# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 1 Introduction
In the simplest form of position computation, as described in standard point positioning (SPP) [7], the pre-fit residuals of all satellites in view are fit by ordinary least squares, OLS, to three coordinates and one time-offset unknowns in an over-determined system of equations. The severity of outlier mis-detection depends on the magnitude of the outlier, on satellite geometry and on the user application. Safety of life (SoL) applications such as civil aviation or autonomous driving require specialized techniques to ensure positioning integrity (i.e., the measure of trust that can be placed in the correctness of the information supplied by a navigation system). The qualitative distinction of outliers from the majority of the sample is immediately remarked when values are graphically represented in an appropriate way. For example, with reference to Figure 1, the 'V-shaped' set of points in the lower-left part is promptly perceived as 'distinct from the majority of the sample'; the appropriate graphical representation of data remains today, when possible, an instrument for _outlier detection_ by visual screening. When visual screening is not possible, or a fine-drawn decision has to be taken, automated techniques must be used. [PERSON], in 1953, coined the term _robust_ in relation to this need, but this (certainly older) concept was formalized in a proper discipline only later, starting from the mid-1960s. Roughly speaking, _robust statistics_ can now be seen as an operational approach to the treatment of outliers; treatment that can range from the detection and elimination of outliers from the sample to some degree of mitigation of their effects on the choice and computation of estimators. The standard example of _resistance to outliers_ of an estimator is given by comparing the effect of a single diverging value \(y\rightarrow\infty\) on the sample mean and the sample median: the mean being immediately influenced by this value of \(y\), while the median is _resistant_. The _degree of resistance_ to the outliers of an estimator is represented by a parameter called _breakdown point_. The breakdown point of the sample mean is 0% while the breakdown point for the median is 50% since it can tolerate up to 50% of 'gross errors' in data before diverging. The use of robust statistics techniques for GNSS positioning is a growing field both in applications and in methodological aspects. Examples of the applications of robust statistics to positioning problems can be found in artificial intelligence [9] and geodesy [10]. Concerning the methodological aspects, an example of improvement of the SPP algorithm by using Huber M-estimators can be found in [11] while in [12] a form of robust Kalman filtering is used in the kinematic positioning component. These two methodological aspects can be referred to the MODELLING and FILTER modules of gLAB. To our knowledge,there are no other attempts in the literature to improve the position solution with robust statistics by exploiting the pre-fit residuals, i.e., by working at the connection between the MODELLING and the FILTER modules, as in the gLAB architecture. The Flexible Statistics and Data Analysis (FSDA) toolbox [13] is a MATLAB module gathering a comprehensive set of statistical functions for the analysis of data. The original aim of this library of functions was to extend the MATLAB Statistics Toolbox with _forward search_ analysis [14]. FSDA was subsequently extended and now includes all the main _robust_ multivariate and regression tools, complemented by instruments for robust transformations and the interactive graphical exploration of data; it is available for Windows, Unix and MacOS platforms. (_FSDA is copyright of the European Commission and the University of Parma. It is protected under European Union Public Licence (EUPL) [15]_, which is a free software licence granting recipients rights to modify and redistribute the code.