# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 3 Analysis with Time Windows  
In this section, we further deepened our investigation by applying the previous methods to a sliding time window of ten epochs. The dataset used in this case belongs to the permanent station named HUEG, located in Germany, for DoY 060 of 2019 (i.e., March 1 st). Although the receiver is a Javad 'TRE_G3 TH', the same model as the one used in the preliminary analysis of Section 2 above, the clock of the receiver (i.e., \(\delta t_{rec}\) in Equation (4)) is not steered to GPS time, presenting a rapid drift as illustrated in the left plot of Figure 3. According to the interface control document (ICD) [19], the receiver clock offset can have any value within a \(\pm 0.5\,\mathrm{ms}\) (i.e., \(\pm 150\) km). When it reaches the minimum (or maximum) limit, it must reset to remain within the prescribed range of \(\pm 0.5\,\mathrm{ms}\) Therefore, in order to analyze the pre-fit residuals containing such clock behaviors, a _detrending_ procedure must be applied. Moreover, as a step towards real-time applications, in this section, we use a sliding window rather than analyzing the complete _time window_ of 24 h at once. Thus, in order to eliminate the clock drift prior to outlier detection, we performed a linear detrending to all pre-fit residuals. This detrending is obtained by subtracting the instantaneous median value at each epoch. The right plot of Figure 3 illustrates the result of such detrending. We can observe few outliers at some particular satellites that were hidden by the clock drift in the original 'raw' pre-fit residuals depicted in the left plot of Figure 3. While most of the outliers range within 100 m, GPS satellite 6 presents an outlier that reaches more than 400 m. This outlier is simultaneously observed in other permanent stations, thus it is attributable to some satellite anomaly. Indeed, the Notice Advisory to  
Figure 3: Actual pre-fit residuals recorded by station HUEG during 1 March 2019. The left plot depicts the raw pre-fit residuals, whereas the right plot depicts the pre-fit residuals linearly detrended. All satellites are depicted in blue color, while pre-fit residuals with outliers are depicted with colors. Navstar Users (NANU) number 2019029 [20] was issued, reporting that the GPS06 satellite was unusable that day. We now apply the outlier detection algorithms based on LMS, LTS and FS to the HUEG dataset. With the aim of developing an efficient mechanism for outlier detection of pre-fit values, we chose a relatively small time window of ten epochs. That is, the window considers five minutes of data for each computation, sliding along the whole 24 h of observations. This choice is arbitrary and is done only as a kind of compromise between the need for a decision about unreliable satellites within 'few' minutes and having a meaningful set of data. The outliers, computed by using data of the whole window, are assigned to the epoch 'entering' the window itself. Figure 4 shows the comprehensive results of outlier detection for the defined time window with LMS, LTS and FS and the evolution of mean and median in each case. The plots of mean and median monitoring do not show big differences. This is not surprising since the outliers are similar in the three cases. FS shows'more centered' means and medians, meaning a better selection of outliers, and possibly, a cleaner output. Moreover, the mean shows, in the three cases, a 'displacement' towards negative values; a comment on this will be given in Section 4.2 below. Figure 5 shows the respective computing times for LMS, LTS and FS for each time window lasting 10 epochs on a Quad Core Intel i7, 2.9 GHz machine. Table 2 reports the respective computing times--cumulative and median--for all 10 epochs windows, and shows a clear computational advantage in using FS in spite of its greater theoretical complexity over LMS and LTS. This computational advantage is due to the efficient implementation of the FS algorithm in FSDA.