长度测量值数字特征的估计方法研究

发布时间：2018-01-31 19:00

本文关键词： 泰勒级数展开式测量偏差牛顿插值最小二乘法　出处：《长安大学》2017年硕士论文　论文类型：学位论文

【摘要】：长度测量值的数字特征分析是地理信息系统和制图操作中的一个重要问题。长度测量同时也可以看作是两个端点之间的距离,从而研究两个端点之间距离测量值的数字特征是当前地理测量学的重要问题之一。目前,关于测量值数字特征的估计方法通常是多次测量,以平均值近似期望,然后通过期望来估计方差;这种方法必须在样本量足够大时才有意义。然而,在实际应用中,样本量通常无法达到足够大,这就使得采用传统方法得到的观测值数字特征有很大的不确定性。基于此,从理论的角度深入研究样本量不大时长度测量值数字特征的估计方法显得尤为重要。为了克服现有估计方法的不足和文献中存在的问题,本文在已有理论和方法的基础上,依据数学理论和统计方法,通过严格的数学推导,给出了基于泰勒级数展开式的长度测量值偏差估计的数学表达式以及基于牛顿插值多项式的测量值数字特征的改进估计方法。研究了泰勒级数展开式在长度测量值偏差估计中的应用,推导出了基于高阶泰勒级数展开式的长度测量偏差的表达式和距离方程的非线性误差传播公式;同时,运用Monte Carlo生成随机样本模拟验证,结果表明了该方法在偏差估计中具有较高的计算精度和参考价值;偏差估计的精度与所测量的距离大小有关,泰勒级数展开式的展开阶数也会影响到偏差估计的精度。而对于较短距离的测量,基于泰勒级数展开式的估计方法无法达到预期的计算效果,表现出了不确定性。为了进一步研究超短距离测量值的数字特征,本文提出了基于牛顿插值多项式近似与最小二乘法优化相结合的数字特征估计方法,构造插值多项式,即通过已知观测值来构造数字特征的估计公式。该方法估计的数字特征的精度与构造的方差矩阵密切相关,且构造矩阵影响着数字特征估计是否可解,并通过选取较短距离的测量值数字特征的估计来验证该方法的有效性。
[Abstract]:The digital feature analysis of length measurement is an important problem in geographic information system and cartographic operation. Length measurement can also be regarded as the distance between two endpoints. Therefore, it is one of the most important problems in geodesy to study the digital feature of the distance measurement between two endpoints. At present, the estimation method of the digital feature of the measured value is usually measured many times, and the average value is approximate to the expectation. Then the variance is estimated by expectation. This method only makes sense when the sample size is large enough. However, in practical application, the sample size is usually not large enough. This makes the numerical features of the observed values obtained by the traditional method have a lot of uncertainty. In order to overcome the shortcomings of the existing estimation methods and the problems in the literature, it is particularly important to study the digital feature estimation method of the length measurement value when the sample size is small from the theoretical point of view. On the basis of the existing theories and methods, according to the mathematical theory and statistical methods, through strict mathematical derivation. The mathematical expression of the deviation estimation of the length measurement value based on the Taylor series expansion and the improved estimation method of the digital characteristic of the measurement value based on Newton interpolation polynomial are given. The Taylor series expansion in the length measurement is studied. Application of value deviation estimation. The expression of the length measurement deviation based on the higher-order Taylor series expansion and the nonlinear error propagation formula of the distance equation are derived. At the same time, Monte Carlo is used to generate random samples for simulation verification. The results show that the method has a high accuracy and reference value in deviation estimation. The accuracy of the deviation estimation is related to the distance measured, and the expansion order of the Taylor series expansion will also affect the accuracy of the deviation estimation. The estimation method based on Taylor series expansion can not achieve the expected calculation effect and shows uncertainty. In order to further study the digital characteristics of ultrashort measurements. In this paper, a digital feature estimation method based on Newton interpolation polynomial approximation and least square optimization is proposed to construct interpolation polynomial. The accuracy of the digital feature estimation is closely related to the variance matrix, and the construction matrix affects the solvability of the digital feature estimation. The validity of the proposed method is verified by the estimation of the digital features of the measured values over a short distance.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P208

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