【作者】 胡宏昌；

【导师】 孙海燕；

【作者基本信息】 武汉大学 ， 大地测量学与测量工程， 2004， 博士

【摘要】 随着测绘科学技术的发展,测绘科学本身及其它相关科学都对现代测量数据处理提出了更高的要求,而现有的数据处理理论已经无法解决在测量实践中遇到的一些新问题,限制和束缚了测绘技术的发展与应用。因此必须进一步研究、改进数据处理理论,并提出和发展新的理论与方法。 在测量数据处理中,人们常常采用参数模型,是因为其结构简单、易于处理,而且在火多数情形下(如常规大地测量的各种静态问题),由于大部分系统误差可在数据处理前补偿、消除或在参数模型中表达,故所建立的参数模型与客观实际是比较一致的,能满足实际需要。但在有些情形下(如大地测量的一些动态问题),观测值中存在既不能消除又无法参数化的系统误差,从而导致了参数模型与客观实际存在不可忽视的偏差。 另一方面,系统误差总是作为有害成分设法予以消除或补偿,这并不一定是很科学的处理方法。实际上,系统误差中含有影响观测值的各种因素的信息,如能正确的识别、提取,则不仅能够提高参数估计精度,而且能为其它学科的研究提供资料。 另外,如果影响观测值的因素可分为两个部分:主要部分是线性关系,另一部分是某种干扰因素,它同观测量的关系是完全未知的,也没有理由将其归入误差项。此时,如用非参数模型(尽管它有较大的适应性)加以处理,则会失去太多的信息,如采用线性模型加以处理,则拟合效果很差。 鉴于以上问题,需要考虑其它的数据处理模型——半参数模型 l_i=A_i~TX+s(t_i)+△_i (i=1,2,…,n),它是八十年代发展起来的一种重要的统计模型。由于它既含有参数分量(描述了观测量中函数关系已知的成分),又含有非参数分量(专门表示函数关系未知的模型偏差),可以概括和描述众多实际问题,更接近于真实,因而引起广泛重视,其研究日益成熟。 一般来说,测量数据处理问题最终归结为参数或非参数估计问题。迄今为止,对半参数模型的研究已存在大量的估计方法,如:早期将非参数分量参数化的思想;两步估计,包括近邻估计、权估计、核估计、小波估计等;两阶段估计;抗差或稳健估计;补偿最小二乘估计法等等。但在数学等理论领域,其研究几乎是理论估计及其大样本性质,很难将它们转化为应用;而在测绘等应用领域,对半参数模型的研究大多数结果存在理论研究不透彻及方法单一等不足。笔者试图在二者之间建起一座桥梁,以便弥补二者的不足。 本论文将结合数学界的理论研究工作与测绘界的实际需要,系统地研究了半参数模型的各种估计方法(补偿最小二乘法、小波估计法、泛最小二乘法、累积法、稳健估计法、迭代法、两阶段估计法等等)及其在测量数据处理中的应用。具体地说,主要研究了如下内容: 在第二章里,阐明了半参数模型的补偿最小二乘估计方法,基于使最小二乘极值问题可以求解及对非参数估计曲线起平滑作用的原因,而提出的补偿最小二乘准则为 V~TPV+aS~TRS=min在该准则下,得到了参数、非参数分量的估计值及观测值改正值的表达式,并用三次样条函数插值法得到了非参数分量的推估表达式。研究了估计量的有偏性、分布、误差大小等统计特性。较为系统地讨论了平滑因子a及正规矩阵R的选取。通过模拟的算例及坐标变换、GPS定位、重力测量等实际应用,说明了该法的成功性及实用性。并从理论上,将流行的自然样条估计方法归结为补偿最小二乘方法,从而把前者作为后者的特例来研究。 在第三章里,以小波估计为例研究两步估计。两步估计的思想是:先基于假设参数已知,更多还原

【Abstract】 Along with scientific and technological development of surveying and mapping, science of surveying and mapping oneself and other related science put forward the higher request to the modern surveying data processing. However, the current theories of data processing can’t resolve the some new problems met in surveying practice, so the technical development and application of surveying and mapping are limited and tied up. Hence, the theories of data processing must be further studied and improved, and new theories and method are put forward and developed.In the course of surveying data processing, many researchers use the parametric model because its construction is simple and apt to be processed. Further more, under a majority of situations (for instance, kinds of static problems of conventional geodetic surveying) , it is in accordance with objective fact and can satisfy practical needs because a majority of system errors are be compensated and cleared up and can be expressed in the parameter model before data processing. However, under some situations (for instance, some dynamic issues of geodetic surveying) , since observed values include system errors which can’t be cleared up and parametrized, it brings on non-neglectable difference between the parametric model and objective practicality.On the other hand, it is not always very scientific that the system errors are always tried to eliminate or compensate as the harmful composition. In fact, the system errors contain lots information that influence observed values, if they can be identified and withdrawn rightly, not only the accuracy of parameter estimate can be increased, but also the data can be provided for the research of the other subjects.In addition, if factor of impacting observed values can be divided into two parts: main part is linear relation, another part is a certain interference factor, relation to observation values is complete unknown, it also fall under error item without any reason. Here, too many information will be lost if the non-parametric model (though it has bigger flexibility) is used, imitated result is bad if the linear model is adopted.Whereas the above problems, other data processing models need consider, that is the smeiparametric modelIt is a kind of important statistical model developed in 1980’s. Because it not only contain the parameter weight (described known composition of function relation in observation values), but also contain the non-parameter weight (exclusively show the model deviation unknown in function relation), can generalize and describe numerous actual problems, and it even near to true thing. As a result, the model is extensively thought, its research is increasingly mature.Generally speaking, the problem of surveying data processing is ultimately come down to the one of parametric or non-parametric estimate. Up to the present, the research of the smeiparametric model has existed a lot of estimate methods, such as earlier period parametrization thought, two stages estimate (including neighbor estimate, weighted estimate, kernel estimate, wavelet estimate etc.), two stages estimate, robust estimate, penalized least squares method etc.. However, in the mathematics etc. theoretical field, its research is almost theoretical estimates and big sample properties, thus it is very difficult to apply them. Moreover, at surveying and mapping etc. applied field, the majority of research results in semiparametric model exist some shortagessuch as no thorough theories research and single method etc.. The writer try to set up a bridge at the two field such that their shortages are offset.In the paper, combining the theoretical research work of mathematics field with practical requirement of surveying and mapping field, estimation methods of the semiparametric model are systematically investigated, including penalized least squares method, wavelet estimate method, universal least squares method, robust estimate method, iterative method, accumulation method, two stages estimate method etc. Their applications in the surveying data processing are studied. Say in a specific way, the main researched contents are as follows:In chapter 2, the penalized least squares method of the semiparametric model is clarified. In order to get only one minimal solution and smooth the curve of non-parametric estimate, the penalized least squares principle is put forward as followsUnder the principle, the parametric and nonparameteric estimators and correct values of observed values are got. Using cubic splines interpolation, the nonparameteric estimator is attained. Some statistical properties are studied. The choices of smooth factor a and regular matrix R are systematically discussed. By some simulating examples and actual applications (for example, coordinate systems transformation, GPS position, gravity measuring etc.), success and validity of the method are illuminated. The popular cubic nature splines method is regarded as one of penalized least squares method, thus the former is researched as special case of the latter.In chapter 3, wavelet estimation method is taken as an example of the two steps estimate method. The thought of two steps estimate is: First, the nonparametric estimator is defined according to the assumption parameter known; Then parametric estimator is attained by using least squares method, thereby the non-parametric estimator is attained. When random error sequences are martingale difference sequences, the semiparametric model is studied. Not only the parametric and nonparametric estimators are attained, but also their asymptotic normality and strong consistency and moment consistency are studied. And an application of wavelet analysis in gross errors processing of the surveying data is preliminarily studied.In chapter 4, the universal least squares method is proposed for the very first time, its standard is given byThe method is an expansion of some methods, such as penalized least squares method, rank-deficient (weighting) minimum norm, ridge estimation. The research of the universal least squares method is begun from a linear model. Combined penalized least squares method and two steps estimation, the universal least squares method of the semiparametric model is researched. At the same time, after the universal least squares method is compared with the penalized least squares method and two steps method and ridge estimate method, the method is better than past estimate methods under square mean meaning and some appropriate conditions. In addition, these methods and results are still verified and explained with some examples.In chapter 5, the writer use accumulation estimates of the parameter model for investigating the semiparametric model. Combining with the penalized least squares method and two steps estimate method, the pilot study is carried through in semiparametric model, and the parametric and non parametric estimators are attained. The method is showed by example.In chapter 6, the robust estimation of the semiparametric model is researched. Using equivalent weight principle, some expressions are got, such as robust two steps estimate, robust penalized least squares estimation, robust universal least squares estimation etc. Their influence functions are derived, which explain that two steps estimate and penalized least squares estimation and universal least squares estimation are not robustness. The robust penalized least squares estimation and the robust universal least squares estimation have robustness by imitated example. M-estimation based on two steps estimation is studied, the linear representation is attained.In chapter 7, the iterative method of semiparametric model is systematically studied, and iterative equations are constructed as follows:Under the certain conditions, the iterative equations converge to the parameters and non-parameters of the semiparametric model, rigorous and theoretical foundation of iterative method are provided. The difference estimate and its statistical characteristics are studied, so theoretical foundation is provided for the actual application. Two stages estimation of the semiparametric model is studied in brief. The thought of robust two stages estimation is putting forward, and the free level net is calculated by using it. The parametrization of the semiparametric model is described in brief.In addition, principal research work and creativeness of the paper are summarized; at the same time, some problems of deserving discussion are put forward for further study the semiparametric model.更多还原