Self-tuning robust adjustment within multivariate regression time series models with vector-autoregressive random errors

Abstract

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).

The variance inflation factor to account for correlations in likelihood ratio tests: deformation analysis with terrestrial laser scanners

“…A second approach to estimate the AT is to model the time series: the AR process as introduced in Box et al (2016) is a prominent example of modelling, see Kargoll et al (2020) and the references inside for applications in geodesy. In that case, a value from a time series is regressed on previous values from that same time series, i.e.…”

The Generalized Method of Wavelet Moments with eXogenous inputs: a fast approach for the analysis of GNSS position time series

“…Other methods use the Markov chain Monte Carlo (Olivares and Teferle 2013 ) or the expectation-maximization (EM) algorithm (Kargoll et al. 2020 ).…”

“…The joint estimation of both deterministic and stochastic models is often based on the maximum likelihood estimator (MLE) and has been implemented in various software packages such as CATS (Williams 2008), Est_noise (Langbein 2008) and Hector (Bos et al 2008). Other methods use the Markov chain Monte Carlo (Olivares and Teferle 2013) or the expectationmaximization (EM) algorithm (Kargoll et al 2020).…”

The Generalized Method of Wavelet Moments with Exogenous Inputs: a Fast Approach for the Analysis of GNSS Position Time Series

“…The joint estimation of both deterministic and stochastic models is often based on the Maximum Likelihood Estimator (MLE) and has been implemented in various software packages such as CATS (Williams, 2008), Est noise (Langbein, 2008) and Hector (Bos et al, 2008). Other methods use the Markov chain Monte-Carlo (Olivares and Teferle, 2013) or the Expectation-Maximization (EM) algorithm (Kargoll et al, 2020).…”