Nonlinear Autoregressive Time Series Research Articles

On the use of estimating functions in monitoring time series for change points

A large class of estimators including maximum likelihood, least squares and M-estimators are based on estimating functions. In sequential change point detection related monitoring functions can be used to monitor new incoming observations based on an initial estimator, which is computationally efficient because possible numeric optimization is restricted to the initial estimation. In this work, we give general regularity conditions under which we derive the asymptotic null behavior of the corresponding tests in addition to their behavior under alternatives, where conditions become particularly simple for sufficiently smooth estimating and monitoring functions. These regularity conditions unify and extend a large amount of existing procedures in the literature, while they also allow us to derive monitoring schemes in time series that have not yet been considered in the literature including non-linear autoregressive time series and certain count time series such as binary or Poisson autoregressive models. We do not assume that the estimating and monitoring functions are equal or even of the same dimension, therefore, we allow for example to combine a non-robust but more precise initial estimator with a robust monitoring scheme. Some simulations and data examples illustrate the usefulness of the described procedures.

Distance Prediction for Commercial Serial Crime Cases Using Time Delay Neural Networks

The prediction of the next serial criminal time is important in the field of criminology for preventing the recurring actions of serial criminals. In the associated dynamic systems, one of the main sources of instability and poor performances is the time delay, which is commonly predicted based on nonlinear methods. The aim of this study is to introduce a dynamic neural network model by using nonlinear autoregressive time series with exogenous (external) input (NARX) and Back Propagation Through Time (BPTT), which is verified intensively with MATLAB to predict and model the crime times for the next distance of serial cases. Recurrent neural networks have been extensively used for modeling of nonlinear dynamic systems. There are different types of recurrent neural networks such as Time Delay Neural Networks (TDNN), layer recurrent networks, NARX, and BPTT. The NARX model for the two cases of input- output modeling of dynamic systems and time series prediction draw more attention. In this study, a comparison of two models of NARX and BPTT used for the prediction of the next serial criminal time illustrates that the NARX model exhibits better performance for the prediction of serial cases than the BPTT model. Our future work aims to improve the NARX model by combining objective functions.

Asymptotics of the Lp-Norms of Density Estimators in the Nonlinear Autoregressive Models

This article investigates the asymptotic behavior of the error density function in nonlinear autoregressive stationary time series regression models. For any 1 ⩽ p < ∞, the kernel density estimator of residuals is shown to be consistent for the error estimator concerning the Lp-distance, which extends the result developed by Cheng and Sun (2008) in L2-norm. Moreover, the result developed in this article is extended the results of Horváth and Zitikis (2003) to nonlinear autoregressive models.

Non Standard Behavior of Density Estimators for Functions of Independent Observations

Densities of functions of two or more independent random variables can be estimated by local U-statistics. Frees (1994) gave conditions under which they converge pointwise at the parametric root-n rate. Uniform convergence at this rate was established by Schick and Wefelmeyer (2004b) for sums of random variables. Giné and Mason (2007) gave conditions under which this rate also holds in Lp-norms. We present several natural applications in which the parametric rate fails to hold in Lp or even pointwise. 1. The density estimator of a sum of squares of independent observations typically slows down by a logarithmic factor. For exponents greater than two, the estimator behaves like a classical density estimator. 2. The density estimator of a product of two independent observations typically has the root-n rate pointwise, but not in Lp-norms. An application is given to semi-Markov processes and estimation of an inter-arrival density that depends multiplicatively on the jump size. 3. The stationary density of a nonlinear or nonparametric autoregressive time series driven by independent innovations can be estimated by a local U-statistic (now based on dependent observations and involving additional parameters), but the root-n rate can fail if the derivative of the autoregression function vanishes at some point.

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.

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).

PIGGYBACKING THRESHOLD PROCESSES WITH A FINITE STATE MARKOV CHAIN

The state-space representations of certain nonlinear autoregressive time series are general state Markov chains. The transitions of a general state Markov chain among regions in its state-space can be modeled with the transitions among states of a finite state Markov chain. Stability of the time series is then informed by the stationary distributions of the finite state Markov chain. This approach generalizes some previous results.

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.

Boosting nonlinear additive autoregressive time series

Several methods for the analysis of nonlinear time series models have been proposed. As in linear autoregressive models the main problems are model identification, estimation and prediction. A boosting method is proposed that performs model identification and estimation simultaneously within the framework of nonlinear autoregressive time series. The method allows one to select influential terms from a large number of potential lags and exogenous variables. The influence of the selected terms is modeled by an expansion in basis function allowing for a flexible additive form of the predictor. The approach is very competitive in particular in high dimensional settings where alternative fitting methods fail. This is demonstrated by means of simulations and two applications to real world data.

Neural networks for bandwidth selection in local linear regression of time series

The problem of automatic bandwidth selection in nonparametric regression is considered when a local linear estimator is used to derive nonparametrically the unknown regression function. A plug-in method for choosing the smoothing parameter based on the use of the neural networks is presented. The method applies to dependent data generating processes with nonlinear autoregressive time series representation. The consistency of the method is shown in the paper, and a simulation study is carried out to assess the empirical performance of the procedure.

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 &amp; 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.

Non-Linear Autoregressive Time Series with Multivariate Gaussian Mixtures as Marginal Distributions

SUMMARY A new form of non-linear autoregressive time series is proposed to model solar radiation data, by specifying joint marginal distributions at low lags to be multivariate Gaussian mixtures. The model is also a type of multiprocess dynamic linear model, but with the advantage that the likelihood has a closed form.

Time-series forecasting using GA-tuned radial basis functions

Abstract In this paper we provide a nonlinear auto-regressive (NAR) time-series model for forecasting applications. The nonlinearity is introduced by using radial basis functions. RBF networks are widely used in time-series analysis. Three main parameter sets are involved in RBF learning process. They are the centers and widths of the radial functions, and their weights. Although the selection of the RBF centers and widths is important, most reported research has dealt only with the problem of weight optimization by making assumptions about the centers and widths. Therefore, there is no guarantee for finding the global optimum with respect to all sets of parameters. In this paper we use genetic algorithms (GAs) to simultaneously optimize all of the RBF parameters so that an effective time-series is designed and used for forecasting. An example is presented with promising results.

Nonparametric Lag Selection for Time Series

A nonparametric version of the Final Prediction Error (FPE) is analysed for lag selection in nonlinear autoregressive time series under very general conditions including heteroskedasticity. We prove consistency and derive probabilities of incorrect selections that have been previously unavailable. Since it is more likely to overfit (have too many lags) than to underfit (miss some lags), a correction factor is proposed to reduce overfitting and hence increase correct fitting. For the FPE calculation, the local linear estimator is introduced in addition to the Nadaraya‐Watson estimator in order to cover a very broad class of processes. To achieve faster computation, a plug‐in band‐width is suggested for the local linear estimator. Our Monte‐Carlo study corroborates that the correction factor generally improves the probability of correct lag selection for both linear and nonlinear processes and that the plug‐in bandwidth works at least as well as its commonly used competitor. The proposed methods are applied to the Canadian lynx data and daily returns of DM/US‐Dollar exchange rates.

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.