# Self-tuning parameter estimation in linear models with autocorrelated errors and outliers

### What is it about?

To facilitate simultaneous robustness against outliers and statistical as well as computational efficiency in light of auto-correlatedness of measured geodetic time series, the optimization principle of expectation maximization (EM) is employed here. This enables a computationally convenient estimation of the parameters of a comprehensive data model, which includes a linear (functional) observation model, an autoregressive (AR) colored noise model, and a heavy-tailed error distribution in the form of a scaled (Student's) t-distribution. The latter involves a degree of freedom and a scale factor, which parameters can be estimated from the data and thus in a self-tuning manner. By estimating these quantities, the shape and in particular the tail characteristics of the probability density function are adapted to the actual error and outlier characteristics present in the data. The EM algorithm is shown to take the form of iteratively reweighted least squares. The good convergence and bias behaviour of this method are demonstrated by means of a Monte Carlo simulation. Furthermore, the algorithm is applied for the purpose of estimating amplitudes from accelerometer data within an oscillation experiment. It is shown that the oscillation model in terms of a sum of sinusoids, the colored noise model, and the shape parameters of the underying t-distribution adapt themselves well to a practically satisfactory solution.

### Why is it important?

Modern geodetic sensors often produce time series which contain huge numbers of measurements, numerous outliers, and random deviations that are characterized by considerable auto-correlations (i.e., colored noise). Applications that involve such sensors are found for instance in satellite geodesy (e.g., earth gravity field determination) or engineering geodesy (e.g., vibration monitoring), in which typically the desired target parameters must be estimated with a very high precision and insignificant bias. Since the aforementioned data characteristics are not taken into account by current geodetic data analysis methods in their entirety, they might not always be optimal in this regard. Consequently, it is important to develop estimators which take colored noise and outliers properly into account in order to obtain the most accurate results. Moreover, prior knowledge about the probabilistic or outlier characteristics is typically lacking, so that it is desirable to adapt the used error law (which can be expected to be heavy-tailed in the presence of outliers) to the measured data. The implementation of such a self-tuning method can help avoiding inadequate error laws and thus improving the accuracy of the estimates.

http://dx.doi.org/10.1007/s00190-017-1062-6

The following have contributed to this page: Prof. Dr.-Ing. Boris Kargoll