状态空间模型的正规核方法

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 881-890.

状态空间模型的正规核方法

王超(),李波*(),王磊

^{华中师范大学数学与统计学院武汉 430071}

收稿日期:2021-01-19 出版日期:2022-06-26 发布日期:2022-05-09
通讯作者: 李波 E-mail:WangChaoAnHui@126.com;haoyoulibo@163.com
作者简介:王超, E-mail: WangChaoAnHui@126.com
基金资助:
国家自然科学基金(61877023);中央高校基本科研专项资金(CCNU19TD009);湖北省科技创新基地（平台）专项(2020DFH002)

Regular Kernel Method for State Space Model

Chao Wang(),Bo Li*(),Lei Wang

^{Department of Mathematics and Statistics, China Central Normal University, Wuhan 430071}

Received:2021-01-19 Online:2022-06-26 Published:2022-05-09
Contact: Bo Li E-mail:WangChaoAnHui@126.com;haoyoulibo@163.com
Supported by:
the NSFC(61877023);the Fundamental Research Funds for the Central Universities(CCNU19TD009);the Hubei Provincial Science and Technology Innovation Base (Platform) Special Project(2020DFH002)

摘要/Abstract

摘要：

状态空间模型（SSMs）为随机过程研究提供了一个通用框架，在被应用到诸如揭示单一经济现象背后潜藏的经济过程、手机信号识别、雷达屏幕中干扰源条件下的飞机位置检测等等具体实践中表现出了较高的应用价值.其中，马尔可夫状态空间模型在具备较高应用价值的同时，还具备良好的理论性质.再生核希尔伯特空间由于具备良好的学习性，能够适应更加广泛的规律拟合包括线性和非线性拟合，因此尝试从再生核空间中寻找一个能够拟合状态空间中状态转移规律的函数具备很好的研究意义.为了保证解的唯一性，正则化方法被添加到最优化问题中.其中，解的存在性和唯一性是探讨预测误差的前提条件，在此基础上，误差上界被给出.最后，通过机场能见度数据（全国研究生数学建模竞赛—中国学位研究生教育发展中心提供），将正规化核方法的状态空间模型和传统ARMA模型以及Kalman Filter（Kalman）模型进行进行对比，用以验证正规化核方法下状态空间模型的有效性.

关键词: 状态空间模型, 函数拟合, 核方法, 自回归模型

Abstract:

State space models(SSMs) provide a general framework for studying stochastic processes, which has been applied in revealing the true underlying economic processes of an economy, recognizing cellphone signals, detecting the loaction of an airplane on a radar screen, et al. In this paper, we study the Markov state space models by modeling the space transformation with reproducing kernel Hilbert space. Not only the existence and uniqueness of solutions are given, but also the error is estimate in L² spaces. We applied our method in Visibility prediction at airport in National Post-Graduate Mathematical Contest in Modeling supported by China Academic Degrees & Graduate Education Development Center.

Key words: State Space Model, Function Reconstruction, Kernel Method, Autoregression

中图分类号:

O212.5

王超,李波,王磊. 状态空间模型的正规核方法[J]. 数学物理学报, 2022, 42(3): 881-890.

Chao Wang,Bo Li,Lei Wang. Regular Kernel Method for State Space Model[J]. Acta mathematica scientia,Series A, 2022, 42(3): 881-890.

图/表 2

参考文献 12

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Method	MSE	SE	SS	MAPE
Kalman	0.00497	7.150	277.96	0.0257
ARMA($ p=1, q=1$)	0.00246	3.545	277.96	0.0127
KSSM($ \gamma=0.2$)	0.00065	0.933	277.96	0.0034

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