Two-stage mixed discrete-continuous identification of radial basis function (RBF) neural models for nonlinear systems

The identification of nonlinear dynamic systems using radial basis function (RBF) neural models is studied in this paper. Given a model selection criterion, the main objective is to effectively and efficiently build a parsimonious compact neural model that generalizes well over unseen data. This is achieved by simultaneous model structure selection and optimization of the parameters over the continuous parameter space. It is a mixed-integer hard problem, and a unified analytic framework is proposed to enable an effective and efficient two-stage mixed discrete-continuous; identification procedure. This novel framework combines the advantages of an iterative discrete two-stage subset selection technique for model structure determination and the calculus-based continuous optimization of the model parameters. Computational complexity analysis and simulation studies confirm the efficacy of the proposed algorithm.

Moving window kernel PCA for adaptive monitoring of nonlinear processes

This paper discusses the monitoring of complex nonlinear and time-varying processes. Kernel principal component analysis (KPCA) has gained significant attention as a monitoring tool for nonlinear systems in recent years but relies on a fixed model that cannot be employed for time-varying systems. The contribution of this article is the development of a numerically efficient and memory saving moving window KPCA (MWKPCA) monitoring approach. The proposed technique incorporates an up- and downdating procedure to adapt (i) the data mean and covariance matrix in the feature space and (ii) approximates the eigenvalues and eigenvectors of the Gram matrix. The article shows that the proposed MWKPCA algorithm has a computation complexity of O(N2), whilst batch techniques, e.g. the Lanczos method, are of O(N3). Including the adaptation of the number of retained components and an l-step ahead application of the MWKPCA monitoring model, the paper finally demonstrates the utility of the proposed technique using a simulated nonlinear time-varying system and recorded data from an industrial distillation column.

On the treatment of singularities of the Watson Hamiltonian for nonlinear molecules

The applicability of the Watson Hamiltonian for the description of nonlinear molecules—especially triatomic ones—has always been questioned, as the Jacobian of the transformation that leads to the Watson Hamiltonian, vanishes at the linear configuration. This results in singular behavior of the Watson Hamiltonian, giving rise to serious numerical problems in the computation of vibrational spectra, with unphysical, spurious vibrational states appearing among the physical vibrations, especially in the region of highly excited states. In this work, we analyze the problem and propose a simple way to confine the nuclear wavefunction in such a way that the spurious solutions are eliminated. We study the water molecule and observe an improvement compared with previous results. We also apply the method to the van der Walls molecule XeHe2.

Three-dimensional pattern formation, multiple homogeneous soft modes, and nonlinear dielectric electroconvection

Patterns forming spontaneously in extended, three-dimensional, dissipative systems are likely to excite several homogeneous soft modes (approximate to hydrodynamic modes) of the underlying physical system, much more than quasi-one- (1D) and two-dimensional (2D) patterns are. The reason is the lack of damping boundaries. This paper compares two analytic techniques to derive the pattern dynamics from hydrodynamics, which are usually equivalent but lead to different results when applied to multiple homogeneous soft modes. Dielectric electroconvection in nematic liquid crystals is introduced as a model for 3D pattern formation. The 3D pattern dynamics including soft modes are derived. For slabs of large but finite thickness the description is reduced further to a 2D one. It is argued that the range of validity of 2D descriptions is limited to a very small region above threshold. The transition from 2D to 3D pattern dynamics is discussed. Experimentally testable predictions for the stable range of ideal patterns and the electric Nusselt numbers are made. For most results analytic approximations in terms of material parameters are given. [S1063-651X(00)09512-X].

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein--Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe--Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org, Oxford University Press.

Linear and nonlinear properties of Rao-dust-Alfven waves in magnetized plasmas

The linear and nonlinear properties of the Rao-dust-magnetohydrodynamic (R-D-MHD) waves in a dusty magnetoplasma are studied. By employing the inertialess electron equation of motion, inertial ion equation of motion, Ampere's law, Faraday's law, and the continuity equation in a plasma with immobile charged dust grains, the linear and nonlinear propagation of two-dimensional R-D-MHD waves are investigated. In the linear regime, the existence of immobile dust grains produces the Rao cutoff frequency, which is proportional to the dust charge density and the ion gyrofrequency. On the other hand, the dynamics of amplitude modulated R-D-MHD waves is governed by the cubic nonlinear Schrodinger equation. The latter has been derived by using the reductive perturbation technique and the two-timescale analysis which accounts for the harmonic generation nonlinearity in plasmas. The stability of the modulated wave envelope against non-resonant perturbations is studied. Finally, the possibility of localized envelope excitations is discussed. (C) 2004 American Institute of Physics.

Nonlinear modulation of transverse dust lattice waves in complex plasma crystals

The occurrence of the modulational instability in transverse dust lattice waves propagating in a one-dimensional dusty plasma crystal is investigated. The amplitude modulation mechanism, which is related to the intrinsic nonlinearity of the sheath electric field, is shown to destabilize the carrier wave under certain conditions, possibly leading to the formation of localized envelope excitations. Explicit expressions for the instability growth rate and threshold are presented and discussed. (C) 2004 American Institute of Physics.

Nonlinear theory of solitary waves associated with longitudinal particle motion in lattices - Application to longitudinal grain oscillations in a dust crystal

The nonlinear aspects of longitudinal motion of interacting point masses in a lattice are revisited, with emphasis on the paradigm of charged dust grains in a dusty plasma (DP) crystal. Different types of localized excitations, predicted by nonlinear wave theories, are reviewed and conditions for their occurrence (and characteristics) in DP crystals are discussed. Making use of a general formulation, allowing for an arbitrary (e.g. the Debye electrostatic or else) analytic potential form phi(r) and arbitrarily long site-to-site range of interactions, it is shown that dust-crystals support nonlinear kink-shaped localized excitations propagating at velocities above the characteristic DP lattice sound speed v(0). Both compressive and rarefactive kink-type excitations are predicted, depending on the physical parameter values, which represent pulse- (shock-)like coherent structures for the dust grain relative displacement. Furthermore, the existence of breather-type localized oscillations, envelope-modulated wavepackets and shocks is established. The relation to previous results on atomic chains as well as to experimental results on strongly-coupled dust layers in gas discharge plasmas is discussed.

Weakly nonlinear vertical dust grain oscillations in dusty plasma crystals in the presence of a magnetic field

The weakly nonlinear regime of transverse paramagnetic dust grain oscillations in dusty (complex) plasma crystals is discussed. The nonlinearity, which is related to the sheath electric/magnetic field(s) and to the intergrain (electrostatic/magnetic dipole) interactions, is shown to lead to the generation of phase harmonics and, in the case of propagating transverse dust-lattice modes, to the modulational instability of the carrier wave due to self-interaction. The stability profile depends explicitly on the form of the electric and magnetic fields in the plasma sheath. The long term evolution of the modulated wave packet, which is described by a nonlinear Schrodinger-type equation, may lead to propagating localized envelope structures whose exact forms are presented and discussed. Explicit suggestions for experimental investigations are put forward. (C) 2004 American Institute of Physics.

Lagrangian description of nonlinear dust-ion acoustic waves in dusty plasmas

An analytical model is presented for the description of nonlinear dust-ion-acoustic waves propagating in an unmagnetized, collisionless, three component plasma composed of electrons, ions and inertial dust grains. The formulation relies on a Lagrangian approach of the plasma fluid model. The modulational stability of the wave amplitude is investigated. Different types of localized envelope electrostatic excitations are shown to exist.