The iteratively reweighted least-squares approach to self-tuning robust adjustment of parameters in linear regression models with autoregressive (AR) and t-distributed random errors, previously established in Kargoll et al. (in J Geod 92(3):271–297, 2018. 10.1007/s00190-017-1062-6), is extended to multivariate approaches. Multivariate models are used to describe the behavior of multiple observables measured contemporaneously. The proposed approaches allow for the modeling of both auto- and cross-correlations through a vector-autoregressive (VAR) process, where the components of the white-noise input vector are modeled at every time instance either as stochastically independent t-distributed (herein called “stochastic model A”) or as multivariate t-distributed random variables (herein called “stochastic model B”). Both stochastic models are complementary in the sense that the former allows for group-specific degrees of freedom (df) of the t-distributions (thus, sensor-component-specific tail or outlier characteristics) but not for correlations within each white-noise vector, whereas the latter allows for such correlations but not for different dfs. Within the observation equations, nonlinear (differentiable) regression models are generally allowed for. Two different generalized expectation maximization (GEM) algorithms are derived to estimate the regression model parameters jointly with the VAR coefficients, the variance components (in case of stochastic model A) or the cofactor matrix (for stochastic model B), and the df(s). To enable the validation of the fitted VAR model and the selection of the best model order, the multivariate portmanteau test and Akaike’s information criterion are applied. The performance of the algorithms and of the white noise test is evaluated by means of Monte Carlo simulations. Furthermore, the suitability of one of the proposed models and the corresponding GEM algorithm is investigated within a case study involving the multivariate modeling and adjustment of time-series data at four GPS stations in the EUREF Permanent Network (EPN).

中文翻译：

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具有向量自回归随机误差的多元回归时间序列模型中的自调整稳健调整

在具有自回归 (AR) 和 t 分布随机误差的线性回归模型中，通过迭代重新加权最小二乘法对参数进行自调整稳健调整，之前在 Kargoll 等人中建立。（在 J Geod 92(3):271–297, 2018. 10.1007/s00190-017-1062-6），扩展到多元方法。多变量模型用于描述同时测量的多个可观察量的行为。所提出的方法允许通过向量自回归 (VAR) 过程对自相关和互相关进行建模，其中白噪声输入向量的分量在每个时间实例都被建模为随机独立的 t 分布（此处为称为“随机模型 A”）或作为多元 t 分布随机变量（此处称为“随机模型 B”）。两种随机模型是互补的，因为前者允许 t 分布的组特定自由度 (df)（因此，传感器组件特定的尾部或异常值特征）但不允许每个白噪声向量内的相关性，而后者允许这种相关性，但不允许不同的 dfs。在观察方程中，通常允许使用非线性（可微分）回归模型。导出了两种不同的广义期望最大化 (GEM) 算法来估计回归模型参数与 VAR 系数、方差分量（在随机模型 A 的情况下）或辅因子矩阵（对于随机模型 B）和 df(s ）。为了能够验证拟合的 VAR 模型并选择最佳模型顺序，应用多元混合检验和 Akaike 信息准则。算法和白噪声测试的性能通过蒙特卡罗模拟来评估。此外，在涉及 EUREF 永久网络 (EPN) 中四个 GPS 站的时间序列数据的多变量建模和调整的案例研究中，研究了其中一个建议模型和相应 GEM 算法的适用性。