Polynomial Autoregressive Model Research Articles

Nonlinearity compensation based on identified NARX polynomials models

This paper deals with the compensation of nonlinearities in dynamical systems using Nonlinear polynomial AutoRegressive models with eXogenous inputs (NARX) identified from data. The compensation approach is formulated for static and dynamical contexts for the general case and is also adapted for systems with hysteresis. Both simulated and experimental results are presented to illustrate the method. In the experimental case, the proposed method is compared to other approaches and was found to be competitive. This method yielded a maximum tracking error of \(3.9\%\) while the corresponding value for the uncompensated system was \(21.0\%\). Furthermore, the presented technique typically results in compensation signals with lower energy requirements. The results also show that the proposed methodology can provide compensation signals that practically linearize the systems using simple nonlinear models with very few terms.

Enabling Invariant Models to Describe Time-Varying Dynamics: A Case Study

This work addresses the problem of identifying nonlinear and time-varying behaviors using Nonlinear polynomial AutoRegressive models with eXogenous inputs (NARX). Two approaches are investigated. The first one is based on a recursive algorithm with an adaptive forgetting factor to update the parameters of black-box models. The second approach aims to include a certain class of regressors to identify invariant models, i.e. with constant parameters, to describe such behaviors. An experimental case study of a pH neutralization process is carried out, which indicated the presence of behaviors that resemble varying dynamics features. Time-invariant models with a specific class of regressors are proposed and seem to be a promising way to deal with such dynamics.

Identification and nonlinearity compensation of hysteresis using NARX models

This paper deals with two problems: the identification and compensation of hysteresis nonlinearity in dynamical systems using nonlinear polynomial autoregressive models with exogenous inputs (NARX). First, based on gray-box identification techniques, some constraints on the structure and parameters of NARX models are proposed to ensure that the identified models display a key feature of hysteresis. In addition, a more general framework is developed to explain how hysteresis occurs in such models. Second, two strategies to design hysteresis compensators are presented. In one strategy, the compensation law is obtained through simple algebraic manipulations performed on the identified models. In the second strategy, the compensation law is directly identified from the data. Both numerical and experimental results are presented to illustrate the efficiency of the proposed procedures. Also, it has been found that the compensators based on gray-box models outperform the cases with models identified using black-box techniques.

Daily Heart Rate Variability Indices in Subjects with and Without Metabolic Syndrome Before and After the Elimination of the Influence of Day-Time Physical Activity.

Background and Objectives: The available research shows conflicting data on the heart rate variability (HRV) in metabolic syndrome (MetS) subjects. The discrepancy suggests a methodical shortcoming: due to the influence of physical activity, the standard measuring of HRV at rest is not comparable with HRV assessment based on 24 h Holter monitoring, which is preferred because of its comprehensiveness. To obtain a more reliable measure and to clarify to what extent HRV is altered in MetS, we assessed a 24 h HRV before and after the elimination of the influence of physical activity. Materials and Methods: We investigated 69 metabolic syndrome (MetS) and 37 control subjects, aged 50–55. In all subjects, 24 h monitoring of electrocardiogram, blood pressure, and actigraphy profiles were conducted. To eliminate the influence of day-time physical activity on RR intervals (RRI), a linear polynomial autoregressive model with exogenous terms (ARX) was used. Standard spectral RRI analysis was performed. Results: Subjects with MetS had blunted HRV; the diurnal SDNN index was reliably lower in the MetS group than in control subjects. The elimination of the influence of physical activity did not reveal a significant HRV change in long-term indices (SDNN, SDANN, and SD2), whilst adjacent RRI values (RMSSD, pNN50, and SD1) and SDNN index significantly increased (p < 0.001). An increase in the latter indices highlighted the HRV difference between the MetS and control groups; a significant (p < 0.001) decrease of all short-term HRV variables was found in the MetS group (p < 0.01), and low-frequency spectral components were less pronounced in the MetS group. Conclusion: The application of a polynomial autoregressive model in 24 h HRV assessment allowed for the exclusion of the influence of physical activity and revealed that MetS is associated with blunted HRV, which reflects mitigated parasympathetic tone.

One‐day ahead wind speed/power prediction based on polynomial autoregressive model

Wind has been one of the popular renewable energy generation methods in the last decades. Foreknowledge of power to be generated from wind is crucial especially for planning and storing the power. It is evident in various experimental data that wind speed time series has non-linear characteristics. It has been reported in the literature that nonlinear prediction methods such as artificial neural network (ANN) and adaptive neuro fuzzy inference system (ANFIS) perform better than linear autoregressive (AR) and AR moving average models. Polynomial AR (PAR) models, despite being non-linear, are simpler to implement when compared with other non-linear AR models due to their linear-in-the-parameters property. In this study, a PAR model is used for one-day ahead wind speed prediction by using the past hourly average wind speed measurements of Cesme and Bandon and performance comparison studies between PAR and ANN-ANFIS models are performed. In addition, wind power data which was published for Global Energy Forecasting Competition 2012 has been used to make power predictions. Despite having lower number of model parameters, PAR models outperform all other models for both of the locations in speed predictions as well as in power predictions when the prediction horizon is longer than 12 h.

Study of the SAR signature of internal waves by nonlinear parametric autoregressive models

In this paper, we propose a method for estimating the nonlinear differential equation governing the propagation of internal waves (IWs) in a stratified water column of the ocean. This estimation of the kind of nonlinear differential equation and the coefficients of this equation is performed from profiles of the synthetic aperture radar signature at the ocean surface. Theoretically, the equation is based upon the Korteveg-de Vries (KdV) equation or one of its improved versions. The estimation of the differential equation version is based on nonlinear polynomial autoregressive models up to the third order. The estimation of the parameters of these models is performed simply by solving a set of linear equations using higher order statistics. This general approach allows us to identify the best model of propagation, i.e., the kind of nonlinearity as well as the structure of each kernel (linear, quadratic, cubic), without determining an analytic solution to the equation of propagation. Moreover, the physical oceanic parameters (such as the thermocline depth) are deduced from the estimated coefficients of the KdV equation (assuming an underlying model of the water column). Results on simulated profiles generally lead to an exact model identification (i.e., the one used for simulation with the exact order of nonlinearity and the exact structure of each kernel) and lead to satisfying geophysical parameter estimates. Finally, the geophysical parameters estimated from three sets of ERS-1 profiles are generally coherent with the parameters observed in the IW propagation mechanism but have not been validated by "in situ" measurements.

Model based method for estimating an attractor dimension from uni/multivariate chaotic time series with application to Bremen climatic dynamics

In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding dimension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of necessary variables to model the dynamics. Therefore, the optimality of this dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor. The method of this paper relies on testing this property by locally fitting a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The corresponding algorithms are developed in uni/multivariate form and some probable advantages of using information from other time series are discussed. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic benchmark systems. Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension.

Application of linear and nonlinear time series modeling to heart rate dynamics analysis

The linear autoregressive (AR) model is often used to investigate the pathophysiologic mechanisms controlling heart rate (HR) dynamics. This study implemented parametric models new to this field to determine if a more appropriate HR dynamics modeling structure exists. The linear AR and autoregressive-moving average (ARMA) models, and the nonlinear polynomial autoregressive (PAR) and bilinear (BL) models were fit to instantaneous HR time series obtained from nine subjects in the supine position. Model orders were determined by the Akaike Information Criteria (AIC). Model residual variance was used as the primary intermodel comparison criterion, with significance evaluated by a chi 2 distributed statistic. The BL model best represented the HR dynamics, as its residual variance was significantly (p < 0.05) smaller than that of the corresponding AR model for nine out of nine data sets. In all cases, the BL model had a smaller residual variance than either the ARMA or PAR models. The bilinear model was ineffective at data forecasting, however, we show that this cannot reflect BL model validity because poor prediction is inherent to the BL model structure. The apparent superiority of the nonlinear bilinear model suggests that future heart rate dynamics studies should put greater emphasis on nonlinear analyses.

Correlation analysis of nonstationary processes using prior information

The paper examines the applicability of correlation analysis to identification of some classes of nonstationary processes. Consistency conditions are obtained for the time-dependent 1st order autoregressive (AR) process and for the polynomial AR model of an arbitrary order. These results are applied for stable AR estimation and for linear prediction on given classes of nonstationary processes.

Nonlinear autoregressive model based on fuzzy relation

A nonlinear autoregressive model that has a regressive function approximated by fuzzy relations is proposed. The internal parameters of the fuzzy relational matrix describe the characteristics of this function. The estimation method of these parameters is shown from the viewpoint of maximum likelihood estimation. As a result of the numerical experiment on artificial data compared with the ordinary autoregressive model and the polynomial autoregressive model, it is shown that the proposed model is able to estimate the more precise regressive function from the observed data.