【作者】 陈伟；

【导师】 尹晖； 花向红；

【作者基本信息】 武汉大学 ， 大地测量学与测量工程， 2013， 博士

【摘要】 变形是自然界普遍存在的现象,全世界每年因为变形灾害给人民和社会带来的危害和损失不计其数。因此,科学、准确、及时地对变形体进行变形监测,对变形情况进行分析与预报就显得尤为重要。时间序列分析方法是一种动态的数据处理方法,能够利用观测数据之间的自相关特性建立相应的参数模型,科学地分析与处理所观测到的数据,并对未来的变化趋势作出估计。考虑到时间序列分析方法中的AR模型相比其它模型的定阶和参数估计都要相对简单的多,而且理论上已证明了即便真实模型是ARMA模型或MA模型,也可以用高阶AR模型近似描述,所以本文选择对自回归AR模型估计理论进行系统地研究具有重要的理论意义和实用价值。本文较系统全面地研究了自回归模型的估计理论,包括一维AR(p)模型、多维AR(p)模型和模糊AR(p)模型,深入探讨了AR模型估计理论的一系列问题,包括数据的检验与预处理、模型的初步识别、定阶、参数估计、预测等问题,基于matlab编写了相应的建模和预测程序,结合实例探讨了自回归模型估计理论在变形监测数据处理中的具体应用。本文的主要研究内容和成果有：1、系统地研究了一维AR(p)模型的估计理论,详细地探讨了一维AR(p)模型估计理论从数据的检验与预处理、模型的初步识别、定阶、参数估计、模型的适用性检验到模型的预测等一系列问题。在参数估计方面,介绍了一维AR(p)模型参数估计的矩估计法和最小二乘法,在此基础上推导了一维AR(p)模型参数估计的岭估计法；考虑到最小二乘估计法在求解自回归模型参数时只考虑了向量Y的误差,而忽略了系数矩阵X的误差,而总体最小二乘法能同时考虑系数矩阵X和观测向量Y的偶然误差,对两者都进行改正。基于此思想,将总体最小二乘平差准则应用到AR(p)模型参数的解算中,并给出了具体的解算步骤。2、鉴于一维时间序列无法表示多因素间复杂的作用关系,而多维时间序列则能够描述多个因素间的作用关系,多维时序比一维时序包含更丰富的信息,更能反映数据的真实情况。因此,对多维时间序列进行研究意义重大。本文研究了多维AR(p)模型的平稳性条件,详细讨论了多维AR(p)模型参数估计的几种方法,如最小二乘估计法、Yule-Walker估计法、Levinson递推算法,在此基础上对传统的最小二乘估计法进行了改进,使之更易于编程实现；并将卡尔曼滤波算法应用到多维AR(p)模型的参数估计中；探讨了多维AR模型的几种定阶方法；在此基础上给出了多维AR(p)模型的最小方差预测和精度分析。3、考虑到测量数据的不确定性不仅具有随机性,也具有模糊性。对现实世界中这些不仅具有随机性,又具有模糊性的现象,本文提出可以尝试用模糊AR(p)模型进行预测。给出了模糊AR(p)模型的形式,提出了构造模糊数的方法；介绍了模糊AR(p)模型的参数估计的两种方法,线性规划法和模糊最小二乘估计法；给出了模糊AR(p)模型的F检验定阶法以及快速F检验定阶法,最后给出了模糊AR(p)模型的预测公式。4、基于matlab平台,编写了三种AR(p)模型的建模和预测程序,通过应用实例探讨了这三种模型在变形监测数据处理中的具体应用,对预测精度进行了分析比较,得出了一些有用的结论。更多还原

【Abstract】 Deformation is a common phenomenon in nature and every year the harm and loss caused by deformation disaster are countless in the whole world.Therefore, scientific, accurate and timely deformation monitoring and analysis and prediction of deformation are particularly important. Time series analysis method is a method of dynamic data processing. It can use parametric model which is established according to the auto-correlation of observed data to scientifically analyze and handle dynamic data and predict the future trend of data. Considering that compared with other models the order selection and parameter estimation of AR model among the time series analysis methods are relatively simple and it has been proved in theory that even if the true model is MA or ARMA sequences, can also be described as the high order AR model, systematically researching on the AR model has important theoretical significance and practical value. This paper systematically studies estimation theory of the stable autoregressive model including one-dimensional AR (P) model, multi-dimensional AR(P) model and the fuzzy AR (P) model.A series of problems of estimation theory of AR model are deeply discussed including inspection and preprocessing of data, preliminary identification of model,order selection of model,parameter estimation of model and prediction of model etc. Based on matlab the corresponding modeling and forecasting programs are written. Finally based on examples this paper discusses the application of autoregressive model estimation theory in the field of deformation monitoring data processing.Followings are the main research contents and achievements of this paper:1. This paper systematically studies estimation theory of the one-dimensional AR (p) model and discusses a series of problems of the one-dimensional AR (p) model in detail including inspection and preprocessing of data, preliminary identification of model,order selection of model,parameter estimation of model and prediction of model etc. In the parameter estimation of the one-dimensional AR (p) model,the moment estimation and the least squares estimation are introduced and the ridge estimation is derived.Least squares estimation only considered the error of vector Y and ignored the error of coefficient matrix X.But the total least squares method can consider both the accidental error of coefficient matrix X and observation vector Y, and carry on the correction at the same time.Based on this idea, the total least squares estimation criterion is applied to the parameter estimation of AR(p) model. The specific calculation steps are given in the paper. 2. Considering that one-dimensional time series cannot represent complex interaction among multiple factors and the multidimensional time series can describe the interaction among multiple factors and better reflect the real data, researching on multidimensional time series has great significance.The paper studies the stationary condition of multidimensional autoregressive model and discusses the methods of parameter estimation of VAR(p) model in detail such as the least squares estimation, Yule-Walker method and Levinson method etc.To make it more easily to program,the improved least squares estimation is proposed. And the caiman filter algorithm is applied to parameter estimation of multidimensional AR (P) model.Then the paper introduces several methods of order selection of multidimensional model and the minimum variance prediction and accuracy analysis are given.3. Since the uncertainty of the measurement data has not only the randomness, but also the fuzziness,the paper presents fuzzy AR(p) model to predict the fuzzy data.Fuzzy AR(p) model is given and the method of constructing fuzzy numbers is proposed.Two kinds of methods of parameter estimation of fuzzy AR(p) model such as linear programming method and fuzzy least squares estimation are introduced.The F test order selection method and rapid F test order selection method are given.Finally,the prediction formula of fuzzy AR(p) model is proposed.4. Based on matlab platform, modeling and forecasting process of the three AR (P) models are programmed. At last,by application examples, the specific application of three models in data processing of deformation monitoring are discussed and the prediction precision are analyzed and compared.Some useful conclusions are given.更多还原