基于极大Lq似然估计的P范分布混合模型的估计理论研究

With the development of science and technology, the new technology can make us to get many data quickly and efficiently, but the observed data may be affected by different kinds of error source, what we want to do is to separate these errors, and get the high precision data.In recent years, scholars in statistics field proposed the P-norm distribution mixture model, this mixture model provide a good idea to solve the problems we described above. We think this P-norm distribution mixture model have some advantages as the traditional models. In this project, we introduced the Maximum Lq estimation method to improve the accuracy of the data processing. Then we use the maximum Lq estimation method to solve the systematic errors, gross error and random components in P-norm distribution mixture model. We try to use kernel function, maximum Lq estimation method, Local likelihood estimation and EM algorithm into earth geosciences data processing. Through the above steps, we also hope to establish a reasonable and comprehensive non-normality of observational data maximum Lq estimation theory; use the Penalized likelihood distance and local influence analysis to realize the excavation and extraction of each component of the mixture model; establish gross error test statistics, reliability measurement, test of hypothesis statistic formulate and so on. After we establish the theory of P-norm distribution mixture model, in order to perfect the model and obtain the more precise data, we want to select the optimal algorithm, and separate the parametric and nonparametric components in P-norm distribution mixture model. Through the practical application, we can compare it with the traditional methods, and get the advantages and disadvantages of themselves. We hope the model we will establish is more scientific, more reliable when we use it to process the original observational data, and to improve the existing data processing theory, get a higher precise data. Nowadays, more and more measurement problems can be summarized as non-normal observed data problems, in this project, our goal is to get the optimal parametric solution, and carry out the theory and applied research of P-norm distribution mixture model, which is based on the maximum Lq estimation method. We apply the new theories and methods to the measured data processing. We think, the research will be to promote the data processing theory and analysis of the theory in geophysics, geological disasters, terrestrial magnetism, geodesy and many other fields of earth sciences, it can also rich and perfect the need of earth sciences.

针对包含系统误差和粗差的非正态观测数据构成的平差问题，利用核函数、极大似然Q估计等扩展极大似然平差方法，然后利用局部似然估计，进行系统误差的质量评定，深入研究P范分布混合模型的关键理论和方法,建立一般误差分布模式下能够识别并分离出各种影响成分的P范分布混合模型的参数估计、算法设计、适应范围、质量评定标准等。重点研究：建立非正态观测数据的P范分布混合模型的极大似然Q估计理论；利用局部似然估计方法分离系统误差和粗差；设计最优算法，增强算法的稳定性，并给出易于程序化的公式和算法；将建立的新理论应用到变形监测和航空重力测量等地球科学数据处理理论中，开展对比分析，验证新理论和新方法的可靠性与正确性。 可以概括为非正态观测数据的测量问题越来越多，本项目以得到参数最优解为目标，开展基于极大似然Q估计的P范分布混合模型的理论和应用研究，其研究成果对提高测绘数据处理水平具有重要的理论意义和应用价值。

对于包含系统误差和粗差的非正态分布观测数据，引入核函数，利用极大似然Q估计方法和矩估计方法，详细推导了相关参数估计量的计算公式，给出了相应的解算步骤，并将研究成果应用到GPS测量、BDS钟差预测、变形监测、航空重力测量、航空摄影测量和海洋测深等实测数据的数据处理过程中，建立了相关理论、推导了相应公式、给出了参数的解算步骤，得到了一些有用的结果。主要成果如下：.构造补偿最小二乘函数，利用泰勒展式，导出了观测数据中剔除第 个观测数据前后参数分量和非参数分量相应估计值之间的关系式，给出了混合Cook距离参数Q和C的一些常用形式，通过合理选择相应参数，实现了粗差的定位，并将系统误差和粗差从观测数据中区别开来。.从观测值数据误差的真实情况出发，引入了二/四阶矩估计法、对数期望矩估计法、对数矩估计法，推导了基于矩估计法的P范分布参数解算公式。.结合已有矩估计方法，提出了一元P范分布的实数阶矩估计方法，推导出矩阶数r与形状参数p的关系表达式，填补了一元P范分布参数估计此前只能使用整数阶矩估计理论的空白，为P范分布参数矩估计理论的进一步扩充提供了新的思路。利用模拟数据对理论的正确性进行了验证，同时辅以一组GIS地图标准化误差实测数据和一组GPS观测值残差实测数据对理论的可用性与实用性进一步加以说明。通过P范分布矩估计理论在GPS伪距单点定位中的应用，说明了P范分布参数估计理论能够解决测量实际问题。.将传统的时间序列模型观测误差推广至服从P范分布的情况，将连分式迭代应用于P范分布参数估计中，针对AR模型的特点得出适用于AR模型的总体最小p范估计公式。并通过实际工程算例来证明该估计方法充分考虑了误差的分布模式，可以建立更加符合实际情况的模型，进而提高了模型的拟合和预测精度。.将半参数模型引入到航空摄影测量中，建立了半参数光束法混合模型。研究了地形对航空重力向下延拓的影响，建立了基于半参数模型的航空重力向下延拓理论及半参数岭估计延拓模型.提出了基于灰关联度的数据筛选方法，建立了加权半参数灰色模型，研究了背景值参数的选取方法，建立了基于半参数核估计方法的灰色预测模型。建立了海洋测深数据粗差分离的半参数趋势面模型。