International Journal of Bifurcation and Chaos

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.

We analyze prediction schemes for stochastic time series data. We propose that under certain conditions a scalar time series, obtained from a vector-valued Markov process, can be modeled as a finite memory Markov process in the observable. The transition rules of the process are easily computed using simple nonlinear time series predictors originally proposed for deterministic chaotic signals. The optimal time lag entering the embedding procedure is shown to be significantly smaller than in the deterministic case. The same concept can be extended to nonstationary stochastic processes, where an increase of the embedding dimension can help to identify instantaneous dynamical properties, and where redundant information in the past is exploited in an optimal way.

In the past few years a new learning method called Support Vector Machines (SVMs) has enjoyed increasing popularity. Based on statistical learning theory it shows very good generalization abilities. Though SVMs are mainly used for classification tasks, they are also applicable to regression problems and thus to modeling the dynamics of a time series. However when regression techniques are used to build dynamical models caution is advisable if the data are noisy. Due to correlations between data points, estimates of model parameters deviate systematically from the true values. An approach is presented to reduce such bias in SVM parameters.

We discuss analysis tools of spatiotemporal patterns. These tools are based on nearest-neighbor considerations similar to cellular automata. Application of these methods to a spatiotemporal data set means selecting certain scales in space and in time. Focusing on spatial length we show that the dependence of the results on this scale can be used to quantify separately the contribution to the dynamics of measurement noise and of dynamical (internal) noise, respectively. In particular, we test the spatiotemporal filters using sample data generated with a network of coupled Sel'kov oscillators. Possible application of our results to biological systems are briefly discussed.

The task of locating an arcing-fault on overhead line using sampled measurements obtained at a single line terminal could be classified as a practical nonlinear system identification problem. The practical reasons impose the requirement that the solution should be with maximum possible precision. Dynamic behavior of an arc in open air is influenced by the environmental conditions that are changing randomly, and therefore the useful practically application of parametric modeling is out of question. The requirement to identify only one parameter is yet another specific of this problem. The parameter we need is the one that linearly correlates the voltage samples with the current derivative samples (inductance). The correlation between the voltage samples and the current samples depends on the unpredictable arc dynamic behavior. Therefore this correlation is reconstructed using nonparametric regression. A partially linear model combines both, parametric and nonparametric parts in one model. The fit of this model is noniterative, and provides an efficient way to identify (pull out) a single linear correlation from the nonlinear time series.

A new, parameter-free approach based on information theoretical tools is presented which allows the detection of dependences in the dynamics between two point processes. The crucial point is the definition of sequences of inter-event intervals between the events of two stochastic point processes where these sequences are ordered to only one common time index. This is an enhancement of the concept of event intervals of a single point process and makes the analysis of the process dynamics of more than one point processes possible. An application of this method is also illustrated using a model consisting of two synaptically coupled Hindmarsh–Rose neurons.

Traffic flow in large and complex freeway networks is a highly nonlinear phenomenon, which makes traffic forecast a difficult task. In this article an approach to traffic forecast is presented, which uses a micro-simulator for traffic flow combined with current and historical traffic data. The micro-simulator and locally measured current traffic data are used to reconstruct the current network-wide traffic state. Then, this state is combined with historical traffic data to forecast the traffic development. This framework is applied to the freeway network of the German state North Rhine-Westphalia. The micro-simulator uses an advanced cellular-automaton model for traffic flow, and the data are supplied from more than 4,000 locally installed loop-detectors, which deliver information on the (local) traffic state online minute by minute.

We show that based on the mathematics of Markov processes and particularly based on the definition of Kramers–Moyal coefficients, it is possible to estimate the deterministic part of the dynamics for a broad class of nonlinear noisy systems. In particular, we show that for different kinds of noise perturbations, including non-Langevin force with finite correlation time and independent measurement noise, the deterministic part can be reconstructed.

Experimental analysis of spatially extended processes is based on measured data of characteristic dynamic variables in space and time. The basic aim of the analysis is to extract a field evolution law from data thus recorded. We present a nonparametric statistical modeling of field evolution, following a state space reconstruction technique. For this purpose a novel state reconstruction is proposed that properly describes chaotic field evolution on short- as well as statistically on long-term basis. From the reconstructed state vectors, deterministic and random parts of field evolution are then approximated by employing a conditional average estimator. The performance of such statistical modeling is demonstrated by predicting simulated stochastic fields, and experimental fields generated by cutting and welding manufacturing processes. The results presented indicate that the proposed method of modeling can be successfully utilized for experimental characterization of different stochastic processes.

This article describes how to use statistical data analysis to obtain models directly from data. The focus is put on finding nonlinearities within a generalized additive model. These models are found by means of backfitting or more general algorithms, like the alternating conditional expectation value one. The method is illustrated by numerically generated data. As an application, the example of vortex ripple dynamics, a highly complex fluid-granular system, is treated.

In this work we propose a method that exploits the feedback between empirical and theoretical knowledge of a complex macroscopic system in order to build a nonlinear model. We apply the method to the monthly earth's mean surface temperature time series. The problems of contamination and stationarity are considered noting the importance of observation and modeling scales. We construct a dynamical system of ordinary differential equations where the vector field relating the relevant degrees of freedom and their variations in time is expressed in terms of a polynomial base orthonormal to the measure associated to the time series under study. The optimal size of the model and the values of its parameters are estimated with the principle of minimum description length and the Adams–Molton predictor–corrector method. This procedure is self-consistent because it does not use any external parameter or assumption. We then present a first approach to find the closest chaotic dynamical system corresponding to the earth's mean surface temperature and compare it with scale consistent theoretical or phenomenological models of the lower atmosphere. This comparison allows us to obtain an explicit functional form of the heat capacity of the earth's surface as a function of the earth's mean surface temperature.

Many physical systems exhibiting nonlinear spatiotemporal dynamics can be modeled by partial differential equations. Although information about the physical properties for many of these systems is available, normally not all dynamical parameters are known and, therefore, have to be estimated from experimental data. We analyze two prominent approaches to solve this problem and describe advantages and disadvantages of both methods. Specifically, we focus on the dependence of the quality of the parameter estimates with respect to noise and temporal and spatial resolution of the measurements.

Recently, it has been shown experimentally [Mainen & Sejnowski, 1995] that, in contrast to the lack of precision in spike timing associated with flat (dc) stimuli, neocortical neurons of rats respond reliably to weak input fluctuations resembling synaptic activity. This has led authors to suggest that, in spite of the high variability of interspike intervals found in cortical activity, the mechanism of spike generation in neocortical neurons has a low level of intrinsic noise. In this work we approach the problem of spike timing by using the well-known FitzHugh–Nagumo (FHN) model of neuronal dynamics and find that here also, fluctuating stimuli allow a more reliable temporal coding than constant suprathreshold signals. This result is associated with the characteristics of a phenomenological stochastic bifurcation taking place in the noisy FHN model.

During the past decades the components involved in cellular signal transduction from membrane receptors to gene activation in the nucleus have been studied in detail. Based on the qualitative biochemical knowledge, signalling pathways are drawn as static graphical schemes. However, the dynamics and control of information processing through signalling cascades is not understood. Here we show that based on time resolved measurements it is possible to quantitatively model the nonlinear dynamics of signal transduction. To select an appropriate model we performed parameter estimation by maximum likelihood and statistical testing. We apply this approach to the JAK-STAT signalling pathway that was believed to represent a feed-forward cascade. We show by comparison of different models that this hypothesis is insufficient to explain the experimental data and suggest a new model including a delayed feedback.

On analyzing data of biochemical reaction dynamics monitored by time-resolved spectroscopy, one faces the problem that the concentration time courses of the involved components are not directly observed, but the superposition of their absorption spectra. Furthermore the single spectra are often unknown, because the corresponding reagents cannot be isolated. We propose a method based on Bock's multiple shooting algorithm to estimate the rate constants and individual spectra simultaneously. Applying this procedure to a biochemical reaction we identify the specific rate constants characterizing the reaction dynamics as well as the nonobservable absorption spectra. The results lead to a better understanding of the kinetics of a novel modification reaction which was used as trapping reaction in disulfide bond mediated protein folding reactions.

We present a derivation of the unscented Kalman filter (UKF) as an approximation to the optimal Bayesian filter equations. The potentials of the UKF are then demonstrated for the problem of simultaneous estimation of states and parameters from noise corrupted data of nonlinear dynamical systems. The UKF even works for the chaotic Chua system which includes nondifferentiable terms.

Fourier surrogate data are artificially generated time series, that — based on a resampling scheme — share the linear properties with an observed time series. In this paper we study a statistical surrogate hypothesis test to detect deviations from a linear Gaussian process with respect to asymmetry in time (Q-statistic). We apply this test to a Fourier representable function and obtain a representation of the asymmetry in time of the sample data, a characteristic for nonlinear processes, and the significance in terms of the Fourier coefficients. The main outcome is that we calculate the expected value of the mean and the standard deviation of the asymmetries of the surrogate data analytically and hence, no surrogates have to be generated. To illustrate the results we apply our method to the saw tooth function, the Lorenz system and to measured X-ray data of Cygnus X-1.

An algorithm is presented which generates pairs of oscillatory random time series which have identical periodograms but differ in the number of oscillations. This result indicates the intrinsic limitations of spectral methods when it comes to the task of measuring frequencies. Other examples, one from medicine and one from bifurcation theory, are given, which also exhibit these limitations of spectral methods. For two methods of spectral estimation it is verified that the particular way end points are treated, which is specific to each method, is, for long enough time series, not relevant for the main result.

There is an important interest in compensating thermally induced errors of modular tool systems to improve the manufacturing accuracy. In this paper, we test the hypothesis whether we can predict such thermal displacements by using a nonlinear regression analysis, namely the alternating conditional expectation algorithm (ACE [Breiman & Friedman, 1985]), reliably. The data analyzed were generated by two different finite element spindle models of modular tool systems. As the main result, we find that the ACE-algorithm is a powerful tool to model the relation between temperatures and displacements. The maximal correlation is larger than 0.999 in both cases, which demonstrates the suitability of the ACE algorithm. Furthermore, preconditions for the applicability of this approach, such as the length and the support of measured data sets, are studied. Hence, this approach seems to be promising for the application to real modular tool systems.

OPCL autosynchronization ([1997] Phys. Rev.E56, 3709–3711) is used for determination of parameters in 1-D models. Numerical results are given for noisy Duffing and logistic systems. The proposed algorithm works also for models that contain parameters nonlinearly. Numerical results show that the error function has a global minimum.