Time series analysis of complex phenomena and a study of %paragon property from the viewpoint of stochastic processes

本研究的目的是开发一种方法，通过该方法我们可以提取时间序列数据的非线性结构，并推导出时间演化的数学模型。我们的基本思想是“从数据到数学定律和模型”，这意味着在将数学定理应用于数据分析之前，必须检查它们的前提条件是否满足。这可以通过KM 20-Langevin方程的理论来实现。到目前为止，我们已经提出了几种类型的测试：测试（S）-平稳性测试，测试（ABN）-异常性测试，测试（D）-确定性测试。将这些方法应用于深低频地震时间序列，发现了深低频地震的特征性质之一，即分离性。因此，从数学的角度来刻画“分离性”是十分重要的。本文将“分离性”表述为一个数学概念，并证明了如果随机过程的有限维分布是对称的，则该过程满足分离性。此外，我们还导出了离散时间随机过程的一种表示定理.与“分离特性”有关，检测时间序列的异常是极其重要的。然而，我们的异常测试测试（ABN）不足以达到这一目的。因此，我们新提出了一种方法，用于检测异常，称为测试（RSK），通过利用非线性预测误差。验证了该方法的有效性.我们还分析了不满足分离性的股票价格时间序列，发现二次多项式变换在描述这些时间序列的动态特性方面起着重要作用。对这一事实的数学解释是未来的任务。

The aim of this research project is to develop a method by which we can extract nonlinear structures of time series data and derive mathematical models of time evolutions. Our basic idea is "from data to mathematical laws and models", which means that it is necessary to examine whether the preconditions of the mathematical theorems are satisfied before applying them to data analysis. This can be realized by the theory of the KM20-Langevin equations. So far, we have proposed several types of tests: Test (S)- stationarity test, Test (ABN)-abnormality test, Test (D)-determinacy test. We have applied these methods to seismic time series of deep low frequency earthquakes and discovered the so called "separation property", which can be seen as one of the characteristic properties of deep low frequency earthquakes. Therefore it is quite important to characterize "separation property" from a mathematical viewpoint.In this research project, we have obtained the following results.1. We have formulated "separation property" as a mathematical concept and proved that if the finite dimensional distributions of a stochastic process are symmetric, the process satisfies separation property. Moreover, we have derived a kind of expression theorem of discrete time stochastic processes.2. In connection with "separation properties", it is extremely important to detect abnormalities of time series. However, our abnormality test Test (ABN)is not sufficient enough for this purpose. Therefore, we have newly proposed a method for detecting abnormality, called Test (RSK), by utilizing nonlinear prediction errors. We also verified the effectiveness of the method.3. We have also analyzed time series of stock prices, which do not satisfy separation property, and found that polynomial transformations of degree two play an important role in describing the dynamics of these time series. Mathematical interpretation of this fact is a future task.