Bilinear Time Series Models Research Articles

Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

Abstract. Here we obtain difference equations for the higher order moments and cumulants of a time series {Xt} 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.

Estimation of Some Bilinear Time Series Models with Time Varying Coefficients

We discuss the problem of estimating the coefficients of the time-varying bilinear model where (ξ t ) t∈ℤ is an independent white noise process, with zero mean and variance and a t (a), b t (b) are the coefficients of the model. The consistency and asymptotic normality of the least squares estimates are established. Applications to seasonal and conditionally heteroscedastic bilinear models are considered.

CLS asymptotic variance for a particular relevant bilinear time series model

The main purpose of this paper is to gain inference about the parameter estimated via the CLS method for a particular bilinear model. We propose a new CLS estimator which is strongly consistent and the CLT and LIL hold with milder conditions on the moments. Furthermore, we derive a closed form expression for the asymptotic variance of this CLS estimator.

On the Covariance Structure of Time Varying Bilinear Models

In this paper, a class of bilinear time series models with time varying coefficients is considered. In this nonstationary and nonlinear framework, our aim is to study the structure of usual time series analysis tools, in particular the sample autocovariance function which has been developed for analyzing stationary linear time series. We use appropriately defined Markovian representations to derive a necessary and sufficient condition for the existence and uniqueness of a solution with bounded first and second order moments (BFSM). A more explicit sufficient condition for the existence of a BFSM solution is provided. An explicit expression of the autocovariance function is obtained. The existence of a weak time-varying ARMA representation of the bilinear model with time varying coefficients is shown. We also discuss the existence of higher order moments. Several subclasses of the model are shown to be quasi-stationary. Under this assumption of quasi-stationarity, the asymptotic distributions of the sample mean and sample covariances are obtained.

On the Selection of Subset Bilinear Time Series Models: a Genetic Algorithm Approach

This paper explores the idea of using a Genetic algorithm (GA) to solve the problem of subset model selection within the class of bilinear time series processes. The research is based on the concept of evolution theory as well as that of survival of the fittest. We use the AIC, BIG or SBC criteria as the adaptive functions to measure the degree of fitness. During the GA process, the best-fitted population is selected and certain characteristics are translated into the next generation. Simulation results demonstrate that genetic-based learning can effectively work out a pattern of the underlying time series. Finally, we illustrate how the GA can be applied successfully to subset selection in a bilinear time series via several examples and a simulation study.

On some distributional properties of a first-order nonnegative bilinear time series model

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

On some distributional properties of a first-order nonnegative bilinear time series model

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

Markovian representation of a bilinear time series model and maximum likelihood estimation of the parameters

A bilinear time series (BLTS) model is expressed in the form of Akaike's Markovian representation in order to use the Kalman recursive estimation approach. It is shown that Akaike's Markovian representation of autoregressive moving average models of orderp and q (ARMA(p,q)) and that of the bilinear model are equivalent. This equivalence facilitates the maximum likelihood estimation of the parameters involved in the bilinear model, which otherwise is an unwieldy problem. The present approach can easily be extended to take into account missing observations

Bilinear models in adaptive control

A key issue, which is addressed in the paper, is the development of an efficient adaptive control algorithm for disturbed, non-linear objects. A generalized non-linear model of the controlled plant is proposed. Model of disturbances is assumed to be a bilinear time-series model. Minimum-variance control of non-linear systems with bilinear model of disturbances is considered. For a generalized system model simple, weighed and generalized adaptive control algorithms are proposed. A comparison of different non-linear adaptive control algorithms is performed. Copyright © 2000 John Wiley & Sons, Ltd.

Optimization of bilinear time series models using fast evolutionary programming

This letter presents a new algorithm, fast evolutionary programming (FEP), for determining the model orders and parameters of reduced parameter bilinear (RPBL) models used for predicting nonlinear and chaotic time series. FEP is a variant of the conventional evolutionary programming (EP) algorithm with a new mutation operator. This new mutation operator enhances EP's ability to escape from local minima resulting in a significantly faster convergence to the optimal solution. Both the model order and the parameters are evolved simultaneously. Experimental results on the sunspot series and Mackey-Glass series show that FEP is capable of determining the optimal model order and, in comparison with conventional evolutionary programming, evolves models with lower normalized mean squared error.

Robust estimation of bilinear time series models

Outliers in time series seriously affect conventional least squares (LS) procedures badly. We propose to study the problem of fitting a bilinear model to time series data in presence of additive outliers (AO). We investigate a modification of some robustified versions of methods used in linear time series models. A Monte Carlo study is performed to investigate the robustness properties of the proposed class of estimates.

Extremes of bilinear time series models

The class of bilinear time series models is an obvious generalization of linear ARMA models and has found many applications in time series modeling. It is known that the sample paths of even the simplest bilinear process may have sudden bursts of large negative and positive values that vary in form and amplitude depending on the model parameters. Yet, little is known about the extremal properties of this class. In this paper, we look at the extremal properties of bilinear processes and explain how model parameters affect the extremal behavior.

A CONDITIONAL LEAST SQUARES APPROACH TO BILINEAR TIME SERIES ESTIMATION

Abstract.In this paper a conditional least squares (CLS) procedure for estimating bilinear time series models is introduced. This method is applied to a special superdiagonal bilinear model which includes the classical linear autoregressive moving‐average model as a particular case and it is proven that the limiting distribution of the CLS estimates is Gaussian and that the law of the iterated logarithm holds.

Potential problems in estimating bilinear time-series models

Abstract We present evidence of undesirable statistical properties of the bilinear model. We show that in certain regions of the parameter space the expected likelihood function exhibits bimodality. The true optimum is often characterized by a long, narrow spike that becomes more pronounced with increases in the sample size. Consequently, conventional optimization routines would frequently miss a global optimum with this feature. Moreover, although statistical tests would have strong power against alternatives near the global optimum, they would have almost zero power against the local minimum. This phenomenon also continues to persist in extremely large samples. In addition, we show that the distributional properties of the parameter estimates and the standard t -statistic also do not have desirable properties in finite samples.

Adaptive Prediction of Bilinear Time Series - Problems of Identification and Numerical Realization

Prediction has applications in feedforward control, time series analysis, and many other situations in which knowledge of the future behavior of process is desired. In this paper stochastic bilinear predictor for the general form of bilinear time series model is considered. To build the minimum variance prediction algorithm first one have to identify the bilinear model of the time series and then design the prediction algorithm using one of the rules proposed in the paper. In the paper a new version of adaptive bilinear predictor for the general bilinear model of time series is proposed. The predictor minimizes the variance of a global prediction error. Asymptotic properties of the algorithm are discussed. A numerical realization of proposed algorithm is described. Results of direct linear and bilinear prediction of bilinear time series are compared. Minimum variance adaptive linear prediction of the bilinear time series is compared with the bilinear one.

On the identification problem for bilinear time series models

In recent years there has been a growing interest in studying bilinear time series models. However, there are difficult problems related to the order identification of these models. In this paper, we consider the bilinear time series models and propose some methods of order identification based on the structure of autocovarance of and the third-order-automoment of {X t}. Decision rules as well as simulated bilinear time series are compared. An advantage of our methods is its simplicity of implementation.

Recursive estimation of bilinear time series models

In this paper we are concerned with the recursive estimation of bilinear models. Some methods from linear time invariant systems are adapted to suit bilinear time series models. The time-varying Kalman filter and associated parameter estimation algorithm is carried on the bilinear time series models. The methods are illustrated with examples.

NONPARAMETRIC TESTS FOR SERIAL DEPENDENCE

Abstract. A nonparametric test statistic based on the distance between the joint and marginal densities is developed to test for the serial dependence for a given sequence of time series data. The key idea lies in observing that, under the null hypothesis of independence, the joint density of the observations is equal to the product of their individual marginals. Histograms are used in constructing such a statistic which is nonparametric and consistent. It possesses high power in capturing subtle or diffuse dependence structure. A bilinear time series model is used to illustrate its performance with the classical correlation approach.

Bayesian inferences and forecasting in bilinear time series models

In this paper, we propose a Bayesian approach to the analysis of bilinear time series which is an extension of Broemeling and Shaarawy's work (1988) on linear time series. The conjugate prior for parameters is used to derive the predictive distribution and the marginal posterior distribution of the bilinear parameters, by which we make inferences about the parameters and for a future observation. Our results are illustrated using the Wolf sunspot numbers from Box and Jenkins (1976) and are compared with a linear time series.

Bayesian analysis of bilinear time series models : a gibbs sampling approach

Nonlinear time series analysis plays an important role in recent econometric literature, especially the bilinear model. In this paper, we cast the bilinear time series model in a Bayesian framework and make inference by using the Gibbs sampler, a Monte Carlo method. The methodology proposed is illustrated by using generated examples, two real data sets, as well as a simulation study. The results show that the Gibbs sampler provides a very encouraging option in analyzing bilinear time series.