Research on Nemerical Methods in Time Series Analysis

研究了基于非线性非高斯状态空间模型的时间序列分析方法。特别是发展了用于非线性非高斯状态空间模型状态估计的蒙特卡罗滤波和平滑器。作为这项研究的结果，高维非线性非高斯系统的建模成为可能。我们还发展了一种蒙特卡罗后验平均光滑器，其中几乎所有的粒子都收敛到边际后验分布的平均值。这些新发展的方法被成功地应用于开发新的统计分析方法，如经济时间序列的季节性调整、卫星数据的准周期运动分析和计数数据的分析。考虑到蒙特卡罗滤波和遗传算法的相似性，Higuchi提出了一种过滤非线性非高斯状态空间模型的新算法。在研究过程中，获得了进一步发展的新线索。在状态空间建模中，我们采用了数值优化方法来估计模型的参数。该方法需要进行多次似然估计，有时会使估计变得非常困难，鉴于蒙特卡罗滤波可以应用于高维非线性系统，我们用未知参数对状态向量进行了扩充。通过对状态向量的这种修改，现在可以同时估计未知状态和模型参数。基于这一思想，我们开发了一种具有自动调整未知参数能力的自校正时间序列模型。

Nonlinear non-Gaussian state space model based time series analysis method was investigated. In particular, Monte Carlo filter and smoother for the state estimation of the nonlinear non-Gaussian state space model was developed. As a result of this research, it became possible to model high-dimensional nonlinear non-Gaussian systems. We also developes a Monte Carlo posterior mean smoother in which almost all particles converge to the mean value of the marginal posterior distribution. These newly developed methods were successfully applied to the development of new statistical analysis method, such as the seasonal adjustment of economic time series, analysis of quasi-periodic motion of data obtained from a satelite and the analysis of a count data.Taking into account of the similarity between Monte Carlo filter and Genetic Algorithm, Higuchi developed a new algorithm for filtering nonlinear non-Gaussian state space model.During the course of this research, a new clew for further deveropment was obtained. In the state space modeling, we used to apply numerical optimization method to estimate the parameters of the model. In this method, evaluation of the likelihoods for many times was required and it sometimes make the estimation very difficult.In view of the fact that our Monte Carlo filter can be applied to high-dimensional nonlinear system, we augmented the state vector with the unknown parameter. With this modification of the state vector it now became possible to estimate the unknown state and the parameter of the model simultaneously. Based on this idea we were lead to develop a self-tuning time series model which has an ability to adjust its unknown parameters automatically.