Nonlinear analysis of spatio-temporal analog discrete events and its application

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Several methods have been proposed to analyze complex behavior produced from nonlinear dynamical systems. If we predict behavior produced from such nonlinear dynamical systems, we have to consider nonlinear prediction methods rather than linear prediction methods.In this report, we propose a novel nonlinear modeling framework to analyze complex, possible chaotic event series. We construct a state space by involving the event timing and its amplitude information simultaneously. Thus, we consider that the important and essential information of the dynamical system is not only event sizes or event timings but both of observed event sizes and timings.In addition, we propose a new nonlinear prediction method to realize high predictability for the observed time series of event sizes and timings. Although the nonlinear prediction methods are classified into a global method and a local method, we focused on the local methods in this report. In particular, we adopted the Jacobian matrix estimat … More ion method, one of the local linear methods. Generally, the local linear methods use information of movements of nearby trajectories of a prediction target on an attractor of the nonlinear dynamical systems. Then, in the local linear method, it is very important to estimate the information of the movements as accurately as possible from observed time series data, because if the estimated information is poor, it is difficult to predict future states correctly.To resolve such an important issue, we proposed a new local linear prediction method that introduces the bootstrap replication method, which is called nonlinear bootstrap prediction. The bootstrap method is one of the statistical techniques to estimate statistics of a population from small number of observed data. We propose a new prediction method by combining the basic local linear prediction and the bootstrap method. Then, we showed that the proposed bootstrap nonlinear prediction method is very effective by performing numerical simulations. To evaluate the validity of the prediction method, we generally use a root mean square error. However, we are often asked to estimate a prediction interval in which true future points might be included. We proposed a new method to estimate prediction intervals using a distribution of nonlinear bootstrap predicted points. Then, we evaluate the validity of the proposed interval estimation comparing to an ensemble prediction which is the conventional interval estimation. As results, we show that the bootstrap method is more reasonable to make efficient prediction intervals especially in the case of short term prediction. Less