Nonlinear Autoregressive Time Series Models Research Articles

Nonlinear semiparametric autoregressive models based on a class of two-piece distributions

In this study, we examined the well-known nonlinear autoregressive time series model in which innovations follow the flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family. The mentioned class of distributions is a rich class of distributions family that covers the robust symmetric/asymmetric light/heavy-tailed distributions. The nonlinear part of the autoregressive time series model is estimated via the semiparametric and nonparametric curve estimation based on the conditional least square method and nonparametric kernel approach. The maximum likelihood (ML) estimates of the model parameters, using a suitable hierarchical representation of the TP-SMN family on the model are obtained via an expectation-maximization type algorithm. Performances and usability of the proposed model and estimates are shown through simulation studies and a real dataset.

Penalized estimation of threshold auto-regressive models with many components and thresholds.

Thanks to their simplicity and interpretable structure, autoregressive processes are widely used to model time series data. However, many real time series data sets exhibit non-linear patterns, requiring nonlinear modeling. The threshold Auto-Regressive (TAR) process provides a family of non-linear auto-regressive time series models in which the process dynamics are specific step functions of a thresholding variable. While estimation and inference for low-dimensional TAR models have been investigated, high-dimensional TAR models have received less attention. In this article, we develop a new framework for estimating high-dimensional TAR models, and propose two different sparsity-inducing penalties. The first penalty corresponds to a natural extension of classical TAR model to high-dimensional settings, where the same threshold is enforced for all model parameters. Our second penalty develops a more flexible TAR model, where different thresholds are allowed for different auto-regressive coefficients. We show that both penalized estimation strategies can be utilized in a three-step procedure that consistently learns both the thresholds and the corresponding auto-regressive coefficients. However, our theoretical and empirical investigations show that the direct extension of the TAR model is not appropriate for high-dimensional settings and is better suited for moderate dimensions. In contrast, the more flexible extension of the TAR model leads to consistent estimation and superior empirical performance in high dimensions.

On stability Conditions of Pareto Autoregressive model

This paper concerned with studding a stability conditions of the proposed non-linear autoregressive time series model Known as Pareto Autoregressive model, acronym is defined by Pareto . A dynamical method Known as local linearization approximation method was used to obtain the stability condition of a non-zero singular point of Pareto model. In addition, we obtain the orbital stability condition of a limit cycle in terms of model parameters when the Pareto possesses a limit cycle with period .

An autoregressive model based on the generalized hyperbolic distribution

AbstractWe define a nonlinear autoregressive time series model based on the generalized hyperbolic distribution in an attempt to model time series with non‐Gaussian features such as skewness and heavy tails. We show that the resulting process has a simple condition for stationarity and it is also ergodic. An empirical example with a forecasting experiment is presented to illustrate the features of the proposed model.

On the Dynamics of Complex Valued Nonlinear Autoregressive Time Series Models and Stochastic Limit Cycles

On the Dynamics of Complex Valued Nonlinear Autoregressive Time Series Models and Stochastic Limit Cycles

On the complex dynamics of functional-coefficients nonlinear autoregressive time series models

In this article, a new chaotic functional-coefficient nonlinear autoregressive time series model is formulated. Asymptotic stability of the equilibria of the skeleton is studied. Complex dynamics of the proposed model are investigated by means of the bifurcations, time series diagrams, and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation. Two chaotic indicators, namely, the fractal dimension and the Lyapunov exponent methods are investigated for the model.

City Fire Forecasts and Analysis Based on Nonlinear Auto-Regressive Time-Series Model

The forecasting to future developments of the city fire time series is a challenging task that has been addressed by many researchers due to the importance. In this paper, a Nonlinear Auto-Regressive (NAR) prediction model is applied to forecast the city fire data based on support vector regression. The performances of the NAR prediction model in city fire forecasting are compared with the BP neural network method. The experimental results show that the proposed model performs best.

Variance estimation in nonlinear autoregressive time series models

This paper considers the variance estimation in nonlinear autoregressive time series models. The estimator, based on the residuals, is shown to be consistent (with n rate) for the error variance. The asymptotic distribution of the estimator is obtained to be normal.

General expression for linear and nonlinear time series models

The typical time series models such as ARMA, AR, and MA are founded on the normality and stationarity of a system and expressed by a linear difference equation; therefore, they are strictly limited to the linear system. However, some nonlinear factors are within the practical system; thus, it is difficult to fit the model for real systems with the above models. This paper proposes a general expression for linear and nonlinear auto-regressive time series models (GNAR). With the gradient optimization method and modified AIC information criteria integrated with the prediction error, the parameter estimation and order determination are achieved. The model simulation and experiments show that the GNAR model can accurately approximate to the dynamic characteristics of the most nonlinear models applied in academics and engineering. The modeling and prediction accuracy of the GNAR model is superior to the classical time series models. The proposed GNAR model is flexible and effective.

Distortion Correction Method Based on Mathematic Model in Machine Vision Measurement System

It is necessary to correct the nonlinear distortions in images of the machine vision measurement system which are vital to measuring precision. The method adopting a time-unvarying parameter model to correct the image distortions is difficult to achieve precise workpiece dimensions due to the stochastic variation of the distortions in production process. So a new nonlinear distortion correction method is proposed. The target part edge data series is described by a general expression for the linear and nonlinear auto-regressive time series model (GNAR model) and with the least square method and the minimum loss function criteria integrated with the computational complexity, the parameter estimation and order determination is realized. The distortion-free edges are obtained after model filtering and the workpiece dimensions are calculated through the vertical distances of the distortion-free edges. Experimental results show that the algorithm can be applied to machine vision measurement system for on-line measurement in machining process.

A goodness-of-fit test of the errors in nonlinear autoregressive time series models

This paper considers the problem of fitting an error density to the goodness-of-fit test of the errors in a nonlinear autoregressive stationary time series regression model. The test statistic is based on the integrated squared error of the nonparametric error density estimate and the null error density. Without knowing the nonlinear autoregressive function, we can show that the test statistic behaves asymptotically the same as the one based on the true errors.

A forecasting procedure for nonlinear autoregressive time series models

AbstractForecasting for nonlinear time series is an important topic in time series analysis. Existing numerical algorithms for multi‐step‐ahead forecasting ignore accuracy checking, alternative Monte Carlo methods are also computationally very demanding and their accuracy is difficult to control too. In this paper a numerical forecasting procedure for nonlinear autoregressive time series models is proposed. The forecasting procedure can be used to obtain approximate m‐step‐ahead predictive probability density functions, predictive distribution functions, predictive mean and variance, etc. for a range of nonlinear autoregressive time series models. Examples in the paper show that the forecasting procedure works very well both in terms of the accuracy of the results and in the ability to deal with different nonlinear autoregressive time series models. Copyright © 2005 John Wiley & Sons, Ltd.

Adaptive R-Estimation in Autoregressions

In this paper we explore the local asymptotic normality property with a ranked residual central sequence in autoregression to give locally asymptotically minimax estimators of the autoregressive parameter. Then, we use the kernel estimator method described by Koul and Schick [Koul, H. L., Schick, A. (1997). Efficient estimation in nonlinear autoregressive time series models. Bernoulli 3:247–277] to construct adaptive estimators. A simulation experiment is carried out to illustrate the performance of the proposed adaptive estimators.

Convergence Theory of a Numerical Method for Solving the Chapman--Kolmogorov Equation

A convergence theory has been established for a new numerical method for solving the Chapman--Kolmogorov equation [Y. Cai, A Numerical Forecasting Procedure for Nonlinear Autoregressive Time Series Model, manuscript, Department of Mathematics and Statistics, University of Surrey, Surrey, UK, 2001]. The theory has been applied to many types of nonlinear time series models in order to obtain m-step ahead predictive probability density function, predictive cumulative distribution function, predictive mean, and predictive variance.

A simple nonlinear time series model with misleading linear properties

This paper gives an example of a first-order nonlinear autoregressive time series model with short memory such that autocorrelations estimated from data generated by the model point at a long-memory model.

EXACT LEAST SQUARES MULTI‐STEP PREDICTION FROM NONLINEAR AUTOREGRESSIVE MODELS

Abstract. We obtain an integral equation recurrence relation for the optimal least squares predictor of a nonlinear autoregressive time series model. Numerical solutions are given for a first order threshold model up to three steps ahead.