Noisy Dynamical Systems Research Articles

Identification of noisy dynamical systems with parameter estimation based on Hopfield neural networks

An algorithm for estimating time-varying parameters of dynamical systems is proposed, within the large family of prediction error methods. The algorithm is based on the ability of Hopfield neural networks to solve optimisation problems, since its formulation can be summarized as minimisation of the prediction error by means of a continuous Hopfield network. In previous work, it was proved, under mild assumptions, that the estimates converge towards the actual values of parameters and the estimation error remains asymptotically bounded in the presence of measurement noise. The novelty of this work is the advance in the robustness analysis, by considering deterministic disturbances, which do not fulfil the usual statistical hypothesis such as normality and uncorrelatedness. A model of HIV epidemics in Cuba is used as suitable benchmark, which is confirmed by the computation of the sensitivity matrix. The results show a promising performance, in comparison to the conventional Least Squares Estimator. Indeed, the estimation error is almost always lower in the proposed method that in least squares, and it is never significantly higher. Further, from a qualitative point of view, the estimate provided by the Hopfield estimator is smoother, with no overshoot that could eventually destabilize a closed control loop. A significant finding is the fact that the form of the perturbation affects critically the dynamical behaviour and magnitude of the estimation, since the estimation error asymptotically vanishes when the disturbances are additive, but not when they are multiplicative. To summarize, we can conclude that the proposed estimator is an efficient and robust method to estimate time-varying parameters of dynamical systems.

A Fresh Look at the Kalman Filter

In this paper, we discuss the Kalman filter for state estimation in noisy linear discrete-time dynamical systems. We give an overview of its history, its mathematical and statistical formulations, and its use in applications. We describe a novel derivation of the Kalman filter using Newton's method for root finding. This approach is quite general as it can also be used to derive a number of variations of the Kalman filter, including recursive estimators for both prediction and smoothing, estimators with fading memory, and the extended Kalman filter for nonlinear systems.

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ∼τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t → 2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.

A phase-synchronization and random-matrix based approach to multichannel time-series analysis with application to epilepsy.

We present a general method to analyze multichannel time series that are becoming increasingly common in many areas of science and engineering. Of particular interest is the degree of synchrony among various channels, motivated by the recognition that characterization of synchrony in a system consisting of many interacting components can provide insights into its fundamental dynamics. Often such a system is complex, high-dimensional, nonlinear, nonstationary, and noisy, rendering unlikely complete synchronization in which the dynamical variables from individual components approach each other asymptotically. Nonetheless, a weaker type of synchrony that lasts for a finite amount of time, namely, phase synchronization, can be expected. Our idea is to calculate the average phase-synchronization times from all available pairs of channels and then to construct a matrix. Due to nonlinearity and stochasticity, the matrix is effectively random. Moreover, since the diagonal elements of the matrix can be arbitrarily large, the matrix can be singular. To overcome this difficulty, we develop a random-matrix based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides a powerful way to assess changes in synchrony. The method is tested using a prototype nonstationary noisy dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced cortico-thalamic synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization for which enhanced local synchrony is similarly presumed.

Statistical inference for noisy nonlinear ecological dynamic systems

Chaotic ecological dynamic systems defy conventional statistical analysis. Systems with near-chaotic dynamics are little better. Such systems are almost invariably driven by endogenous dynamic processes plus demographic and environmental process noise, and are only observable with error. Their sensitivity to history means that minute changes in the driving noise realization, or the system parameters, will cause drastic changes in the system trajectory. This sensitivity is inherited and amplified by the joint probability density of the observable data and the process noise, rendering it useless as the basis for obtaining measures of statistical fit. Because the joint density is the basis for the fit measures used by all conventional statistical methods, this is a major theoretical shortcoming. The inability to make well-founded statistical inferences about biological dynamic models in the chaotic and near-chaotic regimes, other than on an ad hoc basis, leaves dynamic theory without the methods of quantitative validation that are essential tools in the rest of biological science. Here I show that this impasse can be resolved in a simple and general manner, using a method that requires only the ability to simulate the observed data on a system from the dynamic model about which inferences are required. The raw data series are reduced to phase-insensitive summary statistics, quantifying local dynamic structure and the distribution of observations. Simulation is used to obtain the mean and the covariance matrix of the statistics, given model parameters, allowing the construction of a 'synthetic likelihood' that assesses model fit. This likelihood can be explored using a straightforward Markov chain Monte Carlo sampler, but one further post-processing step returns pure likelihood-based inference. I apply the method to establish the dynamic nature of the fluctuations in Nicholson's classic blowfly experiments.

RELATIVE KOLMOGOROV ENTROPY OF A CHAOTIC SYSTEM IN THE PRESENCE OF NOISE

The mixing property is characterized by the metric entropy that is introduced by Kolmogorov for dynamical systems. The Kolmogorov entropy is infinite for a stochastic system. In this work, a relative metric entropy is considered. The relative metric entropy allows to estimate the level of mixing in noisy dynamical systems. An algorithm for calculating the relative metric entropy is described and examples of the metric entropy estimation are provided for certain chaotic systems with various noise intensities. The results are compared to the entropy estimation given by the positive Lyapunov exponents.

Modelling multivariate data systems: application to El Niño and the Annual Cycle

We propose a general procedure for the analysis of noisy high dimensional dynamical systems, potentially with time delay. This procedure combines novel dimensionality reduction techniques, nonlinear time series analysis for dynamical systems and nonparametric statistical estimation of functional dependencies. To check the feasibility of our method, we apply it to the sea surface temperature (SST) field in the tropical Pacific Ocean, in order to build a model for the interaction of El Niño/Southern Oscillation (ENSO) and the Annual Cycle coupled system. This dynamical representation is shown to be reducible to three dimensions by applying Isomap, a recent method of dimensionality reduction. From the resulting time series, we construct a stochastic dynamical system by using nonparametric estimation. This dynamical system is numerically integrated and compared with measured data a posteriori. The use for prediction of the joint system of ENSO and the Annual Cycle is discussed.

Use of particle filters in an active control algorithm for noisy nonlinear structural dynamical systems

The problem of active control of nonlinear structural dynamical systems, in the presence of both process and measurement noises, is considered. The focus of the study is on the use of particle filters for state estimation in feedback control algorithms for nonlinear structures, when a limited number of noisy output measurements are available. The control design is done using the state-dependent Riccati equation (SDRE) method. The stochastic differential equations (SDEs) governing the dynamical systems are discretized using explicit forms of Ito–Taylor expansions. The Bayesian bootstrap filter and that based on sequential important sampling (SIS) are employed for state estimation. The simulation results show the feasibility of using particle filters and SDRE techniques in control of nonlinear structural dynamical systems.

Nonlinear State Estimation Using Fuzzy Kalman Filter

In this paper, the authors have presented an approach for designing a nonlinear observer to estimate the states of a noisy dynamic system. The nonlinear observer design procedure involves representation of the nonlinear system as a family of local linear state space models; the state estimator for each linear local state space model uses standard Kalman filter theory and then a global state estimator is developed that combines the local state estimators. The effectiveness of the proposed fuzzy Kalman filter (nonlinear observer) has been demonstrated on a continuously stirred tank reactor (CSTR) process. The performances of the fuzzy Kalman filter (FKF) and the extended Kalman filter (EKF) have been compared in the presence of initial model/plant mismatch and input and output disturbances. Simulation studies also include an estimation of reactor concentration (inferential measurement), based only on the measured variable temperature of the reactor.

Cooperative dynamics in coupled noisy dynamical systems near a critical point: The dc superconducting quantum interference device as a case study

Dynamical systems that operate near the onset of coupling-induced oscillations can exhibit enhanced sensitivity to external perturbations under suitable operating parameters. This cooperative behavior and the attendant enhancement in the system response (quantified here via a signal-to-noise ratio at the fundamental of the coupling-induced oscillation frequency) are investigated in this work. As a prototype, we study an array of dc superconducting quantum interference device (SQUID) rings locally coupled, unidirectionally as well as bidirectionally, in a ring configuration; it is well known that each individual SQUID can be biased through a saddle-node bifurcation to oscillatory behavior. We show that biasing the array near the bifurcation point of coupling-induced oscillations can lead to a significant performance enhancement.

A study on stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point with the moment method

Abstract This paper analyzes the stochastic resonance induced by a novel transition of one-dimensional bistable system in the neighborhood of bifurcation point with the method of moment, which refer to the transition of system motion among a potential well of stable fixed point before bifurcation of original system and double-well potential of two coexisting stable fixed points after original system bifurcation at the presence of internal noise. The results show: the semi-analytical result of stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point may be obtained, and the semi-analytical result is in accord with the one of Monte Carlo simulation qualitatively, the occurrence of stochastic resonance is related to the bifurcation of noisy nonlinear dynamical system moment equations, which induce the transfer of energy of ensemble average (Ex) of system response in each frequency component and make the energy of ensemble average of system response concentrate on the frequency of input signal, stochastic resonance occurs.

A comparative evaluation of neural networks and hidden Markov models for monitoring turning tool wear

Condition monitoring of machine tool inserts is important for increasing the reliability and quality of machining operations. Various methods have been proposed for effective tool condition monitoring (TCM), and currently it is generally accepted that the indirect sensor-based approach is the best practical solution to reliable TCM. Furthermore, in recent years, neural networks (NNs) have been shown to model successfully, the complex relationships between input feature sets of sensor signals and tool wear data. NNs have several properties that make them ideal for effectively handling noisy and even incomplete data sets. There are several NN paradigms which can be combined to model static and dynamic systems. Another powerful method of modeling noisy dynamic systems is by using hidden Markov models (HMMs), which are commonly employed in modern speech-recognition systems. The use of HMMs for TCM was recently proposed in the literature. Though the results of these studies were quite promising, no comparative results of competing methods such as NNs are currently available. This paper is aimed at presenting a comparative evaluation of the performance of NNs and HMMs for a TCM application. The methods are employed on exactly the same data sets obtained from an industrial turning operation. The advantages and disadvantages of both methods are described, which will assist the condition-monitoring community to choose a modeling method for other applications.

Averaging of Noisy Nonlinear Systems with Rapidly Oscillating and Decaying Components

We consider a noisy n-dimensional nonlinear dynamical system containing rapidly oscillating and decaying components. We extend the results of Papanicolaou and Kohler and Namachchivaya and Lin; these results state that as the noise becomes smaller, a lower dimensional Markov process characterizes the limiting behavior. Our approach springs from a direct consideration of the martingale problem and considers both quadratic and cubic nonlinearities.

Investigating soil moisture feedbacks on precipitation with tests of Granger causality

Granger causality (GC) is used in the econometrics literature to identify the presence of one- and two-way coupling between terms in noisy multivariate dynamical systems. Here we test for the presence of GC to identify a soil moisture ( S) feedback on precipitation ( P) using data from Illinois. In this framework S is said to Granger cause P if F(P t|Ω t− Δt )≠F(P t|Ω t− Δt −S t− Δt ) where F denotes the conditional distribution of P, Ω t− Δt represents the set of all knowledge available at time t−Δ t, and Ω t− Δt −S t− Δt represents all knowledge except S. Critical for land–atmosphere interaction research is that Ω t− Δt includes all past information on P as well as S. Therefore that part of the relation between past soil moisture and current precipitation which results from precipitation autocorrelation and soil water balance will be accounted for and not attributed to causality. Tests for GC usually specify all relevant variables in a coupled vector autoregressive (VAR) model and then calculate the significance level of decreased predictability as various coupling coefficients are omitted. But because the data (daily precipitation and soil moisture) are distinctly non-Gaussian, we avoid using a VAR and instead express the daily precipitation events as a Markov model. We then test whether the probability of storm occurrence, conditioned on past information on precipitation, changes with information on soil moisture. Past information on precipitation is expressed both as the occurrence of previous day precipitation (to account for storm-scale persistence) and as a simple soil moisture-like precipitation-wetness index derived solely from precipitation (to account for seasonal-scale persistence). In this way only those fluctuations in moisture not attributable to past fluctuations in precipitation (e.g., those due to temperature) can influence the outcome of the test. The null hypothesis (no moisture influence) is evaluated by comparing observed changes in storm probability to Monte-Carlo simulated differences generated with unconditional occurrence probabilities. The null hypothesis is not rejected ( p>0.5) suggesting that contrary to recently published results, insufficient evidence exists to support an influence of soil moisture on precipitation in Illinois.

The probabilistic symmetry breaking of periodic regimes rapidly passing through a zone of chaos into the transparency window

The problem of transition of a noisy dynamical system to a periodic oscillatory regime through a zone of chaos is considered. Using the noisy logistics map as an example, domains of attraction of energetically equivalent regimes of period three are found for various transition rates and various noise levels. The fine structure of the domains of attraction under the condition of fast transitions is revealed. It is discovered that the settling time of the stable cycle of period three heavily depends on the initial conditions, i.e., on the structure of the domains of attraction. The critical transition rate that separates the region of the probabilistic symmetry of final states from the region of the dynamic behavior of trajectories is estimated.

Reconstructions and Predictions of Nonlinear Dynamical Systems Weighted by Model Marginal Likelihoods

A hierarchical Bayesian approach is formulated for nonlinear time series prediction problems with neural nets. The proposed scheme consists of several steps : (i) Formulae for posterior distributions of parameters, hyper parameters as well as models via Bayes formula.(ii) Derivation of predictive distributions of future values taking into account model marginal likelihoods.(iii) Using several drastic approximations for computing predictive mean of time series incorporating model marginal likelihoods.The proposed scheme is tested against two examples; (A) Time series data generated by noisy chaotic dynamical system, and (B) Building air-conditioning load prediction problem. The proposed scheme outperforms the algorithm previously used by the authors.

Trace formula for noisy flows

A trace formula is derived for the Fokker–Planck equation associated with Ito stochastic differential equations describing noisy time-continuous nonlinear dynamical systems. In the weak-noise limit, the trace formula provides estimations of the eigenvalues of the Fokker–Planck operator on the basis of the Pollicott–Ruelle resonances of the noiseless deterministic system, which is assumed to be non-bifurcating. At first order in the noise amplitude, the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo-energy of the Onsager–Machlup–Freidlin–Wentzell scheme.

Noise-Induced Resonances in Superparamagnetic Particles

Noise-Induced Resonances in Superparamagnetic Particles

Q2 Symbolic Reasoning about Noisy Dynamic Systems

Symbolic reasoning about continuous dynamic systems requires consistent qualitative abstraction functions and a consistent symbolic model. Classically, symbolic reasoning systems have utilized a box partition of the system space to achieve qualitative abstraction, but boxes can not provide a consistent abstraction. Our Q2 methodology abstracts a provably consistent symbolic representation of noise-free general dynamic systems. However, the Q2 symbolic representation has not been previously evaluated for efficacy in the presence of noise. We evaluate the effects of noise on Q2 symbolic reasoning in the domain of maneuver detection. We demonstrate how the Q2 methodology derives a symbolic abstraction of a general dynamic system model used in evaluating maneuver detectors. Simulation results represented by ROC curves show that the Q2 based maneuver detector is superior to a box-based detector. While no method is consistent in the presence of noise, the Q2 methodology is superior to the classic box’s approach for deriving qualitative decisions about noisy dynamic systems.

Stability and damping in a dynamical model of hillslope hydrology

A stability analysis is performed on the two‐state variable, integral‐balance model of hillslope hydrology developed by Duffy [1996]. The motivation for the research is to develop a physically based, low‐dimensional representation of inflow/outflow behavior within a hillslope‐stream setting. Stability criteria are developed for the model equilibria and are evaluated using the results of numerical solutions of Richards' equation for a convex‐concave hillslope geometry. We show that for homogeneous hillslopes of three widely varying soil types the single moisture equilibrium is classified as a stable node for low precipitation rates and a stable spiral for wet conditions. The spiral equilibrium indicates that the hillslope system is lightly damped, and transient oscillations of the state variables are expected for high precipitation rates. The timescale of these oscillations is of the order of several days to weeks for the model hillslopes examined. Furthermore, we demonstrate that the model contains a Hopf bifurcation from a stable static equilibrium to a stable limit cycle. The amplitude, phase, and frequency of the limit cycle are determined analytically using second‐order averaging. However, this behavior is shown to be nonphysical for the particular homogeneous soils and hillslope geometry investigated. Implications of lightly damped behavior in the hillslope system include moisture oscillations in the field under wet conditions and difficulty in numerical solution of Richards' equation.