最优滤波理论中几种模型问题的研究
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摘要
时间序列作为二十世纪近代统计学的一个分支现在已成为数学界、工程界和经济学界应用最多、最广的课题之一。它包含了丰富的数学内容，并且具有广泛的应用范围，已成为许多学科和工程的一个有力的研究工具。
目前，对时间序列分析的方法主要有以下三种：(1)由Box和Jenkins提出的Box-Jenkins递推预报方法；(2)由Brockwell和Dayis以Hilbert空间的基本理论和方法为基础提出的射影预报方法；(3)最优滤波理论。现代时间序列分析方法是一种新的时间序列分析方法，用它可以直接对时间序列进行研究，也可以和Wiener滤波方法、Kalman滤波方法相结合从而得到新的容易实现的Wiener估值器和Kalman估值器。但用现代时间序列分析方法所得到的Kalman估值器到目前只研究了四种模型，对四种模型分别得到了其相应的稳态的Kalman估值器(即其Kalman增益阵是非时变的)。
鉴于此，本文给出了四种新的模型。对四种新的模型研究方法都是先进行转化，通过转化使模型变成已知模型的一种特殊形式，然后研究模型的ARMA新息和白噪声估值器，然后把状态表示成观测白噪声、输入白噪声和观测的非递推表达式，(后三种模型还需再进行转换)得到非递推估值器，再利用递推射影公式，最后得到稳态的Kalman估值器。
这样采用现代时间序列分析方法对状态进行估计(预报器、平滑器、滤波器)时，我们除了以前的四种模型以外，又可以对四种新的模型进行估计，应用这几种新的模型的估计方法我们可以更好地解决实际中遇到的相关问题。文中并就第一种和第四种模型分别给出一个算例，来说明这几种模型估值算法的应用。
As a branch of pre-modern statistics in 20th century, time series nowadays has become one of the most used and the most widely used topics in math, engineering and economic fields. It contains rich content and has wide application domain, whereas, it has become a powerful research tool in many disciplines and engineering.
Now there're primarily three kinds of methods to analyze the time series. (1) The Box-Jenkins recursive forecasting method advanced by Box and Jenkins, (2) Projection forecasting method based on the fundamental theory and method of Hilbert space, which is raised by Rockwell and Davis. (3) The optimal filtering theory. Modern time series analysis method is a new method which the time series can be directly researched on by it and it can be connected with the Wiener filtering method and Kalman filtering method for obtaining a new and easily realized Wiener estimator and Kalman estimator. But up to now this kind of estimator is just studied for four models, respectively these models have the corresponding steady-state Kalman estimators. (i.e. Kalman gain matrix doesn't change with time.)
Therefore in this thesis it is given four new models. The studying method to the new models is all diverting firstly which makes the model become a particular form of the already-known models. After that ARMA innovation and white noise estimator of models is studied, the state is showed as non-recursive expression of observed white noise, entered white noise and observation( the last three models needs diverting again) for obtaining non-recursive estimator, then recursive projection formula is used to obtain steady-state Kalman estimator.
When adopting modern time series analysis method to estimate the state (smoother, filtering and forecasting), these new models can also be estimated besides the already-known four models. Applying the new method of estimating models, some relative problems encountered in reality can be better resolved. It is given in this thesis a example of calculation respectively for the first and the fourth model to demonstrate the application of the algorithm of the estimator of these kinds of models.
目前，对时间序列分析的方法主要有以下三种：(1)由Box和Jenkins提出的Box-Jenkins递推预报方法；(2)由Brockwell和Dayis以Hilbert空间的基本理论和方法为基础提出的射影预报方法；(3)最优滤波理论。现代时间序列分析方法是一种新的时间序列分析方法，用它可以直接对时间序列进行研究，也可以和Wiener滤波方法、Kalman滤波方法相结合从而得到新的容易实现的Wiener估值器和Kalman估值器。但用现代时间序列分析方法所得到的Kalman估值器到目前只研究了四种模型，对四种模型分别得到了其相应的稳态的Kalman估值器(即其Kalman增益阵是非时变的)。
鉴于此，本文给出了四种新的模型。对四种新的模型研究方法都是先进行转化，通过转化使模型变成已知模型的一种特殊形式，然后研究模型的ARMA新息和白噪声估值器，然后把状态表示成观测白噪声、输入白噪声和观测的非递推表达式，(后三种模型还需再进行转换)得到非递推估值器，再利用递推射影公式，最后得到稳态的Kalman估值器。
这样采用现代时间序列分析方法对状态进行估计(预报器、平滑器、滤波器)时，我们除了以前的四种模型以外，又可以对四种新的模型进行估计，应用这几种新的模型的估计方法我们可以更好地解决实际中遇到的相关问题。文中并就第一种和第四种模型分别给出一个算例，来说明这几种模型估值算法的应用。
As a branch of pre-modern statistics in 20th century, time series nowadays has become one of the most used and the most widely used topics in math, engineering and economic fields. It contains rich content and has wide application domain, whereas, it has become a powerful research tool in many disciplines and engineering.
Now there're primarily three kinds of methods to analyze the time series. (1) The Box-Jenkins recursive forecasting method advanced by Box and Jenkins, (2) Projection forecasting method based on the fundamental theory and method of Hilbert space, which is raised by Rockwell and Davis. (3) The optimal filtering theory. Modern time series analysis method is a new method which the time series can be directly researched on by it and it can be connected with the Wiener filtering method and Kalman filtering method for obtaining a new and easily realized Wiener estimator and Kalman estimator. But up to now this kind of estimator is just studied for four models, respectively these models have the corresponding steady-state Kalman estimators. (i.e. Kalman gain matrix doesn't change with time.)
Therefore in this thesis it is given four new models. The studying method to the new models is all diverting firstly which makes the model become a particular form of the already-known models. After that ARMA innovation and white noise estimator of models is studied, the state is showed as non-recursive expression of observed white noise, entered white noise and observation( the last three models needs diverting again) for obtaining non-recursive estimator, then recursive projection formula is used to obtain steady-state Kalman estimator.
When adopting modern time series analysis method to estimate the state (smoother, filtering and forecasting), these new models can also be estimated besides the already-known four models. Applying the new method of estimating models, some relative problems encountered in reality can be better resolved. It is given in this thesis a example of calculation respectively for the first and the fourth model to demonstrate the application of the algorithm of the estimator of these kinds of models.
引文
[1] 邓自立，最优滤波理论及其应用——现代时间序列分析方法．哈尔滨：哈尔工业大学出版社，2000
[2] 邓自立，最优估值理论及其应用——建模、滤波、信息融合估计．哈尔滨：哈尔滨工业大学出版社，2005
[3] 邓自立，郭一新．现代时间序列分析及其应用——建模、滤波、去卷、预报和控制．北京：知识出版社，1989
[4] 邓自立，卡尔曼滤波与维纳滤波——现代时间序列分析方法．哈尔滨：哈尔滨工业大学出版社，2001
[5] George E.P.Box,Gwilym M.Jenkins,George C.Reinsel. Time Series Analysis Forecasting And Control. Holden-Day, San Francisico, 1970
[6] 程云鹏．矩阵论．西安：西北工业大学出版社，1989
[7] 王耕禄，史荣昌．矩阵理论．北京：国防工业出版社，1988
[8] 安鸿志，陈敏．非线形时间序列分析．上海：上海科学技术出版社，1998
[9] 王燕．应用时间序列分析．北京：中国人民大学出版社，2000
[10] Reinsel G C. Elements of Multivariate Time Series Analysis. Springer-Verlag, New York, 1993
[11] Hamilton J D. Time Series Analysis. Princetion University Press, Princeton, New Jersey,1994
[12] Priestly M B. Spectral Analysis and Time Series. Academic Press,New York,1981
[13] Kailath T. Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, N. J.,1980
[14] 韩京清，何关钰，许可康．线性系统理论代数基础．沈阳：辽宁科学技术出版社，1985
[15] Shieh L S. and Tsay Y T. Transformations of Class of Multivariable Control Systems to Block Companion Forms. IEEE Trans. Automat.Control, 1982
[16] 邓自立．估计MA参数的多维Gevers—Wouters算法及其在构造ARMA新息模型中的应用．控制理论与应用，2001
[17] Darouach M, Zasadzinski M, Mehdi M. State Eatimation of Linear Multivariable Singular Systems. Int. J. Systems Sci., 1992
[18] Gevers M. and Wouters W R E. An Innovations Approach to Discrete-Time Stochastic Realization Problem Journal A, 1978
[19] Anderson B D O, Moore J B. Optimal Filtering. New Jersey:Pretie-Hall, 1979
[20] Lewis F L Optimal Estimation. New York:John Wiley & Sons, 1986
[21] Mendel J M. Optimal Seismic Deconvolution: An Estimation-Based Approach. New York: Academic Press,1983
[22] 王跃云，金钟骥，张钟俊．广义系统的反馈控制与级点配置方法．自动化学报．1988
[23] 邓自立，许燕．基于Kalman滤波的通用的和统一的白噪声估计方法．控制理论与应用，2004
[24] 邓自立．时变系统的统一和通用最优白噪声估值器．控制理论与应用．2003
[25] 邓自立，许燕．基于Kalman滤波的白噪声估值理论．自动化学报．2003
[26] Fahmy MM, and J Observers for Descriptor Systems. Int. J. Control, 1989
[27] Deng Zi-Li and Liu Yu-Mei. Descriptor Kalman Estimators. Int. J. Systems Sci., 1999
[28] Kailath T. Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980
[29] 邓自立，胡萍，吕国英．非递推最优状态估计的几种统一算法．1998中国控制会议论文集．长沙：国防大学出版社．1998
[30] Deng Zi-Li, Zhang Huan-Shui,Liu Shu-Jun, and Zhou Lu. Optimal and Self-tuning White Noise Estimators with Applications to Deconvolution and Filtering Problems.Automatica, 1996
[31] Anderson BDO. and Moor JB. Optimal Filtering. Prentice-Hall, Inc., Englewood Cliffs, New Jersey,1979
[32] Deng Zi-Li and Liu Yu-Mei. Descriptor Kalman Estimators. Int. J. Control, 1989
[33] 邓自立，刘玉梅．广义稳态Kalman估值器．自动化学报，1999
[34] 段广仁．线性系统理论．哈尔滨：哈尔滨工业大学出版社，1996
[35] 中国科学院数学研究所概率组．离散时间系统滤波的数学方法．北京：国防工业出版社．1975
[36] 邓自立，李国英，吴冬梅．一种新的稳态Kalman滤波器．1996年中国控制会议论文集．1996
[37] 邓自立，刘玉梅．一种统一的稳态Kalman估值器．信息与控制．1998
[38] 邓自立，刘玉梅．稳态Kalman滤波的一种统一格式．控制与决策．1999
[39] Tajima K. Estimation of Steady-State Kalman Filter Gain. IEEE Trans. Automatic Control. 1978
[40] Aoki M. Optimization of Stochastic Systems:Topics in Discrete-Time Systems.Academic Press, New York.1989
[41] 陈家鑫．时间序列分析基础．暨南大学出版社．1989
[42] 谢衷洁，时间序列分析．北京大学出版社．1990．
[43] 邓自立．自校正滤波理论及其应用——现代时间序列分析方法．哈尔滨：哈尔滨工业大学出版社，2003
[44] Kalman R E. A New Approach to Linear Filtering and Prediction Theory. Trans. Journal of Basic Eng., 1960
[45] 须田信吴等．自动控制中的矩阵理论。北京：科学出版社，1979
[46] Jianqing Fan, Qiwei Yao. Nonlinear Time Series: Nonparametric and Parametric Method. New York, 2006
[47] Peter J. Brockwell, Richard A. Davis. Time Series: Theory and Methods (Second Edition) Springer-Verlag, New York, 2001
[48] 李裕奇，李永红．随机过程．国防工业出版社．2005
[49] Mendel J M. White Noise Estimator for Seismic Data Processing in Oil Exploration. IEEE Trans. Automatic Control, 1977
[50] 李裕奇，赵联文．概率论与数理统计．国防工业出版社．2005
[51] 邓自立，张焕水．自校正Kalman滤波、预报、去卷、平滑新方法．1998中国控制与决策学术年会论文集．大连：大连海事大学出版社．1998
[2] 邓自立，最优估值理论及其应用——建模、滤波、信息融合估计．哈尔滨：哈尔滨工业大学出版社，2005
[3] 邓自立，郭一新．现代时间序列分析及其应用——建模、滤波、去卷、预报和控制．北京：知识出版社，1989
[4] 邓自立，卡尔曼滤波与维纳滤波——现代时间序列分析方法．哈尔滨：哈尔滨工业大学出版社，2001
[5] George E.P.Box,Gwilym M.Jenkins,George C.Reinsel. Time Series Analysis Forecasting And Control. Holden-Day, San Francisico, 1970
[6] 程云鹏．矩阵论．西安：西北工业大学出版社，1989
[7] 王耕禄，史荣昌．矩阵理论．北京：国防工业出版社，1988
[8] 安鸿志，陈敏．非线形时间序列分析．上海：上海科学技术出版社，1998
[9] 王燕．应用时间序列分析．北京：中国人民大学出版社，2000
[10] Reinsel G C. Elements of Multivariate Time Series Analysis. Springer-Verlag, New York, 1993
[11] Hamilton J D. Time Series Analysis. Princetion University Press, Princeton, New Jersey,1994
[12] Priestly M B. Spectral Analysis and Time Series. Academic Press,New York,1981
[13] Kailath T. Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, N. J.,1980
[14] 韩京清，何关钰，许可康．线性系统理论代数基础．沈阳：辽宁科学技术出版社，1985
[15] Shieh L S. and Tsay Y T. Transformations of Class of Multivariable Control Systems to Block Companion Forms. IEEE Trans. Automat.Control, 1982
[16] 邓自立．估计MA参数的多维Gevers—Wouters算法及其在构造ARMA新息模型中的应用．控制理论与应用，2001
[17] Darouach M, Zasadzinski M, Mehdi M. State Eatimation of Linear Multivariable Singular Systems. Int. J. Systems Sci., 1992
[18] Gevers M. and Wouters W R E. An Innovations Approach to Discrete-Time Stochastic Realization Problem Journal A, 1978
[19] Anderson B D O, Moore J B. Optimal Filtering. New Jersey:Pretie-Hall, 1979
[20] Lewis F L Optimal Estimation. New York:John Wiley & Sons, 1986
[21] Mendel J M. Optimal Seismic Deconvolution: An Estimation-Based Approach. New York: Academic Press,1983
[22] 王跃云，金钟骥，张钟俊．广义系统的反馈控制与级点配置方法．自动化学报．1988
[23] 邓自立，许燕．基于Kalman滤波的通用的和统一的白噪声估计方法．控制理论与应用，2004
[24] 邓自立．时变系统的统一和通用最优白噪声估值器．控制理论与应用．2003
[25] 邓自立，许燕．基于Kalman滤波的白噪声估值理论．自动化学报．2003
[26] Fahmy MM, and J Observers for Descriptor Systems. Int. J. Control, 1989
[27] Deng Zi-Li and Liu Yu-Mei. Descriptor Kalman Estimators. Int. J. Systems Sci., 1999
[28] Kailath T. Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980
[29] 邓自立，胡萍，吕国英．非递推最优状态估计的几种统一算法．1998中国控制会议论文集．长沙：国防大学出版社．1998
[30] Deng Zi-Li, Zhang Huan-Shui,Liu Shu-Jun, and Zhou Lu. Optimal and Self-tuning White Noise Estimators with Applications to Deconvolution and Filtering Problems.Automatica, 1996
[31] Anderson BDO. and Moor JB. Optimal Filtering. Prentice-Hall, Inc., Englewood Cliffs, New Jersey,1979
[32] Deng Zi-Li and Liu Yu-Mei. Descriptor Kalman Estimators. Int. J. Control, 1989
[33] 邓自立，刘玉梅．广义稳态Kalman估值器．自动化学报，1999
[34] 段广仁．线性系统理论．哈尔滨：哈尔滨工业大学出版社，1996
[35] 中国科学院数学研究所概率组．离散时间系统滤波的数学方法．北京：国防工业出版社．1975
[36] 邓自立，李国英，吴冬梅．一种新的稳态Kalman滤波器．1996年中国控制会议论文集．1996
[37] 邓自立，刘玉梅．一种统一的稳态Kalman估值器．信息与控制．1998
[38] 邓自立，刘玉梅．稳态Kalman滤波的一种统一格式．控制与决策．1999
[39] Tajima K. Estimation of Steady-State Kalman Filter Gain. IEEE Trans. Automatic Control. 1978
[40] Aoki M. Optimization of Stochastic Systems:Topics in Discrete-Time Systems.Academic Press, New York.1989
[41] 陈家鑫．时间序列分析基础．暨南大学出版社．1989
[42] 谢衷洁，时间序列分析．北京大学出版社．1990．
[43] 邓自立．自校正滤波理论及其应用——现代时间序列分析方法．哈尔滨：哈尔滨工业大学出版社，2003
[44] Kalman R E. A New Approach to Linear Filtering and Prediction Theory. Trans. Journal of Basic Eng., 1960
[45] 须田信吴等．自动控制中的矩阵理论。北京：科学出版社，1979
[46] Jianqing Fan, Qiwei Yao. Nonlinear Time Series: Nonparametric and Parametric Method. New York, 2006
[47] Peter J. Brockwell, Richard A. Davis. Time Series: Theory and Methods (Second Edition) Springer-Verlag, New York, 2001
[48] 李裕奇，李永红．随机过程．国防工业出版社．2005
[49] Mendel J M. White Noise Estimator for Seismic Data Processing in Oil Exploration. IEEE Trans. Automatic Control, 1977
[50] 李裕奇，赵联文．概率论与数理统计．国防工业出版社．2005
[51] 邓自立，张焕水．自校正Kalman滤波、预报、去卷、平滑新方法．1998中国控制与决策学术年会论文集．大连：大连海事大学出版社．1998