Making sense of a complex world chaotic events modeling

Addresses the identification of nonlinear systems from output time series, which we have called dynamic modeling. We start by providing the mathematical basis for dynamic modeling and show that it is equivalent to a multivariate nonlinear prediction problem in the reconstructed space. We address the importance of dynamic reconstruction for dynamic modeling. Recognizing that dynamic reconstruction is an ill-defined inverse problem, we describe a regularized radial basis function network for solving the dynamic reconstruction problem. Prior knowledge in the form of smoothness of the mapping is imposed on the solution via regularization. We also show that, in time-series analysis, some form of regularization can be accomplished by using the structure of the time series instead of imposing a smoothness constraint on the cost function. We develop a methodology based on iterated prediction to train the network weights with an error derived through trajectory learning. This method provides a robust performance because during learning the weights are constrained to follow a trajectory. The dynamic invariants estimated from the generated time series are similar to the ones estimated from the original time series, which means that the properties of the attractor have been captured by the neural network. We finally raise the question that generalized delay operators may have advantages in dynamic reconstruction, primarily in cases where the time series is corrupted by noise. We show how to set the recursive parameter of the gamma operator to attenuate noise and preserve the dynamics.