Building dynamical models from data and prior knowledge: the case of the first period-doubling bifurcation.

This paper reviews some aspects of nonlinear model building from data with (gray box) and without (black box) prior knowledge. The model class is very important because it determines two aspects of the final model, namely (i) the type of nonlinearity that can be accurately approximated and (ii) the type of prior knowledge that can be taken into account. Such features are usually in conflict when it comes to choosing the model class. The problem of model structure selection is also reviewed. It is argued that such a problem is philosophically different depending on the model class and it is suggested that the choice of model class should be performed based on the type of a priori available. A procedure is proposed to build polynomial models from data on a Poincaré section and prior knowledge about the first period-doubling bifurcation, for which the normal form is also polynomial. The final models approximate dynamical data in a least-squares sense and, by design, present the first period-doubling bifurcation at a specified value of parameters. The procedure is illustrated by means of simulated examples.

Piecewise affine models of chaotic attractors: the Rossler and Lorenz systems.

This paper proposes a procedure by which it is possible to synthesize Rossler [Phys. Lett. A 57, 397-398 (1976)] and Lorenz [J. Atmos. Sci. 20, 130-141 (1963)] dynamics by means of only two affine linear systems and an abrupt switching law. Comparison of different (valid) switching laws suggests that parameters of such a law behave as codimension one bifurcation parameters that can be changed to produce various dynamical regimes equivalent to those observed with the original systems. Topological analysis is used to characterize the resulting attractors and to compare them with the original attractors. The paper provides guidelines that are helpful to synthesize other chaotic dynamics by means of switching affine linear systems.

Síntese de sinais de variabilidade da freqüência cardíaca baseada em modelagem não-linear / Heart rate variability synthesis based on nonlinear modelling

A análise de sinais por meio de índice não-lineares normalmente requer séries de dados estacionários e com um grande número de amostras. Tais requisitos não são facilmente obtidos no caso de sinais biológicos, como a variabilidade da freqüênci cardíaca (VFC). Uma possível maneira de se contornar tais dificuldades seria por meio da geração de um sinal que apresentasse as características dinâmicas do sinal real e que fosse estacionário por construção. Este trabalho apresenta uma técnica de síntese de sinais estacionários, com características semelhantes às de sinais de VFC, baseada na geração de séries de longa duração por meio de modelos NARMAX (Nonlinear AutoRegressive Moving Average with eXogenous inputs). Foram usadas duas representações: polinomial e neural. Um sinal de entrada foi construído considerando informações espectrais do sinal de VFC original. Índices lineares e não-lineares foram utilizados para validar as séries sintéticas, e verificar se essas séries apresentam características, principalmente dinâmicas, próximas daquelas apresentadas pelas séries VFC. Os resultados mostraram que os índices estatísticos do dominio do tempo para as séries sintéticas ficaram distantes dos resultados das séries reais, no caso de séries longas. Os resultados referentes à entropia aproximada se mostraram próximos entre sintéticos e reais, principalmente para as séries geradas por modelos neurais. O índice alfa 1 do método Detrended Fluctuation Analysis (DFA) para as séries sintéticas mostrou-se próximo daqueles apresentados pelas séries reais.

Evaluation of dynamical models: dissipative synchronization and other techniques.

Some recent developments for the validation of nonlinear models built from data are reviewed. Besides giving an overall view of the field, a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model, which is very useful in validating models that should reproduce dominant dynamical features, like bifurcations, of the original system. In order to assess the discriminating power of the procedure, four well-known benchmarks have been used: namely, Duffing-Ueda, Duffing-Holmes, and van der Pol oscillators, plus the Hénon map. The procedure, developed for discrete-time systems, is focused on the dynamical properties of the model, rather than on statistical issues. For all the systems investigated, it is shown that the discriminating power of the procedure is similar to that of bifurcation diagrams--which in turn is much greater than, say, that of correlation dimension--but at a much lower computational cost.

Using data-driven discrete-time models and the unscented Kalman filter to estimate unobserved variables of nonlinear systems.

This paper addresses the problem of state estimation for nonlinear systems by means of the unscented Kalman filter (UKF). Compared to the traditional extended Kalman filter, the UKF does not require the local linearization of the system equations used in the propagation stage. Important results using the UKF have been reported recently but in every case the system equations used by the filter were considered known. Not only that, such models are usually considered to be differential equations, which requires that numerical integration be performed during the propagation phase of the filter. In this paper the dynamical equations of the system are taken to be difference equations--thus avoiding numerical integration--and are built from data without prior knowledge. The identified models are subsequently implemented in the filter in order to accomplish state estimation. The paper discusses the impact of not knowing the exact equations and using data-driven models in the context of state and joint state-and-parameter estimation. The procedure is illustrated by means of examples that use simulated and measured data.

Constraining the topology of neural networks to ensure dynamics with symmetry properties.

This paper addresses the training of network models from data produced by systems with symmetry properties. It is argued that although general networks are global approximators, in practice some properties such as symmetry are very hard to learn from data. In order to guarantee that the final network will be symmetrical, constraints are developed for two types of models, namely, the multilayer perceptron (MLP) network and the radial basis function (RBF) network. In global modeling problems it becomes crucial to impose conditions for symmetry in order to stand a chance of reproducing symmetry-related phenomena. Sufficient conditions are given for MLP and RBF networks to have a set of fixed points that are symmetrical with respect to the origin of the phase space. In the case of MLP networks such conditions reduce to the absence of bias parameters and the requirement of odd activation functions. This turns out to be important from a dynamical point of view since some phenomena are only observed in the context of symmetry, which is not a structurally stable property. The results are illustrated using bench systems that display symmetry, such as the Duffing-Ueda oscillator and the Lorenz system.

Stability analysis of sleep apnea time series using identified models: a case study.

This paper investigates the use of identified nonlinear multivariable autonomous models in the classification of breathing patterns of a patient with sleep apnea. Details about the identification procedure are provided and the results reported for the case study at hand suggest that identified models could be useful in computer-based monitoring.