Nonlinear Autoregressive Systems Research Articles

A gazelle optimization expedition for key term separated fractional nonlinear systems with application to electrically stimulated muscle modeling

Various real-time processes are effectively represented with a fractional nonlinear autoregressive exogenous (F-NARX) model where estimation of model parameters is considered as an essential task. In this study, a population-based gazelle optimization algorithm (GOA) inspired by the evolutionary characteristics of gazelle's is exploited for the parameter estimation of the key term-separated F-NARX system. The Grünwald-Letnikov derivative, a fractional order calculus operator is integrated to develop the F-NARX system from a conventional non-linear autoregressive system. The mean square error-based merit function is developed and the effectiveness of the GOA for the F-NARX system is analyzed in terms of speedy convergence, estimation accuracy, complexity and robustness for different noise scenarios. The extendibility of the GOA is assessed through the estimation of stiff parameters of the electrically stimulated muscle model (ESMM) required for rehabilitation of paralyzed muscles. The efficacy of the GOA is endorsed through Wilcoxon signed rank statistical test in comparison with recent counterparts of Runge Kutta optimization algorithm, Whale optimization algorithm, and Harris Hawks optimization algorithm.

A General Framework for Nonparametric Identification of Nonlinear Stochastic Systems

In this article, nonparametric identification of nonlinear autoregressive systems with exogenous inputs (NARX) is considered; a general criterion function is introduced for estimating the value of the nonlinear function within the system at any fixed point. The criterion function is constructed using a kernel together with a convex objective function. Not only does this framework include the classical kernel-based weighted least-squares estimator but also the kernel-based L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> ,l ≥ 1 criteria as special cases. First, we prove that the minimizer of the general criterion function converges to the true function value with probability 1. Second, recursive algorithms are proposed to find the estimates, which minimize the criterion function, and it is shown that these estimates also converge to the true function value with probability 1. Numerical examples are given, justifying that the framework guarantees the strong consistency of the estimates and exhibits the robustness against outliers in the observations.

A general convergence result for kernel-based nonparametric identification of nonlinear stochastic systems

In this paper we consider nonparametric identification of nonlinear autoregressive systems with exogenous inputs. Using a kernel function, a general criterion is introduced for estimating the values of the nonlinear function within the system at any fixed point. The criterion function includes the classical kernel based Ll, l > 1 criteria for nonparametric identification as special cases. By transforming the optimization of the criterion function into a root-finding problem, it is proved that the zero point of the root-finding function converges to the optimal value of the criterion function with probability one. A numerical example illustrating the convergence result is also given.

Kernel-based local order estimation of nonlinear nonparametric systems

We consider the local order estimation of nonlinear autoregressive systems with exogenous inputs (NARX), which may have different local dimensions at different points. By minimizing the kernel-based local information criterion introduced in this paper, the strongly consistent estimates for the local orders of the NARX system at points of interest are obtained. The modification of the criterion and a simple procedure of searching the minimum of the criterion, are also discussed. The theoretical results derived here are tested by simulation examples.

A Recursive Local Linear Estimator for Identification of Nonlinear ARX Systems: Asymptotical Convergence and Applications

In this paper, we propose a recursive local linear estimator (RLLE) for nonparametric identification of nonlinear autoregressive systems with exogenous inputs (NARX). First, the RLLE is introduced. Next, the strong consistency as well as the asymptotical mean square error properties of the RLLE are established, and then an application of the RLLE to an additive nonlinear system is discussed. The RLLE provides recursive estimates not only for the function values but also their gradients at fixed points. A simulation example is provided to confirm the theoretical analysis.

A Recursive Local Linear Estimator for Identification of Nonlinear ARX Systems

In this work, we propose a recursive local linear estimator (RLLE) for identification of nonlinear autoregressive systems with exogenous inputs, along with an analysis of its strong consistency and asymptotical mean square error properties. The performance of the proposed RLLE is verified by a simulation example.

New results on recursive identification of NARX systems

Recursive identification of nonlinear autoregressive system with exogenous inputs (NARX) yk + 1 = f(yk, ldots, y, uk, …, u) + ek + 1 is considered in this paper. Continuing a work on the same topic by the authors, the present paper covers a larger class of systems, uses less restrictive conditions, and provides deeper results. To be specific, first, by weakening conditions imposed on f(·) the class of systems has significantly been enlarged, for example, it now covers the Hammerstein system as a special case. Second, a technical condition has been removed that was previously imposed on the function coefficient characterizing the rate of convergence to the invariant measure. Third, not only the strong consistency but also the convergence rate of estimates have been established. The behavior of the estimates is demonstrated by some numerical examples. In particular, when applying to a Hammerstein system, the estimate given in this paper is put in a comparison with the Yule–Walker equation-based identification algorithms existing in the literature. Copyright © 2011 John Wiley & Sons, Ltd.

Recursive Identification for Nonlinear ARX Systems Based on Stochastic Approximation Algorithm

The nonparametric identification for nonlinear autoregressive systems with exogenous inputs (NARX) described by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yk</i> +1= <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yk</i> ,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yk</i> +1- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uk</i> ,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uk</i> +1- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> )+ε <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> +1 is considered. First, a condition on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> (·) is introduced to guarantee ergodicity and stationarity of { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yk</i> } . Then the kernel function based stochastic approximation algorithm with expanding truncations (SAAWET) is proposed to recursively estimate the value of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> (φ*) at any given φ* Δ/= [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1)</sup> ,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ), <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1)</sup> ,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> )] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</sup> ∈ <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</b> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . It is shown that the estimate converges to the true value with probability one. In establishing the strong consistency of the estimate, the properties of the Markov chain associated with the NARX system play an important role. Numerical examples are given, which show that the simulation results are consistent with the theoretical analysis. The intention of the paper is not only to present a concrete solution to the problem under consideration but also to profile a new analysis method for nonlinear systems. The proposed method consisting in combining the Markov chain properties with stochastic approximation algorithms may be of future potential, although a restrictive condition has to be imposed on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> (·), that is, the growth rate of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ) should not be faster than linear with coefficient less than 1 as || <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> || tends to infinity.

Nonparametric Algorithms for Identification of Nonlinear Autoregressive Systems with Exogenous Inputs

This paper is concerned with nonparametric identification of nonlinear autoregressive systems with exogenous inputs (NARX), i.e., yk+1 = f(yk, …, yk+1-n0, uk, …, uk+1-n0)+ɛk+1. Kernel functions based stochastic approximation algorithms with expanding truncations are designed for recursively estimating the value of f(·) at any given [y(1), …,y(n0), u(1), …,u(n0)]τ ∈ R2n0. The estimates are shown to be strongly consistent. The NARX systems considered in this paper include the one in Zhao, Chen & Zheng [2008] as a special case. A numerical example is given to justify the theoretical analysis.

Some Simple Remarks on an Autoregressive Scheme and an Implied Problem

Nonminimum phase autoregressive schemes are considered in the context of the problem of determining a stationary process that satisfies a possibly nonlinear autoregressive system of equations. It's noted that in most solutions there is an implicit assumption that the independent random variables generating the process are independent of the past of the process. This is not true of the non-minimum phase schemes. A simple scheme that has a characteristic polynomial with roots inside and outside the unit disc in the complex plane is discussed.

Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression

Abstract We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov Exponent λ1 from time series data generated by a nonlinear autoregressive system with additive noise. For systems with bounded fluctuations, λ1 > 0 is the defining feature of chaos. Thus our procedures can be used to examine time series data for evidence of chaotic dynamics. We show that a consistent estimator of the partial derivatives of the autoregression function can be used to obtain a consistent estimator of λ1. The rate of convergence we establish is quite slow; a better rate of convergence is derived heuristically and supported by simulations. Simulation results from several implementations—one “local” (thin-plate splines) and three “global” (neural nets, radial basis functions, and projection pursuit)—are presented for two deterministic chaotic systems. Local splines and neural nets yield accurate estimates of the Lyapunov exponent; however, the spline method is sensitive to the choice of the emb...