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

已经提出了几种方法来分析非线性动力系统产生的复杂行为。如果我们预测这样的非线性动力系统产生的行为，我们必须考虑非线性预测方法，而不是线性预测方法。在这份报告中，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。我们构造了一个状态空间，同时涉及事件的时间和幅度信息。因此，我们认为，重要的和必要的信息的动力系统不仅是事件的大小或事件的时间，但都观察到的事件的大小和时间，此外，我们提出了一种新的非线性预测方法，实现高可预测性的观察时间序列的事件的大小和时间。虽然非线性预测方法分为全局方法和局部方法，但本文主要讨论局部方法。特别地，我们采用了雅可比矩阵估计， ...更多信息 离子方法，局部线性方法之一。通常，局部线性方法使用预测目标的附近轨迹在非线性动力学系统的吸引子上的移动信息。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很差，就很难正确地预测未来的状态，为了解决这一问题，我们提出了一种新的局部线性预测方法，该方法引入了自举复制方法，称为非线性自举预测。Bootstrap方法是从少量观测数据估计总体统计量的统计技术之一。我们提出了一种新的预测方法，结合基本的局部线性预测和自助方法。然后，我们通过数值模拟表明，所提出的自助非线性预测方法是非常有效的。为了评估预测方法的有效性，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，其中可能包括真实的未来点。我们提出了一种新的方法来估计预测区间使用非线性自助预测点的分布。然后，我们评估建议的区间估计的有效性相比，集合预测，这是传统的区间估计。结果表明，自举方法更能合理地确定有效的预测区间，特别是在短期预测的情况下。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