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Time-varying (TV) nonlinear systems widely exist in various fields of engineering and science. Effective identification and modeling of TV systems is a challenging problem due to the nonstationarity and nonlinearity of the associated processes. In this paper, a novel parametric modeling algorithm is proposed to deal with this problem based on a TV nonlinear autoregressive with exogenous input (TV-NARX) model. A new class of multiple beta wavelet (MBW) basis functions is introduced to represent the TV coefficients of the TV-NARX model to enable the tracking of both smooth trends and sharp changes of the system behavior. To produce a parsimonious model structure, a locally regularized ultraorthogonal forward regression (LRUOFR) algorithm aided by the adjustable prediction error sum of squares (APRESS) criterion is investigated for sparse model term selection and parameter estimation. Simulation studies and a real application to EEG data show that the proposed MBW-LRUOFR algorithm can effectively capture the global and local features of nonstationary systems and obtain an optimal model, even for signals contaminated with severe colored noise.

A global parameter estimation method for a permanent magnet synchronous machines (PMSM) drive system is proposed, where the electrical parameters, mechanical parameters, and voltage-source-inverter (VSI) nonlinearity are regarded as a whole and parameter estimation is formulated as a single parameter optimization model. A dynamic learning estimator is proposed for tracking the electrical parameters, mechanical parameters, and VSI of PMSM drive by using dynamic self-learning particle swarm optimization (DSLPSO). In DSLPSO, a novel movement modification equation with dynamic exemplar learning strategy is designed to ensure its diversity and achieve a reasonable tradeoff between the exploitation and exploration during the search process. Moreover, a nonlinear multiscale based interactive learning operator is introduced for accelerating the convergence speed of the $Pbest$ particles; meanwhile a dynamic opposition-based learning strategy is designed to facilitate the $gBest$ particle to explore a potentially better region. The proposed algorithm is applied to parameter estimation for a PMSM drive system. The results show that the proposed method has better performance in tracking the variation of electrical parameters, and estimating the immeasurable mechanical parameters and the VSI disturbance voltage simultaneously.

The Atlantic meridional overturning circulation (AMOC), an important component of the climate system, has only been directly measured since the RAPID array’s installation across the Atlantic at 26°N in 2004. This has shown that the AMOC strength is highly variable on monthly timescales; however, after an abrupt, short-lived, halving of the strength of the AMOC early in 2010, its mean has remained ~ 15% below its pre-2010 level. To attempt to understand the reasons for this variability, we use a control systems identification approach to model the AMOC, with the RAPID data of 2004–2017 providing a trial and test data set. After testing to find the environmental variables, and systems model, that allow us to best match the RAPID observations, we reconstruct AMOC variation back to 1980. Our reconstruction suggests that there is inter-decadal variability in the strength of the AMOC, with periods of both weaker flow than recently, and flow strengths similar to the late 2000s, since 1980. Recent signs of weakening may therefore not reflect the beginning of a sustained decline. It is also shown that there may be predictive power for AMOC variability of around 6 months, as ocean density contrasts between the source and sink regions for the North Atlantic Drift, with lags up to 6 months, are found to be important components of the systems model.

This paper is concerned with the model selection and model averaging\ud problems in system identification and data-driven modelling for nonlinear\ud systems. Given a set of data, the objective of model selection is to evaluate\ud a series of candidate models and determine which one best presents the data.\ud Three commonly used criteria, namely, Akaike information criterion (AIC),\ud Bayesian information criterion (BIC) and an adjustable prediction error sum\ud of squares (APRESS) are investigated and their performance in model\ud selection and model averaging is evaluated via a number of case studies\ud using both simulation and real data. The results show that APRESS\ud produces better models in terms of generalization performance and model\ud complexity.

In this paper, an accurate parameter estimation model of surface permanent magnet synchronous machines (SPMSMs) is established by taking into account voltage-source-inverter (VSI) nonlinearity. A fast dynamic particle swarm optimization (DPSO) algorithm combined with a receptor editing (RE) strategy is proposed to explore the optimal values of parameter estimations. This combination provides an accelerated implementation on graphics processing unit (GPU), and the proposed method is, therefore, referred to as G-DPSO-RE. In G-DPSO-RE, a dynamic labor division strategy is incorporated into the swarms according to the designed evolutionary factor during the evolution process. Two novel modifications of the movement equation are designed to update the velocity of particles. Moreover, a chaotic-logistic-based immune RE operator is developed to facilitate the global best individual ( gBest particle) to explore a potentially better region. Furthermore, a GPU parallel acceleration technique is utilized to speed up parameter estimation procedure. It has been demonstrated that the proposed method is effective for simultaneous estimation of the PMSM parameters and the disturbance voltage ( $V_{\mathrm { {dead}}})$ due to VSI nonlinearity from experimental data for currents and rotor speed measured with inexpensive equipment. The influence of the VSI nonlinearity on the accuracy of parameter estimation is analyzed.

Influenza and influenza-like illnesses are one of the leading causes of death in the world, resulting in heavy losses to individual families and nations. Accurate and timely forecasts of seasonal influenza would therefore crucially important to inform and facilitate public health decision-making for presenting and intervening influenza epidemics. System identification and data-driven modelling approaches play an indispensable role in analyzing and understanding complex processes including medical, healthcare and environmental time series. This paper aims to present a type of sparse, interpretable and transparent (SIT) model, which cannot only be used for future behavior prediction but more importantly for understanding the dependent relationship between the response variables of a system on potential independent variables (also known as input variables or predictors). An ideal candidate for such a SIT representation is the well-known NARMAX (nonlinear autoregressive moving average with exogenous inputs) model, which can be established based on input and output data of the system of interest, and the final refined model is usually simple, parsimonious and easy to interpret. The general framework of the NARMAX model is presented, and the state-of-the-art algorithms for such a SIT model estimation are described. Two case studies are provided to illustrate how well the SIT-NARMAX model can work for medical, healthcare and related data.

Solvent-based post combustion capture (PCC) is a well-developed technology for CO2 capture from power plants and industry. A reliable model that captures the dynamics of the solvent-based capture process is essential to implement suitable control design. Typically, first principle models are used, however they usually require comprehensive knowledge and deep understanding of the process. System identification approach is adopted to obtain a model that accurately describes the dynamics between key variables in the process. The nonlinear auto-regressive with exogenous (NARX) inputs model is employed to represent the relationship between the input variables and output variables as two Multiple-Input Single-Output (MISO) sub-models. The forward regression with orthogonal least squares (FROLS) algorithm is implemented to select an accurate model structure that best describes the dynamics within the process. The prediction performance of the identified NARX models is promising and shows that the models capture the underlying dynamics of the CO2 capture process.

Data-driven modelling approaches play an indispensable role in analyzing and understanding complex processes. This study proposes a type of sparse, interpretable and transparent (SIT) machine learning model, which can be used to understand the dependent relationship of a response variable on a set of potential explanatory variables. An ideal candidate for such a SIT representation is the well-known NARMAX (nonlinear autoregressive moving average with exogenous inputs) model, which can be established from measured input and output data of the system of interest, and the final refined model is usually simple, parsimonious and easy to interpret. The performance of the proposed SIT models is evaluated through two real healthcare datasets.

Severe geomagnetic disturbances can be hazardous for modern technological systems. The reliable forecast of parameters related to the state of the magnetosphere can facilitate the mitigation of adverse effects of space weather. This study is devoted to the modeling and forecasting of the evolution of the Kp index related to global geomagnetic disturbances. Throughout this work the Nonlinear AutoRegressive with eXogenous inputs (NARX) methodology is applied. Two approaches are presented: i) a recursive sliding window approach, and ii) a direct approach. These two approaches are studied separately and are then compared to evaluate their performances. It is shown that the direct approach outperforms the recursive approach, but both tend to produce predictions slightly biased from the true values for low and high disturbances.

A new transient Granger causality detection method is proposed based on a time-varying\ud parametric modelling framework, and is applied to real EEG signals to reveal the causal information flow\ud during motor imagery (MI) tasks. The time-varying parametric modelling approach employs a nonlinear\ud autoregressive with external input (NARX) model, whose parameters are approximated by a set of multiwavelet\ud basis functions. A regularized orthogonal least squares (ROLS) algorithm is then used to produce\ud a parsimonious or sparse regression model and estimate the associated model parameters. The time-varying\ud Granger causality between nonstationary signals can be detected accurately by making use of both the good\ud approximation properties of multi-wavelets and the good generalization performance of the ROLS in the\ud presence of high-level noise. Two simulation examples are presented to demonstrate the effectiveness of\ud the proposed method for both linear and nonlinear causal detection respectively. The proposed method is\ud then applied to real EEG signals of MI tasks. It follows that transient causal information flow over the\ud time course between various sensorimotor related channels can be successfully revealed during the whole\ud reaction processes. Experiment results from these case studies confirm the applicability of the proposed\ud scheme and show its utility for the understanding of the associated neural mechanism and the potential\ud significance for developing MI tasks based brain-computer interface (BCI) systems.

The methodology based on Nonlinear AutoRegressive Moving Average eXogenous (NARMAX) models is one of the most robust techniques for studying complex dynamical systems. NARMAX was initially employed to deduce models for engineering systems, which could be used in conjunction with control engineering techniques. However, the power of such a technique was quickly realized, and it has been successfully applied to investigate many diverse systems, such as stem cells dynamics, neuroimaging, finance, and the solar-terrestrial system. This chapter reviews the application of the NARMAX methodology to problems in space weather, such as forecasting the geomagnetic indices and fluxes of the high-energy particles in the radiation belts. As NARMAX provides physically interpretable models, this chapter also discusses how the application of the NARMAX methodology can advance physical understanding of space weather processes.

A new iOFR-MF iterative orthogonal forward regression--modulating function algorithm is proposed to identify continuous-time models from noisy data by combining the MF method and the iOFR algorithm. In the new method, a set of candidate terms, which describe different dynamic relationships among the system states or between the input and output, are first constructed. These terms are then modulated using the MF method to generate the data matrix. The iOFR algorithm is next applied to build the relationships between these modulated terms, which include detecting the model structure and estimating the associated parameters. The relationships between the original variables are finally recovered from the model of the modulated terms. Both nonlinear state-space models and a class of higher order nonlinear input–output models are considered. The new direct method is compared with the traditional finite difference method and results show that the new method performs much better than the finite difference method. The new method works well even when the measurements are severely corrupted by noise. The selection of appropriate MFs is also discussed.

The identification of nonlinear time-varying systems using linear-in-the-parameter models is investigated. An efficient common model structure selection CMSS algorithm is proposed to select a common model structure, with application to EEG data modelling. The time-varying parameters for the identified common-structured model are then estimated using a sliding-window recursive least squares SWRLS approach. The new method can effectively detect and adaptively track and rapidly capture the transient variation of nonstationary signals, and can also produce robust models with better generalisation properties. Two examples are presented to demonstrate the effectiveness and applicability of the new approach including an application to EEG data.

A new parametric approach is proposed for nonlinear and non-stationary system identification based on a time-varying nonlinear autoregressive with exogenous input (TV-NARX) model. The time-varying coefficients of the TV-NARX model are expanded using multi- wavelet basis functions and the model is thus transformed into a time-invariant regression problem.\ud An ultra-orthogonal forward regression (UOFR) algorithm aided by mutual information (MI) is designed to identify a parsimonious model structure and estimate the associated model parameters. The UOFR-MI algorithm which uses not only the observed data themselves but also weak derivatives of the signals is more powerful in model structure detection. The proposed approach combining the advantages of both the basis function expansion method and the UOFR-MI algorithm is proved to be capable of tracking the change of time-varying parameters effectively in both numerical simulations and the real EEG data.

Spectral measures of linear Granger causality have been widely applied to study the causal connectivity between time series data in neuroscience, biology, and economics. Traditional Granger causality measures are based on linear autoregressive with exogenous (ARX) inputs models of time series data, which cannot truly reveal nonlinear effects in the data especially in the frequency domain. In this study, it is shown that the classical Geweke's spectral causality measure can be explicitly linked with the output spectra of corresponding restricted and unrestricted time-domain models. The latter representation is then generalized to nonlinear bivariate signals and for the first time nonlinear causality analysis in the frequency domain. This is achieved by using the nonlinear ARX (NARX) modeling of signals, and decomposition of the recently defined output frequency response function which is related to the NARX model.

Background Frequency domain Granger causality measures have been proposed and widely applied in analyzing rhythmic neurophysiological and biomedical signals. Almost all these measures are based on linear time domain regression models, and therefore can only detect linear causal effects in the frequency domain. New method A frequency domain causality measure, the partial directed coherence, is explicitly linked with the frequency response function concept of linear systems. By modeling the nonlinear relationships between time series using nonlinear models and employing corresponding frequency-domain analysis techniques (i.e. generalized frequency response functions), a new nonlinear partial directed coherence method is derived. Results The advantages of the new method are illustrated via a numerical example of a nonlinear physical system and an application to electroencephalogram signals from a patient with childhood absence epilepsy. Comparison with existing methods The new method detects both linear and nonlinear casual effects between bivariate signals in the frequency domain, while the existing measures can only detect linear effects. Conclusions The proposed new method has important advantages over the classical linear measures, because detecting nonlinear dependencies has become more and more important in characterizing functional couplings in neuronal and biological systems.

Objective To introduce a new method of quantitative EEG analysis in the time domain, the error reduction ratio (ERR)-causality test. To compare performance against cross-correlation and coherence with phase measures. Methods A simulation example was used as a gold standard to assess the performance of ERR-causality, against cross-correlation and coherence. The methods were then applied to real EEG data. Results Analysis of both simulated and real EEG data demonstrates that ERR-causality successfully detects dynamically evolving changes between two signals, with very high time resolution, dependent on the sampling rate of the data. Our method can properly detect both linear and non-linear effects, encountered during analysis of focal and generalised seizures. Conclusions We introduce a new quantitative EEG method of analysis. It detects real time levels of synchronisation in the linear and non-linear domains. It computes directionality of information flow with corresponding time lags. Significance This novel dynamic real time EEG signal analysis unveils hidden neural network interactions with a very high time resolution. These interactions cannot be adequately resolved by the traditional methods of coherence and cross-correlation, which provide limited results in the presence of non-linear effects and lack fidelity for changes appearing over small periods of time.

This paper introduces a new approach for nonlinear and non-stationary time-varying system identification based on time-varying nonlinear autoregressive moving average with exogenous variable TV-NARMAX models. The challenging model structure selection and parameter tracking problems are solved by combining a multiwavelet basis function expansion of the time-varying parameters with an orthogonal least squares algorithm. Numerical examples demonstrate that the proposed approach can track rapid time-varying effects in nonlinear systems more accurately than the standard recursive algorithms. Based on the identified time domain model, a new frequency domain analysis approach is introduced based on a time-varying generalised frequency response function TV-GFRF concept, which enables the analysis of nonlinear, non-stationary systems in the frequency domain. Features in the TV-GFRFs which depend on the TV-NARMAX model structure and time-varying parameters are investigated. It is shown that the high-dimensional frequency features can be visualised in a low-dimensional time–frequency space.

A new frequency-domain analysis framework for nonlinear time-varying systems is introduced based on parametric time-varying nonlinear autoregressive with exogenous input models. It is shown how the time-varying effects can be mapped to the generalized frequency response functions (FRFs) to track nonlinear features in frequency, such as intermodulation and energy transfer effects. A new mapping to the nonlinear output FRF is also introduced. A simulated example and the application to intracranial electroencephalogram data are used to illustrate the theoretical results.

SummaryBackgroundRespiratory disease may cause profound hypoxaemia during flight. Previously derived linear equations poorly predict the need for supplemental oxygen during air travel. The current gold standard assessment is the hypoxic challenge test (HCT). Recent guidelines recommend HCT is performed for those patients with SpO2

A new NARX-based Granger linear and nonlinear casual influence detection method is presented in this paper to address the potential for linear and nonlinear models in data with applications to human EEG data analysis. Considering two signals initially, the paper introduces four indexes to measure the linearity and nonlinearity of a single signal, and one signal influencing the second signal. This method is then extended to the time-varying and multivariate cases. An adaptation of an Orthogonal Least Squares routine is employed to select the significant terms in the models. A numerical example is provided to illustrate the effectiveness of the new algorithms together with the application to real EEG data collected from 4 patients.

It is known that the least-squares (LS) class of algorithms produce unbiased estimates providing certain assumptions are met. There are many practical problems, however, where the required assumptions are violated. Typical examples include non-linear dynamical system identification problems, where the input and output observations are affected by measurement uncertainty and possibly correlated noise. This will result in biased LS estimates and the identified model will exhibit poor generalisation properties. Model estimation for this type of error-in-variables problem is investigated in this study, and a new identification scheme based on a bootstrap algorithm is proposed to improve the model estimates for non-linear dynamical system identification.

In non-linear system identification, the available observed data are conventionally partitioned into two parts: the training data that are used for model identification and the test data that are used for model performance testing. This sort of ‘hold-out’ or ‘split-sample’ data partitioning method is convenient and the associated model identification procedure is in general easy to implement. The resultant model obtained from such a once-partitioned single training dataset, however, may occasionally lack robustness and generalisation to represent future unseen data, because the performance of the identified model may be highly dependent on how the data partition is made. To overcome the drawback of the hold-out data partitioning method, this study presents a new random subsampling and multifold modelling (RSMM) approach to produce less biased or preferably unbiased models. The basic idea and the associated procedure are as follows. First, generate K training datasets (and also K validation datasets), usin...

A new algorithm for determining the output frequency range and the frequency components of Volterra models under multiple inputs is introduced for nonlinear system analysis. For a given Volterra model, the output frequency components corresponding to a multi-tone input can easily be calculated using the new algorithm.

The Dst index is a key parameter which characterises the disturbance of the geomagnetic field in magnetic storms. Modelling of the Dst index is thus very important for the analysis of the geomagnetic field. A data-based modelling approach, aimed at obtaining efficient models from limited input–output observational data, provides a powerful tool for analysing and forecasting geomagnetic activities including the prediction of the Dst index. In this study, the process of the Dst index is treated to be a structure-unknown system, where the solar wind parameter ( VBs ) and the solar wind dynamic pressure ( P ) are the system inputs, and the Dst index is the system output. A novel multiscale RBF (MSRBF) network is introduced to represent such a two-input and single-output system, where the Dst index is related to the solar wind parameter and the dynamic pressure, via a hybrid network model consisting of two submodels: a linear part that reflects the linear relationship between the output and the inputs, and a nonlinear part that captures the effect of the interacting contribution of past observations of the inputs and the output, on the current output. The proposed MSRBF network can easily be converted into a linear-in-the-parameters form and the training of the linear network model can easily be implemented using a forward orthogonal regression (FOR) algorithm. One advantage of the new MSRBF network, compared with traditional single scale RBF networks, is that the new network is more flexible for describing complex nonlinear dynamical systems.

Most nonlinear modelling approaches focus on solving a model selection problem with complete data, and in many cases original data are pre-processed with some nonlinear transforms such as Box-Tidwell and fractional polynomial transformation. Often these approaches can lead to models that are better than traditional models (for example, logistic model and quadratic model). However, in the case of missing data, it is not easy to predict the relationship between the predictor and dependent variables; traditional nonlinear models in some cases of missing data analysis give poor results. This paper explains nonlinear model selection techniques for missing data. It includes an overview of nonlinear model selection with complete data, and provides accessible descriptions of Box-Tidwell and fractional polynomial methods for model selection. In particular, this paper focuses on a fractional polynomial method for nonlinear modelling in cases of missing data and presents analysis examples to illustrate performance of the method.

A great deal of recent methodological research on missing data analysis has focused on model parameter estimation using modern statistical methods such as maximum likelihood and multiple imputation. These approaches are better than traditional methods (for example listwise deletion and mean imputation methods). These modern techniques can lead to unbiased parametric estimation in many particular application cases. However, these methods do not work well in some cases especially for nonlinear systems that have highly nonlinear behaviour. This paper explains the linear parametric estimation in existence of missing data, which includes an overview of biased and unbiased linear parametric estimation with missing data, and provides accessible descriptions of expectation maximization (EM) algorithm and Gauss-Newton method. In particular, this paper proposes a Gauss-Newton iteration method for nonlinear parametric estimation in case of missing data. Since Gauss-Newton method needs initial values that are hard to obtain in the presence of missing data, the EM algorithm is thus used to estimate these initial values. In addition, we present two analysis examples to illustrate the performance of the proposed methods.

This paper introduces a new approach for the identification of coupled map lattice models of complex spatio-temporal patterns from measured data. The nonlinear functionals describing the evolution of the spatio-temporal patterns are constructed using B-spline wavelet and scaling functions. This provides a multiresolution approximation for the underlying spatio-temporal dynamics. An orthogonal least squares algorithm is used to determine the suitable terms from wavelet functions to form an accurate representation of the nonlinear spatio-temporal dynamics. Three examples are used to demonstrate the application of the proposed new approach.

The long term prediction of non-linear dynamical time series, based on identified multiresolution wavelet models, from historically observed data sets is investigated and a new direct prediction approach is introduced. Prediction results based on the new direct scheme are compared with those from iterative methods and it is shown that improved predictions can be obtained using the new approach.

An efficient class of nonlinear models, constructed using cardinal B-spline (CBS) basis functions, are proposed for high tide forecasts at the Venice lagoon. Accurate short term predictions of high tides in the lagoon can easily be calculated using the proposed CBS models, which can also produce good long term (up to 24 hrs ahead) forecasts for normal water levels.

Identification of linear and nonlinear time-varying systems is investigated and a new wavelet model identification algorithm is introduced. By expanding each time-varying coefficient using a multiresolution wavelet expansion, the time varying problem is reduced to a time invariant problem and the identification reduces to regressor selection and parameter estimation. Several examples are included to illustrate the application of the new algorithm.

A linear and nonlinear causality detection method called the error-reduction-ratio causality (ERRC) test is introduced in this paper to investigate if linear or nonlinear models should be considered in the study of human electroencephalograph (EEG) data. In comparison to the traditional Granger methods, one significant advantage of the ERRC approach is that it can effectively detect the time-varying linear and nonlinear causalities between two signals without fitting a complete nonlinear model. Two numerical simulation examples are employed to compare the performance of the new method with other widely used methods in the presence of noise and in tracking time-varying causality. Finally, an application to measure the linear and nonlinear relationships between two EEG signals from different cortical sites for patients with childhood absence epilepsy is discussed.

A basic assumption on the data used for nonlinear dynamic model identification is that the data points are continuously collected in chronological order. However, there are situations in practice where this assumption does not hold and we end up with an identification problem from multiple data sets. The problem is addressed in this paper and a new cross-validation-based orthogonal search algorithm for NARMAX model identification from multiple data sets is proposed. The algorithm aims at identifying a single model from multiple data sets so as to extend the applicability of the standard method in the cases, such as the data sets for identification are obtained from multiple tests or a series of experiments, or the data set is discontinuous because of missing data points. The proposed method can also be viewed as a way to improve the performance of the standard orthogonal search method for model identification by making full use of all the available data segments in hand. Simulated and real data are used in this paper to illustrate the operation and to demonstrate the effectiveness of the proposed method.

A novel type of discrete-time fractional-power nonlinear autoregressive with exogenous input (FPNARX) model is introduced for system identification, modeling and prediction. Parameter estimation of such a model is a nonlinear optimization problem. A harmony search algorithm is then applied to solve such fractional models. Examples of both simulated and real data are provided.

A novel simple approach is presented for solving small sample size problem in nonlinear dynamic system identification where only a very small number of measured input and output data points are available. The technique is built upon an orthogonal least squares algorithm and a new resampling method called output jittering method,. The proposed algorithm is applied to the model identification of bioethanol production and it shows that a robust and reliable model can be obtained with the new system identification approach.

Statistical measures such as coherence, mutual information, or correlation are usually applied to evaluate the interactions between two or more signals. However, these methods cannot distinguish directions of flow between two signals. The capability to detect causalities is highly desirable for understanding the cooperative nature of complex systems. The main objective of this work is to present a linear and nonlinear time-varying parametric modeling and identification approach that can be used to detect Granger causality, which may change with time and may not be detected by traditional methods. A numerical example, in which the exact causal influences relationships, is presented to illustrate the performance of the method for time-varying Granger causality detection. The approach is applied to EEG signals to track and detect hidden potential causalities. One advantage of the proposed model, compared with traditional Granger causality, is that the results are easier to interpret and yield additional insights into the transient directed dynamical Granger causality interactions.

[1] The NARMAX OLS-ERR algorithm, which is widely used in the study of systems dynamics, is able to determine the causal relationship between the input and output variables for nonlinear systems. This technique has been applied to measurements of the solar wind from ACE at L1 and the Dst index in order to find the best solar wind–magnetosphere coupling function, i.e., which combination of solar wind parameters provides the best predictive capabilities of the Dst index. The data-deduced coupling functions were then compared to those suggested in previous analytical and data-based studies. The most appropriate coupling function was found to be n1/2VαBT sin6(θ/2), where the power of velocity, α, was inconclusive but should be in the range 2–3.

[1] The NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) modeling based approach is used to identify the relations for Solar wind Magnetosphere Coupling Functions in order to identify the one with the best Dst index forecasting abilities. It is argued that the Error Reduction Ratio which is a cornerstone concept in the NARMAX system identification methodology is more appropriate for such nonlinear system as the terrestrial magnetosphere than correlation function analysis which has often been used in the past. It is shown that the identified relationships for coupling functions are proportional to , where is the clock angle, and not to as was suggested in earlier studies. It is shown that the factor appears naturally within the analytical model of Kan and Lee (1979) as the dependence on the clock angle.

A new iterative orthogonal least squares forward regression (iOFR) algorithm is proposed to identify nonlinear systems which may not be persistently excited. By slightly revising the classic forward orthogonal regression (OFR) algorithm, the new iterative algorithm provides search solutions on a global solution space. Examples show that the new iterative algorithm is computationally efficient and capable of producing a good model even when the input is not completely persistently excited.

A comparative study of wavelet and polynomial models for non-linear Regime-Switching (RS) systems is carried out. RS systems, considered in this study, are a class of severely non-linear systems, which exhibit abrupt changes or dramatic breaks in behaviour, due to RS caused by associated events. Both wavelet and polynomial models are used to describe discontinuous dynamical systems, where it is assumed that no a priori information about the inherent model structure and the relative regime switches of the underlying dynamics is known, but only observed input-output data are available. An Orthogonal Least Squares (OLS) algorithm interfered with by an Error Reduction Ratio (ERR) index and regularised by an Approximate Minimum Description Length (AMDL) criterion, is used to construct parsimonious wavelet and polynomial models. The performance of the resultant wavelet models is compared with that of the relative polynomial models, by inspecting the predictive capability of the associated representations. It is shown from numerical results that wavelet models are superior to polynomial models, in respect of generalisation properties, for describing severely non-linear RS systems.