A comparative study on global wavelet and polynomial models for non-linear regime-switching systems

Abstract

Abstract-A comparative study of wavelet and polynomial models for nonlinear regime-switching (RS) systems is carried out. Regime-switching systems, considered in this study, are a class of severely nonlinear systems, which exhibit abrupt changes or dramatic breaks in behavior, due to regime switching 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 generalization properties, for describing severely nonlinear regime-switching systems.Keywords-NARX models; Nonlinear system identification; Regime-switching systems; Wavelets. Abstract-A comparative study of wavelet and polynomial models for nonlinear regime-switching (RS) systems is carried out. Regime-switching systems, considered in this study, are a class of severely nonlinear systems, which exhibit abrupt changes or dramatic breaks in behavior, due to regime switching 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 generalization properties, for describing severely nonlinear regime-switching systems.

Time-varying parametric modelling and time-dependent spectral characterisation with applications to EEG signals using multiwavelets

“…Wavelets have excellent approximation properties that outperform many other approximation schemes and are well suited for approximating general nonlinear signals, even those with sharp discontinuities Wei and Billings, 2007). Wavelets have been successfully applied to EEG signal processing and analysis, see for example Schiff et al (1994), Kalayci and Ozdamar (1995), Blanco et al (1998), and Adeli et al (2003), as well as have been widely used in many other fields including nonlinear signal processing and system identification, see for example Billings and Coca (1999), Liu et al (2002), Wei (2005a, 2005b), Wei and Billings (2004a, 2004b, 2006a, and Wei, Billings and Balikhin (2004).…”

Power-law behaviour evaluation from foreign exchange market data using a wavelet transform method

“…Using the embedding theorem and the result here, along with other variable selection methods [56], we can determine the best model variables used for constructing dynamical models that are suitable for predicting individual foreign exchange rates. It is also believed that the reported result has another potential application, that is, it can be used to aid the determination of wavelet scale parameters if dynamical multiscale or multiresolution wavelet models [7][8][9][10][11][12], which have been proved to be very effective for dynamical system modelling, are to be employed to model and forecast foreign exchange rates, where wavelet models for different individual exchange rates will require different wavelet scale parameters. Figure 1 Graphs of the wavelet transform correlation function defined by (8) for the twenty datasets of foreign exchange rates listed in Table 1 where Daubechies' wavelet of order 20 was used.…”

“…For example, we plan to extend and adapt the recently developed nonlinear system and identification methods and algorithms including some wavelet based dynamical modelling approaches [7][8][9][10][11][12]54] to forecast foreign exchange rates. One direct application of the result here is to aid the selection and determination of model variables using the value of the power-law parameter β .…”

“…Similarly, at a fine resolution level, one would get detailed features. This enables us to see both the 'forest' and the 'trees', so to speak, and makes wavelets very useful [3] for data modelling and analysis in diverse fields including dynamical systems modelling [4][5][6][7][8][9][10][11][12], as well as random signal processing and analysis in for example statistical self-similarity detection and fractal property characterization .…”