Bayesian adaptive robust adjustment of multivariate geodetic measurement processeswith data gaps and nonstationary colored noise

现代大地测量传感器通常产生多个空间时间序列，其包含大量测量值、大量异常值以及数据间隙和随机误差，其特征在于相当大的自相关和互相关（即，色噪声）。鉴于这些不利因素，目前的大地测量数据分析工具无法解决的全部，我们打算开发经典和贝叶斯统计调整程序，允许一个强大的和有效的估计参数模型，从这样的时空测量系列。为了同时促进鲁棒性和统计以及计算效率，我们采用一方面的期望最大化（EM）的原则。这使得能够对数据间隙进行插补，并且同时对函数模型的参数、向量自回归移动平均（VARMA）有色噪声模型的系数以及潜在误差分布的形状参数进行自适应估计。后者是由一个多变量，缩放（学生）t-分布，并涉及到一个数据适应度的自由度和比例因子。通过估计这些量，形状，特别是概率密度函数的尾部特征适应于数据中存在的实际误差和离群值特征。在随后的工作步骤中，我们还将允许泛函和噪声模型的参数的动态变化。最后，我们调查贝叶斯过程的基础上平均场变分贝叶斯和马尔可夫链蒙特卡罗（MCMC）技术，允许纳入先验信息的功能模型的参数，VARMA模型和基本的t-分布到自适应鲁棒调整。由于调整产生关于所有未知模型参数的详细概率信息，因此我们还能够严格测试关于假设误差分布、关于可疑自相关/互相关模式以及关于此类模式的时间可变性的假设。我们适用于静态版本的一般观测模型和估计过程的调整问题的基础上，大地测量数据集来自静态多传感器系统的地理参考。它们的参考传感器可以是3D定位传感器，如GNSS设备或测速仪。将动态模型应用于某拱桥的荷载试验数据。由于预期的高水平的灵活性和效率的方法，我们希望他们也适用于其他类型的大地测量传感器数据，如在卫星大地测量。

Modern geodetic sensors often produce multiple spatial time series which contain huge numbers of measurements, numerous outliers as well as data gaps, and random errors that are characterized by considerable auto- andcross-correlations (i.e., colored noise). In view of these adversities, which cannot be resolved by current geodetic data analysis tools in their entirety, we intend to develop both classical and Bayesian statistics in connection with adjustment procedures that allow for a robust and efficient estimation of parametric models from such spatio-temporal measurement series. To facilitate simultaneous robustness and statistical as well as computational efficiency, we employ on the one hand the principle of expectation maximization (EM). This enables an imputation of the data gaps and concurrently an adaptive estimation of the parameters of the functional model, of the coefficients of a vector autoregressive moving-average (VARMA) colored noise model, and of the shape parameters of the underlying error distribution. The latter is defined by a multivariate, scaled (Student) t-distribution and involves a data-adaptable degree of freedom and scale factor. By estimating these quantities, the shape and in particular the tail characteristics of the probability densityfunction is adapted to the actual error and outlier characteristics present in the data. In a subsequent work step, we will also allow for dynamic changes of the parameters of the functional and of the noise model. Finally, we investigate Bayesian procedures based on Mean-Field Variational Bayes and Markov Chain Monte Carlo (MCMC) techniques, which allow for the incorporation of prior information regarding the parameters of the functional model, of the VARMA model and of the underlying t-distribution into the adaptive robust adjustment. Since the adjustment yields detailed probabilistic information regarding all of the unknown model parameters, we will for instance also be able to rigorously test hypotheses about the assumed error distribution, about suspected auto-/cross-correlation patterns, and about the time-variability of such patterns. We apply the static version of the general observation model and estimation procedure to adjustment problems based on geodetic data sets stemming from geo-referencing of static multi-sensor systems. Their referencing sensors can be 3D positioning sensors, like GNSS equipment or tacheometer. The dynamic version is applied to loading test data stemming from an arch bridge. Due to the anticipated high level of flexibility and efficiency of the methods, we expect them to be applicable also to other types of geodetic sensor data, as obtained e.g. in satellite geodesy.