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

已经提出了几种方法来分析由非线性动力系统产生的复杂行为。要预测这种非线性动力系统产生的行为，必须考虑非线性预测方法而不是线性预测方法。在本文中，我们提出了一个新的非线性建模框架来分析复杂的、可能的混沌事件序列。我们通过同时考虑事件时序及其幅度信息来构造状态空间。因此，我们认为动态系统的重要和基本信息不仅是事件大小或事件时间，还包括观测到的事件大小和时间。此外，我们还提出了一种新的非线性预测方法，以实现对事件大小和时间观测时间序列的高可预测性。虽然非线性预测方法分为全局方法和局部方法，但在本报告中我们主要关注局部方法。特别是，我们在…采用了雅可比矩阵估计More ion方法，局部线性化方法之一。通常，局部线性方法利用预测目标在非线性动力系统的吸引子上附近轨迹的运动信息。因此，在局部线性预测方法中，如何从观测到的时间序列数据中尽可能准确地估计出运动的信息是非常重要的，因为如果估计的信息较差，就很难正确地预测未来的状态。Bootstrap方法是从少量观测数据中估计总体统计量的统计技术之一。将基本局部线性预测与Bootstrap方法相结合，提出了一种新的预测方法。通过数值模拟，验证了本文提出的Bootstrap非线性预测方法的有效性。为了评价预测方法的有效性，我们通常使用均方根误差。然而，我们经常被要求估计一个预测区间，其中可能包括真正的未来点。提出了一种利用非线性Bootstrap预测点分布估计预测区间的新方法。然后，我们与传统的区间估计的集成预测进行了比较，评估了所提出的区间估计的有效性。结果表明，Bootstrap方法能更合理地确定有效的预测区间，特别是在短期预测的情况下。较少

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