卡尔曼滤波在时间序列分析中的应用.pdf

论文摘要 卡尔曼滤波理论是经典的最优滤波理论之一，在工程控制、信号分析等方面 有着广泛的应用。近些年来，把卡尔曼滤波的方法应用到经济金融模型中进行 分析也有了极大的发展。本文系统的讨论了卡尔曼滤波的方法及其在时间序列 状态空间模型中的应用。 首先对卡尔曼滤波、卡尔曼平滑、参数估计的EM方法及其拓展进行了系统 的分析与概括。接着，对卡尔曼滤波的进一步发展，如扩展卡尔曼滤波、无迹 卡尔曼滤波和粒子滤波做了讨论，对扩展卡尔曼滤波给出了噪声统计自适应的 改进方法；通过详细的理论阐述，对卡尔曼滤波的各项技术做了比较全面的归 纳。 其次，对时间序列的状态空间表达式做了介绍，对结构时间序列的趋势、季 节、循环分量的建模、判别和分离方法作了详细的剖析；对ARMA、ARIMA和SARIMA 给出直接的状态空间表达式；并对非高斯非线性时间序列模型做了简要的分析。 在此基础之上，把卡尔曼滤波和平滑的方法用于实证分析；通过对GDP数据 的预测，表明状态空间模型合并卡尔曼滤波算法更加准确；接着适当选取各成 分模型建立结构时间序列的状态空间模型，通过EM算法和卡尔曼平滑迭代方程 用极大似然法估计参数，并把分离出的成分和传统的季节调整和滤波方法进行 了比较，总结了各自的优缺点。用卡尔曼滤波对 GARCH 的状态空间形式进行迭 代，检验了结果。 关键词：卡尔曼滤波；状态空间；ARIMA；结构时间序列；实证分析     Abstract Kalman Filtering theory is a classical optimal filtering theory which is widely used in engineering control, signal analysis and many other areas. In recent years, the analysis of economics and finance models with Kalman Filtering theory has become popular. This paper gives some analysis and discuss the fundamental theory of the Kalman filtering and its applications in time series analysis area. First, the fundamental theory of Kalman filter and its extension, EKF, UKF，Particle filtering and modified EKF method has been shown. And also the parameter estimation method of a state space has been discussed. Second, for applying filtering technique in time series analysis, this paper has introduced the state space expressions of the ARMA type models and the structural time series models. And we summarize the trend, seasonal and cycle component modeling expression, give two examples of non-Gaussian non-linear models. At last, this paper applies the state space model and Kalman filtering in the model of GDP data, and we verify that the Kalman filtering method is exacter in forecasting, and uses the Kalman Smoothing method to estimate the trend, cycle and seasonal components in the structural time m