测量数据的建模与半参数估计
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摘要
在测绘科学中,过去人们研究与讨论系统误差或粗差(称之为函数模型误差),主要局限于单个或几个观测量上,其原因在于采用常规测量仪器和测量方法进行各种静态测量时,常规测量受到外界环境变化的影响不大。通过重复性实验,干扰因素(习惯上称为系统误差或粗差)对测量结果的影响规律有比较明确的了解,观测值中的系统误差在平差前就能够得到较好的补偿。而且常规测量观测量较小,在观测过程中又有比较完善的观测程序和检核条件(观测量之间通常满足一定几何条件),粗差也比较容易发现和剔除,残余的系统误差与偶然误差相比达到了可以忽略不计的程度。即使有个别的系统误差或粗差存在,在参数回归数据处理中,有许多方法(如数据探测、稳健估计等)能够解决系统误差或粗差问题。随着现代测量仪器和测绘技术的发展,特别是空间技术的广泛应用,在较短时间内可以获得大量的观测数据,而且观测数据受外部环境的影响较大。由于影响测量结果的因素较多,其函数关系复杂而且对其认识较少,如果不考虑这种系统性的影响,近似的按参数回归建模,将会导致了参数模型与客观实际存在不可忽视的偏差,严重的影响估计量的结果。另外,从某种意义来说,经典参数数据处理方法没有从根本上解决模型误差问题,也没有从根本上解决好系统误差与粗差的区分问题,或者把两者混为一体,当平差函数模型包含有系统误差与粗差时,参数回归中处理模型误差的方法可能会不尽如意。因此,有必要提出一种新的数据处理理论与方法,来完善现有的测量数据处理理论。
综上所述,研究与解决模型误差,以及如何区分系统误差与粗差问题,是现代测量数据处理所研究重要内容之一。20世纪80年代发展起来的一种重要的统计模型—半参数回归模型(Semiparametric regression model),为我们研究上述问题提供了一种新的方法。
半参数回归模型可表达为:
L=BX+s(t)+△式中s(t)是描述未知函数关系的模型误差(这里指函数模型),它是某个量t的函数,也就是所谓的非参数。由于半参数模型引入了非参数,克服传统平差函数模型的局限性,使得数学模型与客观实际更为接近,如果采用某种估计方法,在数值上能够分别求出参数、非参数(系统性的模型误差)和偶然误差,该估计方法将是一种较为理想的数据处理方法,而且必将有着广阔的应用前景。基于上述原因,作者主要基于半参数模型估计的基本理论与方法,研究测量数据建模中如何处理模型误差,以及模型的诊断与粗差检验等问题,同时将半参数模型与测绘的实际结合起来,来解决测绘数据处理的实际问题。
本论文分为七章,第一章绪论主要阐述了在模型误差与半参数模型估计方面的研究现状,以及本论文研究的内容与意义。其它章节研究的主要内容如下:
在第二章里,论述了线性参数模型数据处理的基本理论与分析方法。包括参数估计、参数显著性检验、模型诊断与模型误差假设检验等问题。介绍了残差分析与检验的基本方法,提出了需要解决的问题:1、当平差模型存在明显的系统误差时应如何处理?2、当系统误差与粗差同时存在时应如何探测粗差、估计系统误差?
In the science of surveying and mapping, people researched and discussed the systematic errors or gross errors on one or a few observations in the past, the reasons are that static ground observations are less impacted by outside surrounding environment factors as the static surveying to be accomplished with conventional instruments and methods. Through duplicate experiments impacting factors (usually called systematic errors or outliers) that influence result of observation are much known, and systematic errors can be fairly compensated or be expressed in parameter model before adjustment. Furthermore observation quantity is less, and there are fairly perfect observation procedures and checking conditions (such as definite geometric conditions), so outliers are easily detected or cleared up. It can reach up to be ignored comparing remainder system errors with occasional errors. Though there is individual error, many methods (data snooping, robust estimate etc.) can deal with it. along with development of surveying instruments and technologies, especially extensive application of space technologies, a great deal of surveying data can be gained in the short time, moreover, data is seriously influenced by outside environment, and factors impacting observation results is too much , and its function relation is complicated and unknown. If not considering this the parametric model is approximately built, and it can bring forward not ignored deviation between parametric model and real model, and seriously influenced estimates. In a sense, the classical parametric regression model does not solve essentially model error, and also can not fundamentally distinguish system errors with gross errors, or mix up both. When the adjustment model include systematic errors and outliers, the method dealing with model errors is not quite satisfactory , so it is necessary that the new data processing methods to improve present theories of surveying data processing.From what has been discussed above, researching and resolving model errors, or differentiating systematic error with gross is one of the main contents to be researched nowadays.The eighth decade of the 20~(th) century, a important statistical model ——the semiparametricregression model has been developed, it is a fire-new method for us to research the above problems.The semiparametric model is as follows:L=BX+s(t)+△Where s(t) is the model errors (here called function model errors) describing unknown function relation, and it is the function of variablet, and is called the non-parameter. It can overcome the demerits of conventional adjustment model and get mathematics model close to reality since the non-parametric weight be brought in the semiparametric model. The parameter, non-parameter and accidental error can be got if an estimation method is used, and it will be an ideal method, moreover has very good foreground of application. Duo to these reasons above the semiparametric model is given abroad attentions, and the writer deal with model errors mainly in surveying data modeling with its theories and methods, and research model diagnose and gross test etc. At the same time, surveying data processing be combined with surveying practice to solve practical questions.The paper is divided into seven chapters. In chapter one, introduction expatiate the present state
of research on the model errors and the semiparametric model estimation, and clarify content and signification. The main contents are as follows on the other chapters:In chapter 2, it clarifies the basic theories and methods on the data processing of linear parameter model, Including parameter estimation, test of parameter significance, model diagnose and outlier test. It give methods of residual analysis and test, and bring forward questions to be solved: 1 .. how do you deal with it when there is systematical errors in adjustment model ? 2, how do you snoop outliers and estimate systematical errors when both is at one time?In chapter 3, especially researching estimation methods of the semi-parametric mode based on regular matrix; detailedly deriving the penalized least square formulas of the nonparametric model with natural spline, and giving the penalized least square formulas of the semiparametric model with cubic natural spline; in addition, introducing partial kernel smoothing and partial residual estimate. Giving penalized least square formulas of the semi-parametric model with vector measurements. The choices of regular matrix and smoothing factor are systematically discussed, finally giving some statistical properties for the estimates of the semiparametric model.In chapter 4, based on the diagnose means of parametric model, testifying the estimates are identical using forecasted and deleted models to estimate parameters in the semi-parametric model. Some statistical diagnose are derived and testing methods on gross errors are given. Discussing method on the test of non-parameter significance and giving its testing formulas, and finally it give brief processes and steps in the modeling of generalized linear models.In chapter 5, by analysis and comparison of several adjustment models (such as rank-deficient free network, least square collocation, ridge estimate, semi-parametric model). Putting forward the unified principle of their solutions to be derived and the unified formula of solutions on ill-posed problems. The characters of solution on ridge estimate of ill-posed problems are systematically expounded. Along with basic principle of ridge estimate of parameter, the rule of universal penalized least squares is represented based on penalized least squares of the semi-parametric model, and its formula of estimate are derived. In addition, summing up the basic theories and methods of M-estimation, and the principle of parametric robust estimate is applied on estimate of the semi-parametric model, and robust estimate of the semi-parametric model is put forward, some expressions of estimations are got for robust penalized least squares and robust universal penalized least squares. Finally simulation examples are used to illustrate the effect of related estimation methods.In chapter 6, it is combined with some examples in survey, and researched on the applications of the semi-parametric model on surveying data processing. It verifies the practicality and effect of semiparametric model with computing and analyzing of some examples.In the end, the main research work and innovation in the paper are summarized, and some problems to be further studied in the future are bring forward.
综上所述,研究与解决模型误差,以及如何区分系统误差与粗差问题,是现代测量数据处理所研究重要内容之一。20世纪80年代发展起来的一种重要的统计模型—半参数回归模型(Semiparametric regression model),为我们研究上述问题提供了一种新的方法。
半参数回归模型可表达为:
L=BX+s(t)+△式中s(t)是描述未知函数关系的模型误差(这里指函数模型),它是某个量t的函数,也就是所谓的非参数。由于半参数模型引入了非参数,克服传统平差函数模型的局限性,使得数学模型与客观实际更为接近,如果采用某种估计方法,在数值上能够分别求出参数、非参数(系统性的模型误差)和偶然误差,该估计方法将是一种较为理想的数据处理方法,而且必将有着广阔的应用前景。基于上述原因,作者主要基于半参数模型估计的基本理论与方法,研究测量数据建模中如何处理模型误差,以及模型的诊断与粗差检验等问题,同时将半参数模型与测绘的实际结合起来,来解决测绘数据处理的实际问题。
本论文分为七章,第一章绪论主要阐述了在模型误差与半参数模型估计方面的研究现状,以及本论文研究的内容与意义。其它章节研究的主要内容如下:
在第二章里,论述了线性参数模型数据处理的基本理论与分析方法。包括参数估计、参数显著性检验、模型诊断与模型误差假设检验等问题。介绍了残差分析与检验的基本方法,提出了需要解决的问题:1、当平差模型存在明显的系统误差时应如何处理?2、当系统误差与粗差同时存在时应如何探测粗差、估计系统误差?
In the science of surveying and mapping, people researched and discussed the systematic errors or gross errors on one or a few observations in the past, the reasons are that static ground observations are less impacted by outside surrounding environment factors as the static surveying to be accomplished with conventional instruments and methods. Through duplicate experiments impacting factors (usually called systematic errors or outliers) that influence result of observation are much known, and systematic errors can be fairly compensated or be expressed in parameter model before adjustment. Furthermore observation quantity is less, and there are fairly perfect observation procedures and checking conditions (such as definite geometric conditions), so outliers are easily detected or cleared up. It can reach up to be ignored comparing remainder system errors with occasional errors. Though there is individual error, many methods (data snooping, robust estimate etc.) can deal with it. along with development of surveying instruments and technologies, especially extensive application of space technologies, a great deal of surveying data can be gained in the short time, moreover, data is seriously influenced by outside environment, and factors impacting observation results is too much , and its function relation is complicated and unknown. If not considering this the parametric model is approximately built, and it can bring forward not ignored deviation between parametric model and real model, and seriously influenced estimates. In a sense, the classical parametric regression model does not solve essentially model error, and also can not fundamentally distinguish system errors with gross errors, or mix up both. When the adjustment model include systematic errors and outliers, the method dealing with model errors is not quite satisfactory , so it is necessary that the new data processing methods to improve present theories of surveying data processing.From what has been discussed above, researching and resolving model errors, or differentiating systematic error with gross is one of the main contents to be researched nowadays.The eighth decade of the 20~(th) century, a important statistical model ——the semiparametricregression model has been developed, it is a fire-new method for us to research the above problems.The semiparametric model is as follows:L=BX+s(t)+△Where s(t) is the model errors (here called function model errors) describing unknown function relation, and it is the function of variablet, and is called the non-parameter. It can overcome the demerits of conventional adjustment model and get mathematics model close to reality since the non-parametric weight be brought in the semiparametric model. The parameter, non-parameter and accidental error can be got if an estimation method is used, and it will be an ideal method, moreover has very good foreground of application. Duo to these reasons above the semiparametric model is given abroad attentions, and the writer deal with model errors mainly in surveying data modeling with its theories and methods, and research model diagnose and gross test etc. At the same time, surveying data processing be combined with surveying practice to solve practical questions.The paper is divided into seven chapters. In chapter one, introduction expatiate the present state
of research on the model errors and the semiparametric model estimation, and clarify content and signification. The main contents are as follows on the other chapters:In chapter 2, it clarifies the basic theories and methods on the data processing of linear parameter model, Including parameter estimation, test of parameter significance, model diagnose and outlier test. It give methods of residual analysis and test, and bring forward questions to be solved: 1 .. how do you deal with it when there is systematical errors in adjustment model ? 2, how do you snoop outliers and estimate systematical errors when both is at one time?In chapter 3, especially researching estimation methods of the semi-parametric mode based on regular matrix; detailedly deriving the penalized least square formulas of the nonparametric model with natural spline, and giving the penalized least square formulas of the semiparametric model with cubic natural spline; in addition, introducing partial kernel smoothing and partial residual estimate. Giving penalized least square formulas of the semi-parametric model with vector measurements. The choices of regular matrix and smoothing factor are systematically discussed, finally giving some statistical properties for the estimates of the semiparametric model.In chapter 4, based on the diagnose means of parametric model, testifying the estimates are identical using forecasted and deleted models to estimate parameters in the semi-parametric model. Some statistical diagnose are derived and testing methods on gross errors are given. Discussing method on the test of non-parameter significance and giving its testing formulas, and finally it give brief processes and steps in the modeling of generalized linear models.In chapter 5, by analysis and comparison of several adjustment models (such as rank-deficient free network, least square collocation, ridge estimate, semi-parametric model). Putting forward the unified principle of their solutions to be derived and the unified formula of solutions on ill-posed problems. The characters of solution on ridge estimate of ill-posed problems are systematically expounded. Along with basic principle of ridge estimate of parameter, the rule of universal penalized least squares is represented based on penalized least squares of the semi-parametric model, and its formula of estimate are derived. In addition, summing up the basic theories and methods of M-estimation, and the principle of parametric robust estimate is applied on estimate of the semi-parametric model, and robust estimate of the semi-parametric model is put forward, some expressions of estimations are got for robust penalized least squares and robust universal penalized least squares. Finally simulation examples are used to illustrate the effect of related estimation methods.In chapter 6, it is combined with some examples in survey, and researched on the applications of the semi-parametric model on surveying data processing. It verifies the practicality and effect of semiparametric model with computing and analyzing of some examples.In the end, the main research work and innovation in the paper are summarized, and some problems to be further studied in the future are bring forward.
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