# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 3 Analysis with Time Windows
The three drops of LTS and LMS in computing time are coincident with consecutive epochs containing the lowest number of satellites (i.e., 7) included in the computation. The difference in performance between LMS and LTS on one side and FS in these specific intervals, with a time drop on one hand and almost no change on the other hand, is due to the different computational complexity of the methods, as explained below. Table 2 and Figure 5 show the respective computing times for the three methods on the HUEG dataset (24 h of measurements sampled every 30 s) and the same time window length (of 10 epochs). The clear advantage of the FS might be surprising, given that the method relies on a process that can iterate as many times as the size of the data sample to be analyzed, say \(n\). Several factors determine the advantage over LMS and LTS. The first one is that LMS and LTS need a minimum number of random samples of size \(\lceil n/2\rceil\) selected with computationally intensive'sampling without replacement' methods. This explains why Figure 5 shows a visible drop in computing corresponding to the three intervals where data come from only 7 satellites instead of 9-10 as in the majority of cases. \begin{table}
\begin{tabular}{c c c} \hline
**Estimator** & **Cumulative (s)** & **Median of Ten Epochs Windows (ms)** \\ \hline LMS & 54.16 & 18.65 \\ LTS & 58.26 & 20.12 \\ FS & 30.56 & 10.55 \\ \hline \end{tabular}
\end{table} 
Table 2: Computing times for the three estimators. The sliding window version of the MATLAB script is applied to the same dataset and with the same MATLAB version. Figure 4: Outlier detection against LMS regression (**top**), LTS (**middle**) and FS (**bottom**) with a sliding time window of 10 epochs. The left column depicts the pre-fit residuals (blue) with outliers as marked in red, whereas the right column depicts the evolution of mean (blue) and median (black) in each case. FS shows ‘more centered’ means and medians with respect to LMS and LTS. Instead, the FS is initialized with an LMS/LTS regression estimated on very few data points (those sufficient to fit the regression line, therefore two points as a minimum, in our case) and this is computationally trivial. Then, it progresses with a sequence of order statistics and OLS computations, which are also computationally simple. Finally, FS can start monitoring for potential signals only in the final part of the search, and this additionally reduces its computational burden. As a reference, on the same machine, the computation of the SPP by gLAB with only the default outliers (satellites with a pre-fit residual greater than forty meters and with elevation lower than five degrees) is done, resulting in a median over eleven trials of 1.078 s on the same dataset. In this setting, i.e., the MATLAB-interpreted version of FS, the time overhead for the FS computation in the case of a moving window results in 30.56 s. However, it is noted that although in its interpreted version (i.e., not compiled), FS processes 24 h of data in half a minute, which enables real-time operations.