Nonlinear Autoregressive Moving Average Research Articles

Comparison of non-linear time series models (Beta-t-EGARCH and NARMAX models) with Radial Basis Function Neural Network using Real Data

This paper presents a comparison of three different non-linear time series modelling approaches: NARMAX (Non-linear Autoregressive Moving Average with Exogenous Inputs), Beta-t-EGARCH (Beta t Exponential Generalized Autoregressive Conditional Heteroscedasticity), and Radial Basis Function Neural Networks (RBFNN) applied to weekly stock market index data. We will explain three types of models and compare their compositions and structures. Then, we will show which model gives better predictions. To study series data, the comparison involved analysing the structure of the model and its errors in various time series models and summarising their findings. We divide the data into two parts: training data to structure the time series and testing. The training data tests the model's predictions. Then, we can analyse the model with the errors and the best deterrence predictions. After selecting the NARMAX and Beta-t-EGARCH models, we test them with specific criteria. The best choice is finding the model with the lowest average errors. For this study, we analysed the weekly average closing of the Aramco 2222 index from 15 December 2019 to 16 July 2023 and made 187 observations.

Comparison of non-linear time series models (Beta-t-EGARCH and NARMAX models) with Radial Basis Function Neural Network using Real Data

This paper presents a comparison of three different non-linear time series modelling approaches: NARMAX (Non-linear Autoregressive Moving Average with Exogenous Inputs), Beta-t-EGARCH (Beta t Exponential Generalized Autoregressive Conditional Heteroscedasticity), and Radial Basis Function Neural Networks (RBFNN) applied to weekly stock market index data. We will explain three types of models and compare their compositions and structures. Then, we will show which model gives better predictions. To study series data, the comparison involved analysing the structure of the model and its errors in various time series models and summarising their findings. We divide the data into two parts: training data to structure the time series and testing. The training data tests the model's predictions. Then, we can analyse the model with the errors and the best deterrence predictions. After selecting the NARMAX and Beta-t-EGARCH models, we test them with specific criteria. The best choice is finding the model with the lowest average errors. For this study, we analysed the weekly average closing of the Aramco 2222 index from 15 December 2019 to 16 July 2023 and made 187 observations.

Optical phase encoding in a pulsed approach to reservoir computing.

The exploitation of the full structure of multimode light fields enables compelling capabilities in many fields including classical and quantum information science. We exploit data-encoding on the optical phase of the pulses of a femtosecond laser source for a photonic implementation of a reservoir computing protocol. Rather than intensity detection, data-reading is done via homodyne detection that accesses combinations of an amplitude and a phase of the field. Numerical and experimental results on nonlinear autoregressive moving average (NARMA) tasks and laser dynamic predictions are shown. We discuss perspectives for quantum-enhanced protocols.

Online learning of stable robust adaptive controllers design based on data-dependent feedback linearization with application to rotary inverted pendulum

This study introduces an online (supervised) learning method to design nonlinear auto-regressive moving average (NARMA) controllers for feedback-linearized nonlinear single-input single-output (SISO) systems. The algorithm ensures Schur stability of the overall closed-loop system and provides adaptiveness and robustness for the NARMA controllers. The first stage of the method derives, in a data-dependent way, a feedback-linearized model of the nonlinear plant by using its input and output sample pairs. The method’s second stage, which constitutes the novel part of the presented study, builds up an online learning scheme for the linear auto-regressive moving average (ARMA) controller based on an already learned feedback-linearized model of the nonlinear plant. During online supervised learning, ARMA parameters of the feedback-linearized SISO plant model and the closed-loop ARMA model are computed by minimizing the plant identification and the closed-loop system tracking errors. Both errors are defined as ℓ1,ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\ell}}_{1,{\\varvec{\\varepsilon}}}$$\\end{document}, namely ε-insensitive loss functions that provide NARMA controller the robustness against noise and outliers. The proposed online learning control algorithm is applied to a rotary inverted pendulum model and to a real rotary inverted pendulum setup. The tracking performance of the developed controller is compared with those of the linear quadratic regulator and coupled sliding mode controller in terms of mean square error.

Hysteresis modeling and compensation of a piezoelectric fast steering platform using multiple NARMA-L2 models

The piezoelectric fast steering platform (PFSP) enhances the scanning capabilities of lidar systems for autonomous vehicles. However, hysteresis nonlinearity degrades the scanning accuracy of the PFSP. To meet this challenge, this paper proposes a novel approach for hysteresis modeling and compensation using multiple nonlinear autoregressive moving average (NARMA)-L2 models. First, the hysteresis model is partitioned into a collection of submodels based on the K-means methodology, where the feature vector stems from the classical Bouc–Wen (CBW) model. Each submodel describes the dynamics of a specific hysteresis segment by a single NARMA-L2 model. The composite neural network (CNN), formed by combining two multilayer perceptron (MLP) networks, is trained to estimate the hysteresis output of the single NARMA-L2 model. Compared to the submodels obtained by evenly partitioning according to the input range, the submodels obtained by K-means partitioning offer higher hysteresis modeling accuracy. To compensate for the hysteresis nonlinearity, a novel adaptive Multi-NARMA-L2 (AMNL2) controller is then designed by online adjusting the parameters of the CNN. Furthermore, an adaptive filter is proposed to alleviate the oscillation caused by high-frequency disturbances during switching. Finally, experiments demonstrate that the AMNL2 controller outperforms the traditional MNL2 controller and the proportional–integral–derivative (PID) controller.

Modeling inflation rate factors on present consumption price index in Ethiopia: threshold autoregressive models approach

BackgroundInflation is the industrious and non-stop ascent in the overall prices of any given commodity in an economy. During the global food crisis, Ethiopia experienced an unprecedented increase in inflation ranked the highest in Africa. It is among the most macroeconomic variable described nonlinear behavior.ObjectiveThe main purpose of this study was intended to modeling inflation rate factors on present consumption price index (CPI) in Ethiopia: using the threshold autoregressive (TAR) models.MethodsThe study was utilized the secondary data collected from monthly data of CPI for inflation rate from January 1994 to December 2020 which was obtained from central statistical Agency. The forecast was applied between the nonlinear and linear ARMA models using different techniques. The unit root test of Dickey–Fuller test was made for each variables and applied lag length transformation for the variables that had unit root. A threshold autoregressive models was utilized for data handling technique using least square estimation.ResultsThe results showed that monthly rate of inflation was characterized a non-constant mean and an unstable variance. The outcome of Tsay tests was revealed that non linearity of CPI and SETAR(2,4,4) had the smallest value of AIC under this study. The forecasting performance comparison results were showed that the nonlinear SETAR model outperform the linear ARMA models. Moreover, the out-of-sample forecast indicates that the CPI of inflation has almost a constant trend. The in-sample forecast using the best-fit asymmetric for the SETAR(2,4,4) model the CPI series exhibits an upward trade until 2012; decreases until 2011; slightly increase up to 2018 and then decrease at the end of the study period.ConclusionThe superiority in performance of nonlinear models was attributed to their ability to capture the stochastic nature of the monthly rates as evident in the pattern of the forecast errors. The investigators are recommended that using TAR models policy makers can be able to capture the price volatility persistence and also forecasting can be made.

Turbo-shaft engine adaptive neural network control based on nonlinear state space equation

Intelligent Adaptive Control (AC) has remarkable advantages in the control system design of aero-engine which has strong nonlinearity and uncertainty. Inspired by the Nonlinear Autoregressive Moving Average (NARMA)-L2 adaptive control, a novel Nonlinear State Space Equation (NSSE) based Adaptive neural network Control (NSSE-AC) method is proposed for the turbo-shaft engine control system design. The proposed NSSE model is derived from a special neural network with an extra layer, and the rotor speed of the gas turbine is taken as the main state variable which makes the NSSE model be able to capture the system dynamic better than the NARMA-L2 model. A hybrid Recursive Least-Square and Levenberg-Marquardt (RLS-LM) algorithm is advanced to perform the online learning of the neural network, which further enhances both the accuracy of the NSSE model and the performance of the adaptive controller. The feedback correction is also utilized in the NSSE-AC system to eliminate the steady-state tracking error. Simulation results show that, compared with the NARMA-L2 model, the NSSE model of the turbo-shaft engine is more accurate. The maximum modeling error is decreased from 5.92% to 0.97% when the LM algorithm is introduced to optimize the neural network parameters. The NSSE-AC method can not only achieve a better main control loop performance than the traditional controller but also limit all the constraint parameters efficiently with quick and accurate switching responses even if component degradation exists. Thus, the effectiveness of the NSSE-AC method is validated.

Experimental Demonstration of High‐Performance Physical Reservoir Computing with Nonlinear Interfered Spin Wave Multidetection

Physical reservoir computing, which is a promising method for the implementation of highly efficient artificial intelligence devices, requires a physical system with nonlinearity, fading memory, and the ability to map in high dimensions. Although it is expected that spin wave interference can perform as highly efficient reservoir computing in some micromagnetic simulations, there has been no experimental verification to date. Herein, reservoir computing is demonstrated that utilizes multidetected nonlinear spin wave interference in an yttrium‐iron‐garnet single crystal. The subject computing system achieves excellent performance when used for hand‐written digit recognition, second‐order nonlinear dynamical tasks, and nonlinear autoregressive moving average (NARMA). It is of particular note that normalized mean square errors for NARMA2 and second‐order nonlinear dynamical tasks are 1.81 × 10−2 and 8.37 × 10−5, respectively, which are the lowest figures for any experimental physical reservoir so far reported. Said high performance is achieved with higher nonlinearity and the large memory capacity of interfered spin wave multidetection.

Electrochemical reservoir computing based on surface-functionalized carbon nanotubes

Reservoir computing, a type of neural network computing, is an energy-efficient and fast machine learning method, expected to realize low-power edge processing in sensor devices. This study constructs an electrochemical reservoir using carbon nanotube (CNT) thin-film electrodes on flexible plastic and proves the capability of the electrochemical functionalization of CNT thin films to improve the reservoir's performance. The performance of the time-series data prediction task, called the nonlinear autoregressive moving average (NARMA) task, can be improved using multiple CNT electrodes functionalized under different conditions. Based on the correlation coefficient analysis of the reservoir output, we illustrate that the reservoir dimensionality is enhanced by combining CNT electrodes with different functionalization conditions. Furthermore, the dimensionality and memory capacity of the reservoir can be improved using the displacement current of the electric double layer and the redox current in the electrochemical electrode. We demonstrate the time-series data prediction of blood glucose levels in patients with type 1 diabetes using a physical reservoir for the first time. The use of functionalized CNTs enables flexible and high-performance reservoir computing, leading to the low-power in-sensor edge processing of wearable sensors.

Complex systems modelling of UK winter wheat yield

Wheat is one of the most important global crops, understanding the drivers of wheat yield has significant societal benefits. Climate variables are particularly important in determining interannual variations in wheat yield, either as primary factors which directly influence the stages of wheat growth, or as secondary factors through their influence on pests, diseases and soil conditions. Here we present a new approach to model wheat yield; an empirical method based on nonlinear complex systems identification, known as NARMAX (Nonlinear AutoRegressive Moving Average with eXogenous inputs model). We deploy the NARMAX analytical approach for a specific site, Rothamsted, UK, where detailed meteorological variables are available, together with specific information on site conditions and crop growth stages. NARMAX yield forecasts are compared with those from the WOFOST crop model and nine state-of-the-art machine learning (ML) models; experimental results show that NARMAX outperforms all the compared methods in both prediction accuracy and model interpretability. We also develop regional wheat yield forecasts derived from a new gridded meteorological data product.The NARMAX approach produces skillful forecasts (r = 0.78) of Rothamsted wheat yield for a validation period, with small errors. The NARMAX regional forecasts, based on less specific information than WOFOST, also show a high degree of skill (r = 0.73). In addition, the predictor terms chosen for the model are identifiable and can help to give insight into potential key processes involved in the determination of wheat yield at a specific location. This approach can be extended in principle to other crop types and locations. It is straightforward and inexpensive to implement, using a limited number of meteorological predictor variables, which can be taken from site-based observations, or from gridded meteorological datasets. The method is a new tool to understand the environmental drivers of wheat yields on an annual basis.

Performance Enhancement of a Spin-Wave-Based Reservoir Computing System Utilizing Different Physical Conditions

We numerically study how to enhance reservoir computing performance by thoroughly extracting the spin-wave device potential for higher-dimensional information generation. The reservoir device has a 1-input exciter and 120-output detectors on the top of a continuous magnetic garnet film for spin-wave transmission. For various nonlinear and fading-memory dynamic phenomena distributing in the film space, small in-plane magnetic fields are used to prepare stripe domain structures and various damping constants at the film sides and bottom are explored. The ferromagnetic resonant frequency and relaxation time of spin precession clearly characterizes the change in spin dynamics with the magnetic field and damping constant. The common input signal for reservoir computing is a 1-GHz cosine wave with random 6-valued amplitude modulation. A basic 120-dimensional reservoir output vector is obtained from time-series signals at the 120-output detectors under each of three magnetic field conditions. Then, 240- and 360-dimensional reservoir output vectors are also constructed by concatenating two and three basic ones, respectively. In nonlinear autoregressive moving average (NARMA) prediction tasks, the computational performance is enhanced as the dimension of the reservoir output vector becomes higher and a significantly low prediction error is achieved for the tenth-order NARMA task using the 360-dimensional vector and optimum damping constant. The results are clear evidence that the collection of diverse output signals efficiently increases the dimensionality of the integrated reservoir state vector (i.e. reservoir-state richness) and thereby contributes to high computational performance. This paper demonstrates that performance enhancement through various configuration settings is a practical approach for on-chip reservoir computing devices with small numbers of real output nodes.

An optimal artificial neural network controller for load frequency control of a four-area interconnected power system

In this paper, an optimal artificial neural network (ANN) controller for load frequency control (LFC) of a four-area interconnected power system with non-linearity is presented. A feed forward neural network with multi-layers and Bayesian regularization backpropagation (BRB) training function is used. This controller is designed on the basis of optimal control theory to overcome the problem of load frequency control as load changes in the power system. The system comprised of transfer function models of twothermal units, one nuclear unit and one hydro unit. The controller model is developed by considering generation rate constraint (GRC) of different units as a non-linearity. The typical system parameters obtained from IEEE press power engineering series and EPRI books. The robustness, effectiveness, and performance of the proposed optimal ANN controller for a step load change and random load change in the system is simulated through using MATLAB-Simulink. The time response characteristics are compared with that obtained from the proportional, integral and derivative (PID) controller and non-linear autoregressive-moving average (NARMA-L2) controller. The results show that the algorithm developed for proposed controller has a superiority in accuracy as compared to other two controllers.

A Novel Reservoir Computing Platform Constructed by Laser Controlled GaAs Resonator

In this work, a novel realization of the well known machine learning approach—reservoir computing (RC) using nanoelectromechanical system (NEMS) GaAs resonator is proposed. The specially designed resonator can interact with laser directly without cavity. Thanks to the richness of nonlinearity embedded in the light-mediated dynamical interacting process, the RC platform performs outstandingly even by eliminating the time delay feedback, and the conduction of nonlinear autoregressive moving-average (NARMA) task shows that the normalized mean square error (NMSE) can achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.1\,\,\times10$ </tex-math></inline-formula> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> that is one order of magnitude lower than the recently reported result. Detailed theoretical modeling as well as numerical simulations on device’s nonlinear dynamics and RC’s hyperparameters are given. The interplay between the transient dynamics of the resonator and RC performance is particularly investigated.

Artificial Neural Network-based Neurocontroller for Hydropower Plant Control

In this paper, the behavior of a system dynamics is represented where neuro-controller is designed, trained, and implemented. The development of the mathematical models is based on suggestions and recommendations from the literature issued by the working group of IEEE. According to the mathematical models, simulation is developed in Simulink software. MATLAB/Simulink software was used to represent the difference between the conventional PID controller and artificial neural network (ANN) neuro-controller. Nonlinear autoregressive-moving average (NARMA-L2) has been used for control simulation of the hydro-power plant (HPP) with neuro-controllers on one hand, and conventional PID control on the other hand.

High-Precision Displacement and Force Hybrid Modeling of Pneumatic Artificial Muscle Using 3D PI-NARMAX Model

Pneumatic artificial muscle (PAM) is attractive in rehabilitation and biomimetic robots due to its flexibility. However, there exists a strong hysteretic nonlinearity in PAMs and strong coupling between the output displacement and the output force. At present, most commonly used hysteresis models can be treated as two-dimensional models, which only consider the nonlinearity between the input and the output displacement of the PAM without considering the coupling of the output force. As a result, high-precision modeling and estimation of the PAM’s behavior is difficult, especially when the external load of the system varies significantly. In this paper, the influence of the output force on the displacement is experimentally investigated. A three-dimensional model based on the modified Prandtl–Ishlinskii (MPI) model and the Nonlinear AutoRegressive Moving Average with eXogenous inputs (NARMAX) model is proposed to describe the relationship and couplings among the input, the output displacement, and the output force of the PAM. Experiments are conducted to verify the modeling accuracy of the proposed model when the external load of the PAM varies across a wide range. The experimental results show that the proposed model captures well the hysteresis and couplings of the PAM and can precisely predict the PAM’s behavior.

Inter-event Times Statistic in Stationary Processes: Nonlinear ARMA Modeling of Wind Speed Time Series

The random sequence of inter-event times of a level-crossing is a statistical tool that can be used to investigate time series from complex phenomena. Typical features of observed series as the skewed distribution and long range correlations are modeled using non linear transformations applied to Gaussian ARMA processes. We investigate the distribution of the inter-event times of the level-crossing events in ARMA processes in function of the probability corresponding to the level. For Gaussian ARMA processes we establish a representation of this indicator, prove its symmetry and that it is invariant with respect to the application of a non linear monotonic transformation. Using simulated series we provide evidence that the symmetry disappears if a non monotonic transformation is applied to an ARMA process. We estimate this indicator in wind speed time series obtained from three different databases. Data analysis provides evidence that the indicator is non symmetric, suggesting that only highly non linear transformations of ARMA processes can be used in modeling. We discuss the possible use of the inter-event times in the prediction task.

Iterative learning NARMA-L2 control for turbofan engine with dynamic uncertainty in flight envelope

Nonlinear control of turbofan engines in the flight envelope has attracted much attention in consideration of the inherent nonlinearity of the engine dynamics. Most nonlinear control design techniques rely on the correction theory of reference model parameter to extend the typical flight operations from ground operation. However, dynamic uncertainties in flight envelope lead to the deviation of operating state, and it is negative to control performance. This article is to develop online correction neural network–based speed control approaches for the turbofan engine with dynamic uncertainty in the flight envelope. Two improved online correction nonlinear ways combined with nonlinear autoregressive moving average (NARMA) are proposed, such as gradient search nonlinear autoregressive moving average with feedback linearization (NARMA-L2) control and iterative learning NARMA-L2 control. The contribution of this article is to provide better control quality of fast regulation and less steady errors of engine speed by the proposed methodology in comparison to the conventional NARMA-L2 control. Some important results are reached on both turbofan engine controller design and dynamic uncertainty tolerance at the typical flight operations, and the numerical examples demonstrate the superiority of the proposed control in the flight envelope.

Data-Driven Stability Assessment of Multilayer Long Short-Term Memory Networks

Recurrent Neural Networks (RNNs) are increasingly being used for model identification, forecasting and control. When identifying physical models with unknown mathematical knowledge of the system, Nonlinear AutoRegressive models with eXogenous inputs (NARX) or Nonlinear AutoRegressive Moving-Average models with eXogenous inputs (NARMAX) methods are typically used. In the context of data-driven control, machine learning algorithms are proven to have comparable performances to advanced control techniques, but lack the properties of the traditional stability theory. This paper illustrates a method to prove a posteriori the stability of a generic neural network, showing its application to the state-of-the-art RNN architecture. The presented method relies on identifying the poles associated with the network designed starting from the input/output data. Providing a framework to guarantee the stability of any neural network architecture combined with the generalisability properties and applicability to different fields can significantly broaden their use in dynamic systems modelling and control.

Modeling and Identification of Rate Dependent Hysteresis in Piezoelectric Actuated Nano-Stage: A Gray Box Neural Network Based Approach

A modeling and parameter identification method for rate dependent hysteresis of piezoelectric actuated nano-stage is presented in this work. A system level quasi-static hysteresis model is employed to construct a neural network. To better describe the rate dependent behavior of hysteresis in piezoelectric actuated stage, a Nonlinear AutoRegressive Moving Average with eXogenous input (NARMAX) based dynamic model is incorporated with the quasi-static hysteresis model, where the weights of specifically designed neural network corresponds to the model parameters. To handle the multivalued problem of hysteresis, generalized input gradient is proposed to convert multivalued mapping of hysteresis into one-to-one mapping. The parameters of the nonlinear rate dependent hysteresis in piezoelectric actuated stage is identified by neural network training, taking advantage of their universal function approximation capabilities. The proposed scheme is also compared with conventional black box and particle swarm optimization identification based methods, simulation and experimental results demonstrate significant performance improvement with an error of 20.77nm for proposed method whereas 96.56nm and 31.46nm for black box and particle swarm optimization respectively.

Recurrent-neural-network-based unscented Kalman filter for estimating and compensating the random drift of MEMS gyroscopes in real time

The presence of the stochastic errors in MEMS (Micro Electro Mechanical Systems) gyroscopes makes the improvement of the measurement precision challenging. This paper addresses a novel method to estimate and compensate the random drift of MEMS gyroscopes in real time, combining unscented Kalman filter (UKF) with recurrent neural network (RNN). In the proposed method, the random drift is regarded as a generalized nonlinear autoregressive moving average (NARMA) model, and its optimal predictor is realized by a dynamic RNN. To compensate the random drift in real time, the RNN model is brought into the framework of UKF, for establishing the state equation of the improved UKF. The novelty of this paper is that a strategy is presented to guarantee the validity of the combination of UKF and RNN. The effectiveness and superiorities of the proposed method are verified by experiments.