# Analysis of ‘Pre-Fit’ Datasets of gLAB by Robust Statistical Techniques

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## Abstract

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## 1. Introduction

- The INPUT module: interfaces with the standard input files and the rest of the program. It implements all the reading capabilities and stores the data in the memory structures of gLAB, that contain, as minimum, the raw measurements and the pseudo-Keplerian elements that allow computing the GNSS satellite position and its clock offset.
- The PREPROCESSING module: checks and selects the read data to be further processed. It detects cycle-slips (i.e., discontinuities) in the carrier-phase measurements, decimates the input data to a lower processing rate (if required) and selects which satellites and which constellations are used in the following modules, among other functions.
- The MODELLING module: provides an accurate model of the measurements from the receiver to each tracked satellite. In this regard, the pseudorange measurements ${P}_{rec}^{sat}$ can be written according to [4] as$${P}_{rec}^{sat}={\rho}_{rec}^{sat}+c(\delta {t}_{rec}-\delta {t}^{sat})+Tro{p}_{rec}^{sat}+Io{n}_{rec}^{sat}+IF{B}^{sat}+{\epsilon}_{rec}^{sat}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}sat=1,\dots ,n$$$$prefi{t}_{rec}^{sat}={P}_{rec}^{sat}-Pmode{l}_{rec}^{sat}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}sat=1,\dots ,n$$Considering the model terms described in the standard point positioning (SPP) [7], the pre-fit residuals result in:$$prefi{t}_{rec}^{sat}={\rho}_{rec}^{sat}+c\delta {t}_{rec}+{\epsilon}_{rec}^{sat}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}sat=1,\dots ,n$$
- The FILTER module: implements an extended Kalman filter to obtain the estimations of the receiver PVT from the pre-fit residuals. The filter outputs the values of the estimated unknowns together with its co-variance, as a measure of the uncertainty in the estimation process.
- The OUTPUT module: is in charge of printing intermediate and final results in a structured manner. This allows extracting useful and complete information at any point in the GNSS data processing chain.

## 2. Structure of the Dataset and Preliminary Analysis

`LXS`function. For

`LXS`, we used a very conservative estimate, which was a Bonferroni-corrected [17] confidence level of $1-0.01/N\simeq 0.9999997$ where N is the number of observations—$32,000$ in the present case. The quantity $0.01/N\simeq 3\times {10}^{-7}$ can be considered as the probability to falsely declare a dataset as contaminated by outliers, when one is testing a large number of genuine datasets.

- 1
- The outlier detection by visual screening and LMS regression with Bonferroni correction give comparable results;
- 2
- Many robust methods (including those implemented in FSDA) are based on the assumption that the error component of the data analyzed is normally distributed. As the actual distribution can of course deviate from this common assumption, the confidence intervals are just ‘nominal’ in that the actual proportion of outliers detected can differ from the expected one.
- 3
- The outlier detection algorithm presently used by gLAB, based on an heuristic threshold of 40 m:
`epochs := set of considered epochs;``median := median of all pre-fit values in all epochs;``for each epoch in epochs``for each satellite s in epoch``if | prefit(s) - median | > treshold``then discard satellite s in current epoch.`

would have not detected the obvious outliers.

## 3. Analysis with Time Windows

## 4. Results, Discussion and Proposal for Further Work

#### 4.1. Positioning Results

#### 4.2. Discussion

#### 4.3. Further Work

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Raw actual pre-fit residuals recorded by station KIR0 on 4 November 2018. The pre-fit residuals for all satellites are depicted in blue color, while pre-fit residuals for GPS satellites 3 and 32 are depicted in red and black, respectively, to highlight some outliers.

**Figure 2.**Dataset from 4th November 2018 (day 203 of year 2018). A single application to all data of LMS, re-weighted LTS and FS produces the outliers in red. More precisely, the red ‘+’ symbols are detected by all methods, while LMS also detects the red ‘*’ associated to an absolute pre-fit of about 5 m. FS is applied in the standard form, calibrated to simultaneous $1\%$ confidence level while, to ensure same significance, we corrected the LMS and LTS with Bonferroni.

**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.

**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.

**Figure 6.**3D position errors of KIR0 using the SPP. In red, the original PVT obtained without detecting and removing outliers. In blue, the PVT achieved detecting and filtering the outliers with LMS.

**Figure 7.**3D position errors of HUEG using the SPP. In red, the original PVT obtained without detecting outliers. The PVT obtained after removing the outliers detected by LMS (blue), LTS (green) and FS (black).

**Figure 8.**Zoom of Figure 7 for positioning errors smaller than 10 m. 3D position errors of HUEG using the SPP. In red, the original PVT obtained without detecting outliers. The PVT obtained after removing the outliers detected by LMS (blue), LTS (green) and FS (black).

**Figure 9.**Zoom of Figure 8 for the time period comprised between 18 and 23 h. The top plot depicts the 3D position errors, whereas the bottom plot the GDOP.

**Figure 10.**The FS outlier detection mechanism. The top panel enlarges Figure 4 (third row, first column) in an interesting zone between hours 19 and 22, where three somewhat deviating satellites were detected as outliers. The zone refers to the window of 10 epochs starting at iteration 2470 of 2879, that is at time $20h34m$. The bottom-left panel shows the $n=90$ observations (9 satellites for 10 epochs) monitored in that zone; here, the x axis tick marks are unique observation identifiers (index-numbers) not necessarily ordered in time: they refer to the random order the satellites enter in the sliding window. The red crosses were detected as outliers by the FS, whose progress is shown on the right panel; the x axis ticks of the right panel refer in fact to the FS steps.

**Figure 11.**Monitoring the intercept estimates obtained with the FS on the basis of a moving time window of 10 epochs. Small deviances of the intercept estimates around zero indicate that the estimated pre-fit position remains stable. The time monitoring of the deviances (bottom panel) is complemented by views on the intercept fluctuations distribution (top left histogram) and normality departure (top right normal probability plot). The x axes of the upper panels refer again to the deviance, while the y axes refer, respectively, as it is customary, to the bin frequencies and the cumulative normal distribution function evaluated at the empirical deviance values.

**Table 1.**Excerpt from the pre-fit data table. Epoch in seconds, pre-fit and ranges in meters, elevation of the satellites in degrees.

Year | DoY | Epoch | Const. | Satellite | Pre-Fit | Measured Range | Modeled Range | Elevation |
---|---|---|---|---|---|---|---|---|

2018 | 203 | 30.00 | GPS | 30 | 1.1285 | 20,774,695.8820 | 20,774,694.7535 | 62.830 |

2018 | 203 | 30.00 | GPS | 7 | 0.5062 | 20,902,305.0420 | 20,902,304.5358 | 62.107 |

2018 | 203 | 30.00 | GPS | 5 | 0.6373 | 21,449,496.2580 | 21,449,495.6207 | 48.474 |

2018 | 203 | 30.00 | GPS | 13 | 0.0678 | 22,580,115.9500 | 22,580,115.8822 | 32.131 |

2018 | 203 | 30.00 | GPS | 27 | −0.7367 | 23,106,933.9460 | 23,106,934.6827 | 23.969 |

2018 | 203 | 30.00 | GPS | 21 | 1.5017 | 24,496,859.0780 | 24,496,857.5763 | 19.849 |

2018 | 203 | 30.00 | GPS | 9 | −0.6624 | 23,870,542.7020 | 23,870,543.3644 | 16.482 |

2018 | 203 | 30.00 | GPS | 8 | −0.3127 | 24,198,683.3000 | 24,198,683.6127 | 14.795 |

2018 | 203 | 30.00 | GPS | 28 | −1.0963 | 24,119,609.3400 | 24,119,610.4363 | 13.659 |

2018 | 203 | 30.00 | GPS | 16 | 0.3602 | 24,836,613.8360 | 24,836,613.4758 | 9.066 |

2018 | 203 | 77,520.00 | GPS | 9 | 0.6399 | 20,249,609.7060 | 20,249,609.0661 | 72.297 |

2018 | 203 | 77,520.00 | GPS | 23 | −0.2187 | 21,521,386.7380 | 21,521,386.9567 | 50.623 |

2018 | 203 | 77,520.00 | GPS | 2 | 2.2097 | 22,539,967.2700 | 22,539,965.0603 | 37.848 |

2018 | 203 | 77,520.00 | GPS | 26 | −0.6910 | 22,878,625.4080 | 22,878,626.0990 | 29.222 |

2018 | 203 | 77,520.00 | GPS | 16 | −0.4619 | 22,622,817.8300 | 22,622,818.2919 | 29.130 |

2018 | 203 | 77,520.00 | GPS | 6 | −0.2707 | 22,930,857.2880 | 22,930,857.5587 | 27.353 |

2018 | 203 | 77,520.00 | GPS | 7 | 0.1535 | 23,373,509.6060 | 23,373,509.4525 | 26.341 |

2018 | 203 | 77,520.00 | GPS | 29 | 0.0111 | 23,390,396.5840 | 23,390,396.5729 | 21.900 |

2018 | 203 | 77,520.00 | GPS | 5 | −0.8185 | 23,946,589.7820 | 23,946,590.6005 | 16.100 |

2018 | 203 | 77,520.00 | GPS | 3 | −16.2497 | 25,180,666.6940 | 25,180,682.9437 | 5.478 |

**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.

Estimator | Cumulative (s) | Median of Ten Epochs Windows (ms) |
---|---|---|

LMS | 54.16 | 18.65 |

LTS | 58.26 | 20.12 |

FS | 30.56 | 10.55 |

**Table 3.**Mean, median, 68th and 95th percentiles and RMS of the positioning errors obtained for the two permanent stations, KIR0 and HUEG, applying different outlier detection methods. Note that the ‘NONE’ method only excludes satellites with an elevation lower than 5 degrees.

Station | Outlier Detection Method | 3D Positioning Error (m) | ||||
---|---|---|---|---|---|---|

Mean | 50% | 68% | 95% | RMS | ||

KIR0 | NONE | 2.21 | 1.61 | 2.13 | 6.83 | 2.93 |

KIR0 | LMS | 1.55 | 1.44 | 1.76 | 2.90 | 1.70 |

HUEG | NONE | 5.23 | 2.57 | 3.05 | 6.00 | 15.21 |

HUEG | LMS | 2.23 | 2.15 | 2.67 | 3.84 | 2.41 |

HUEG | LTS | 2.45 | 2.39 | 2.84 | 3.96 | 2.60 |

HUEG | FS | 2.26 | 2.18 | 2.67 | 3.87 | 2.44 |

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**MDPI and ACS Style**

Alonso, M.T.; Ferigato, C.; Ibanez Segura, D.; Perrotta, D.; Rovira-Garcia, A.; Sordini, E.
Analysis of ‘Pre-Fit’ Datasets of gLAB by Robust Statistical Techniques. *Stats* **2021**, *4*, 400-418.
https://doi.org/10.3390/stats4020026

**AMA Style**

Alonso MT, Ferigato C, Ibanez Segura D, Perrotta D, Rovira-Garcia A, Sordini E.
Analysis of ‘Pre-Fit’ Datasets of gLAB by Robust Statistical Techniques. *Stats*. 2021; 4(2):400-418.
https://doi.org/10.3390/stats4020026

**Chicago/Turabian Style**

Alonso, Maria Teresa, Carlo Ferigato, Deimos Ibanez Segura, Domenico Perrotta, Adria Rovira-Garcia, and Emmanuele Sordini.
2021. "Analysis of ‘Pre-Fit’ Datasets of gLAB by Robust Statistical Techniques" *Stats* 4, no. 2: 400-418.
https://doi.org/10.3390/stats4020026