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).     

Bootstrap tests for unit roots in seasonal autoregressive models

This paper proposes bootstrap tests for the presence of unit roots in a seasonal autoregressive model. The asymptotic validity of the proposed bootstrap scheme is established, and Monte Carlo experiments are used to investigate the small-sample performance of the tests.     

A bias-compensated fractional order normalized least mean square algorithm with noisy inputs

Numerical Algorithms - This paper comes up with a stable bias-compensated fractional order normalized least mean square (BC-FONLMS) algorithm with noisy inputs. This kind of bias-compensated...     

Global property of error density estimation in nonlinear autoregressive time series models

This paper considers estimation of the error density function in nonlinear autoregressive stationary time series regression model. The asymptotic distribution of the maximum of a suitably normalized deviation of the density estimator from the expectation of the kernel error density (based on the true error) is obtained to be the same as in the case of the one sample set up, which is given in Bickel and Rosenblatt (Ann Stat 6:1071–1095, 1973).     

Bayesian inferences for several autoregressive processes

Several first-order autoregressive processes are studied using Bayesian analysis principles. These autoregressive processes are the same, except perhaps for the level of the process. A test is developed to detect possible level differences by using the mean value functions of these first-order autoregressive processes. To illustrate the test results, a numerical study is conducted using two time series groups.     

An efficient estimator for the expectation of a bounded function under the residual distribution of an autoregressive process

Consider a stationary first-order autoregressive process, with i.i.d. residuals following an unknown mean zero distribution. The customary estimator for the expectation of a bounded function under the residual distribution is the empirical estimator based on the estimated residuals. We show that this estimator is not efficient, and construct a simple efficient estimator. It is adaptive with respect to the autoregression parameter.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

The discrete-time queue with autoregressive inputs revisited

In this paper, we present an exact queuing analysis of a discrete-time queue whose arrival process is correlated and consists of a discrete autoregressive model of order 1 (DAR(1)). The functional equation describing this DAR(1)/D/1 queuing model, originally derived in Hwang and Sohraby (Queuing Systems 43 (2003)29–41), is manipulated and transformed into a mathematical tractable form. By using simple analytical transform techniques, we show how our proposed approach allows us to derive an equivalent (yet simpler) expression for the steady-state probability generating function (pgf) of the queue length, as originally derived in Hwang and Sohraby (Queuing Systems 43 (2003)29–41). From this pgf, we characterize the distribution of the packet delay. New numerical results related to packet loss ratio and mean delay of the DAR(1)/D/1 queue are also presented. The proposed approach outlines an alternate solution technique and a general framework under which more complex time-series based queuing models can be analyzed. AMS Subject Classifications 60K25     

Sequential fixed accuracy estimation for nonstationary autoregressive processes

For an autoregressive process of order p, the paper proposes new sequential estimates for the unknown parameters based on the least squares (LS) method. The sequential estimates use p stopping rules for collecting the data and presumes a special modification the sample Fisher information matrix in the LS estimates. In case of Gaussian disturbances, the proposed estimates have non-asymptotic normal joint distribution for any values of unknown autoregressive parameters. It is shown that in the i.i.d. case with unspecified error distributions, the new estimates have the property of uniform asymptotic normality for unstable autoregressive processes under some general condition on the parameters. Examples of unstable autoregressive models satisfying this condition are considered.