Multivariate polynomial regression for identification of chaotic time series.

*(English)*Zbl 1117.62099Summary: Multivariate polynomial regression was used to generate polynomial iterators for time series exhibiting autocorrelations. A stepwise technique was used to add and remove polynomial terms to ensure the model contained only those terms that produce a statistically significant contribution to the fit. An approach is described in which datasets are divided into three subsets for identification, estimation, and validation. This produces a parsimonious global model that is can greatly reduce the tendency towards undesirable behaviours such as overfitting or instability.

The technique was found to be able to identify the nonlinear dynamic behaviour of simulated time series, as reflected in the geometry of the attractor and calculation of multiple Lyapunov exponents, even in noisy systems. The technique was applied to times series data obtained from simulations of the Lorenz and Mackey-Glass equations with and without measurement noise. The model was also used to determine the embedding dimension of the Mackey-Glass equation.

The technique was found to be able to identify the nonlinear dynamic behaviour of simulated time series, as reflected in the geometry of the attractor and calculation of multiple Lyapunov exponents, even in noisy systems. The technique was applied to times series data obtained from simulations of the Lorenz and Mackey-Glass equations with and without measurement noise. The model was also used to determine the embedding dimension of the Mackey-Glass equation.

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

37M10 | Time series analysis of dynamical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

##### Keywords:

nonlinear modelling; Lyapunov coefficients; multivariate polynomial regression; chaos; identification
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\textit{D. A. Vaccari} and \textit{H. K. Wang}, Math. Comput. Model. Dyn. Syst. 13, No. 4, 395--412 (2007; Zbl 1117.62099)

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