Chaotic Time Series Prediction with Volterra Series Methods

This paper introduces two methods for chaotic time series prediction: Volterra series single-step prediction and Volterra series multi-step prediction. Chaotic time series prediction is a technique used for forecasting nonlinear time series by analyzing and modeling the dynamic behavior of systems. The Volterra series serves as a fundamental mathematical framework for characterizing the dynamical properties of nonlinear systems. In chaotic time series prediction, we utilize Volterra series to construct predictive models through systematic analysis and computational approaches. The implementation typically involves estimating Volterra kernels from historical data using techniques like least squares estimation or orthogonal least squares methods. For single-step prediction, the model calculates the next time step value using current and past observations through a truncated Volterra series expansion. Multi-step prediction extends this approach by iteratively applying the single-step predictor or developing separate higher-order models for longer forecasting horizons. In code implementation, single-step prediction can be achieved by solving the Volterra series coefficients through matrix operations, where the design matrix contains lagged time series values and their nonlinear combinations. Multi-step prediction requires careful handling of prediction errors accumulation, often implemented through recursive prediction or direct multi-step ahead forecasting strategies. Both methods employ regularization techniques to prevent overfitting when dealing with high-dimensional Volterra representations. By employing these methodologies, we can achieve more accurate predictions of chaotic time series evolution and obtain superior forecasting performance compared to traditional linear approaches.