Exponential Autoregressive Model Research Articles

An Exponential Autoregressive Time Series Model for Complex Data

In this paper, an exponential autoregressive model for complex time series data is presented. As for estimating the parameters of this nonlinear model, a three-step procedure based on quantile methods is proposed. This quantile-based estimation technique has the benefit of being more robust compared to least/absolute squares. The performance of the introduced exponential autoregressive model is evaluated by means of four established goodness-of-fit criteria. The practical utility of the novel time series model is showcased through a comparative analysis involving simulation studies and real-world data illustrations.

Gradient-based Parameter Estimation for a Nonlinear Exponential Autoregressive Time-series Model by Using the Multi-innovation

Gradient-based Parameter Estimation for a Nonlinear Exponential Autoregressive Time-series Model by Using the Multi-innovation

Mathematical modeling and statistical analysis of Moroccan mean annual rainfall using EXPAR processes

In this work, we propose a study of the mean annual rainfall time series in order to evaluate the climate changes pattern over time. If the analysis of this time series is carried out correctly, it can contribute to improve planning and policy development. That is why we consider the problem of mathematical modeling and analysis of the mean annual rainfall of Morocco between 1901 and 2020 using descriptive statistics, structure changes analysis, spectral analysis and a nonlinear Exponential Autoregressive (EXPAR) processes to reproduce the behavior of this time series. The results indicate three main breakpoints and show that the time series has a remarkable cycles about 60, 18 and 6 years with a global decrease tendency about 0.56 mm per year. Furthermore, we have justified the choice of using a non-linear EXPAR processes rather than a linear traditional one and provided a good fitted EXPAR model.

Joint Parameter and Time-Delay Estimation for a Class of Nonlinear Time-Series Models

Nonlinear time-series modeling is fundamental to a wide variety of control and prediction problems. This letter focuses on the joint parameter and time-delay estimation for an extended version of the nonlinear exponential autoregressive (ExpAR) time-series model. To address the difficulties posed by the unknown time-delay and improve the estimation accuracy, we first employ the redundant rule to transform the ExpAR model into an augmented identification model. Then we invoke the multi-innovation theory to enhance data utilization and propose a new algorithm that combines stochastic gradient descent with discrete search for estimating the unknown model parameters and time-delay. The simulation results show that by properly adjusting the innovation length, the estimation accuracy of the proposed multi-innovation algorithm can significantly exceed that of the single-innovation algorithm.

TAC Method for Fitting Exponential Autoregressive Models and Others: Applications in Economy and Finance

There are a couple of purposes in this paper: to study a problem of approximation with exponential functions and to show its relevance for economic science. The solution of the first problem is as conclusive as it can be: working with the max-norm, we determine which datasets have best approximation by means of exponentials of the form f(t)=b+aexp(kt), we give a necessary and sufficient condition for some a,b,k∈R to be the coefficients that give the best approximation, and we give a best approximation by means of limits of exponentials when the dataset cannot be best approximated by an exponential. For the usual case, we have also been able to approximate the coefficients of the best approximation. As for the second purpose, we show how to approximate the coefficients of exponential models in economic science (this is only applying the R-package nlstac) and also the use of exponential autoregressive models, another well-established model in economic science, by utilizing the same tools: a numerical algorithm for fitting exponential patterns without initial guess designed by the authors and implemented in nlstac. We check one more time the robustness of this algorithm by successfully applying it to two very distant areas of economy: demand curves and nonlinear time series. This shows the utility of TAC (Spanish for CT scan) and highlights to what extent this algorithm can be useful.

Parameter estimation for an exponential autoregressive time series model by the Newton search and multi-innovation theory

ABSTRACT This paper focuses on the recursive parameter estimation problem of the exponential autoregressive (ExpAR) model. Applying the Newton search and multi-innovation theory, a multi-innovation Newton recursive algorithm is presented for identifying the ExpAR model. In order to improve the computational efficiency, the hierarchical identification principle is employed to decompose an ExpAR model into two sub-models, and to derive a hierarchical multi-innovation Newton recursive algorithm. A simulation example is provided to demonstrate the effectiveness of the proposed algorithms.

Comparison of Classical and Robust Estimates of Threshold Auto-regression Parameters

The exponential auto-regression model is a discrete analog of the second-order nonlinear differential equations of the type of Duffing and van der Pol oscillators. It is used to describe nonlinear stochastic processes with discrete time, such as vehicle vibrations, ship roll, electrical signals in the cerebral cortex. When applying the model in practice, one of the important tasks is its identification, in particular, an estimate of the model parameters from observations of the stochastic process it described. A traditional technique to estimate autoregressive parameters is the nonlinear least squares method. Its disadvantage is high sensitivity to the measurement errors of the process observed. The M-estimate method largely has no such a drawback. The M-estimates are based on the minimization procedure of a non-convex function of several variables. The paper studies the effectiveness of several well-known minimization methods to find the M-estimates of the parameters of an exponential autoregressive model. The paper demonstrates that the sequential quadratic programming algorithm, the active set algorithm, and the interior-point algorithm have shown the best and approximately the same accuracy. The quasi-Newton algorithm is inferior to them in accuracy a little bit, but is not inferior in time. These algorithms had approximately the same speed and were one and a half times faster than the Nelder-Mead algorithm and 14 times faster than the genetic algorithm. The Nelder-Mead algorithm and the genetic algorithm have shown the worst accuracy. It was found that all the algorithms are sensitive to initial conditions. The estimate of parameters, on which the autoregressive equation linearly depends, is by an order of magnitude more accurate than that of the parameter on which the auto-regression equation depends in a nonlinear way.

Gradient iterative-based kernel method for exponential autoregressive models

Two kernel method-based gradient iterative algorithms are proposed for exponential autoregressive (ExpAR) models in this study. A polynomial kernel function is utilised to transform the ExpAR model into a linear-parameter model. Since the order of the linear-parameter model is large, a momentum stochastic gradient algorithm and an adaptive step-length gradient iterative algorithm are developed. Both these two algorithms can estimate the parameters with less computational efforts. Finally, a simulation example shows that the proposed algorithms are effective.

Gradient iterative-based kernel method for exponential autoregressive models

Gradient iterative-based kernel method for exponential autoregressive models

Adaptive test for periodicity in restrictive EXPAR(p) models

Locally asymptotically most stringent test for testing the periodicity in the restricted exponential autoregressive model EXPAR(1) has been constructed by Merzougui et al. This test is generalized here to the restricted EXPAR of order p. On the other hand, we construct a semi-parametric optimal test of periodicity when the innovation density is unknown. The sufficient conditions of the Local Asymptotic Normality (LAN) property are adapted to the periodic model.

Data filtering‐based recursive identification for an exponential autoregressive moving average model by using the multi‐innovation theory

This study employs the data filtering technique to investigate the recursive identification problems for a non-linear exponential autoregressive model with moving average noise, i.e. the ExpARMA model. Whitening the ExpARMA model by a linear filter, the original identification model is divided into a filtered identification model and a coloured noise model, then a filtering-based extended stochastic gradient algorithm is derived. In order to improve the parameter estimation accuracy, the multi-innovation identification theory is used to develop a filtering-based multi-innovation extended stochastic gradient algorithm for the ExpARMA model. A simulation example is given to demonstrate the superiority of the proposed filtering-based multi-innovation algorithm over the existing algorithms.

Two‐stage recursive identification algorithms for a class of nonlinear time series models with colored noise

SummaryThis article concentrates on the recursive identification algorithms for the exponential autoregressive model with moving average noise. Using the decomposition technique, we transform the original identification model into a linear and nonlinear subidentification model and derive a two‐stage least squares (LS) extended stochastic gradient (ESG) algorithm. In order to improve the parameter estimation accuracy, we employ the multi‐innovation identification theory and develop a two‐stage LS multi‐innovation ESG algorithm. A simulation example is provided to test the effectiveness of the proposed algorithms.

Robust Identification of an Exponential Autoregressive Model

One of the most common nonlinear time series (random processes with discrete time) models is the exponential autoregressive model. In particular, it describes such nonlinear effects as limit cycles, resonant jumps, and dependence of the oscillation frequency on amplitude. When identifying this model, the problem arises of estimating its parameters --- the coefficients of the corresponding autoregressive equation. The most common methods for estimating the parameters of an exponential model are the least squares method and the least absolute deviation method. Both of these methods have a number of disadvantages, to eliminate which the paper proposes an estimation method based on the robust Huber approach. The obtained estimates occupy an intermediate position between the least squares and least absolute deviation estimates. It is assumed that the stochastic sequence is described by the autoregressive equation of the first order, is stationary and ergodic, and the probability distribution of the innovations process of the model is unknown. Unbiased, consistency and asymptotic normality of the proposed estimate are established by computer simulation. Its asymptotic variance was found, which allows to obtain an explicit expression for the relative efficiency of the proposed estimate with respect to the least squares estimate and the least absolute deviation estimate and to calculate this efficiency for the most widespread probability distributions of the innovations sequence of the equation of the autoregressive model

Recursive search‐based identification algorithms for the exponential autoregressive time series model with coloured noise

This study focuses on the recursive parameter estimation problems for the non-linear exponential autoregressive model with moving average noise (the ExpARMA model for short). By means of the gradient search, an extended stochastic gradient (ESG) algorithm is derived. Considering the difficulty of determining the step-size in the ESG algorithm, a numerical approach is proposed to obtain the optimal step-size. In order to improve the parameter estimation accuracy, the authors employ the multi-innovation identification theory to develop a multi-innovation ESG (MI-ESG) algorithm for the ExpARMA model. Introducing a forgetting factor into the MI-ESG algorithm, the parameter estimation accuracy can be further improved. With an appropriate innovation length and forgetting factor, the variant of the MI-ESG algorithm is effective to identify all the unknown parameters of the ExpARMA model. A simulation example is provided to test the proposed algorithms.

Detection of EXPAR nonlinearity in the Presence of a Nuisance Unidentified Under the Null Hypothesis

This work considers the problem of detecting the eventual existence of an exponential component in autoregressive models of order p ≥ 1. This problem comes down to test a linear dependence AR(p) against a nonlinear one of exponential autoregressive model EXPAR(p). Following Le Cam’s asymptotic theory, we have established the local asymptotic normality of EXPAR(p) models in the vicinity of AR(p) ones. Then, we have used pseudo-Gaussian methods to extract a pseudo-Gaussian test which is locally asymptotically optimal under Gaussian densities and valid under a large class of non-Gaussian ones. The main problem arising in this context is the fact that the test statistic’s involves nuisance parameter unidentified under the null hypothesis. Contrary to the simple case of order 1 the test statistic’s depends on this nuisance through a complex function. So as to solve this problem, we suggest a method which consists to take the maximum of the test statistic’s over a specific compact set of the nuisance parameter, then, we use the AR-sieve bootstrap procedure to approximate its asymptotic distribution.

Compensated closed-loop Kalman filtering for nonlinear systems

In this paper, we introduce new state estimation algorithm for nonlinear systems with the measurement loss at output channels. The goal of proposed methodology is to compensate for the output state-measurement losses through an advanced scheme that relies on the nonlinear prediction (NLP) and Kalman filter. Using NLP subsystem, based on Exponential Autoregressive (EXPAR) model, a compensated observation signal is first regenerated and then it is delivered to nonlinear discrete filtering or linearized Kalman filter in the form of closed-loop. Two proposed scheme versions (EXPAR-Substitution and EXPAR-Elimination) are presented in this paper. A two-phase permanent magnet synchronous motor (PMSM) case study is exposed to loss of measurements and then the loss is reconstructed using EXPAR-based schemes to demonstrate some capable results of the proposed algorithm. Simulation results of the proposed algorithm are found to be superior compared to other algorithms in the literature.

Integration & Volatility across Global Indices in different Time Zones: Some Evidence


 This paper examines the association of value instability crosswise over Global Indices of seven securities exchanges. Utilizing every day information of these seven nations situated in various time zones, this paper attempts to call attention to the nearness of nonsynchronous exchanging impacts utilizing open and close logarithmic returns of seven securities exchange files including Indian Indexat the middle. The hilter kilter effect of unpredictability overflow is analyzed by a multivariate exponential general autoregressive restrictive heteroskedastic model utilizing an example of 1742 perceptions taken from Oct 2011 to November 2018. The test outcomes give out many fascinating actualities alongside cost and unpredictability overflow from one market to the next because of time zone impact and additionally, influence impact is seen from the eastern markets' nearby value child Indian file open cost.

On some parameter estimation algorithms for the nonlinear exponential autoregressive model

SummaryModeling an exponential autoregressive (ExpAR) time series is the basis of solving the corresponding prediction and control problems. This paper investigates the hierarchical parameter estimation methods for the ExpAR model. By the hierarchical identification principle, the original nonlinear optimization problem is transformed into the combination of a linear and nonlinear optimization problem, and then, we derive a hierarchical least squares and stochastic gradient (LS‐SG) algorithm. Given the difficulty of determining the step‐size in the hierarchical LS‐SG algorithm, an approach is proposed to obtain the optimal step‐size. To improve the parameter estimation accuracy, the multi‐innovation identification theory is employed to develop a hierarchical least squares and multi‐innovation stochastic gradient algorithm for the ExpAR model. Two simulation examples are provided to test the effectiveness of the proposed algorithms.

Fitting the exponential autoregressive model through recursive search

This paper focuses on the recursive parameter estimation methods for the exponential autoregressive (ExpAR) model. Applying the negative gradient search and introducing a forgetting factor, a stochastic gradient and a forgetting factor stochastic gradient algorithms are presented. In order to improve the parameter estimation accuracy and the convergence rate, the multi-innovation identification theory is employed to derive a forgetting factor multi-innovation stochastic gradient algorithm. A simulation example is provided to test the effectiveness of the proposed algorithms.

Modeling a nonlinear process using the exponential autoregressive time series model

The parameter estimation methods for the nonlinear exponential autoregressive (ExpAR) model are investigated in this work. Combining the hierarchical identification principle with the negative gradient search, we derive a hierarchical stochastic gradient algorithm. Inspired by the multi-innovation identification theory, we develop a hierarchical-based multi-innovation identification algorithm for the ExpAR model. Introducing two forgetting factors, a variant of the hierarchical-based multi-innovation identification algorithm is proposed. Moreover, to compare and demonstrate the serviceability of these algorithms, a nonlinear ExpAR process is taken as an example in the simulation.