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arxiv: 2605.21461 · v1 · pith:QPEZECZ5new · submitted 2026-05-20 · 💻 cs.LG

A Machine Learning Framework for Weighted Least Squares GNSS Positioning based on Activation Functions

Pin-Hsun Lee , Harry Leib This is my paper

Pith reviewed 2026-05-21 05:25 UTC · model grok-4.3

classification 💻 cs.LG
keywords GNSSpositioningweighted least squaresmachine learningactivation functionsurban environmentssignal qualityensemble learning
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The pith

Machine learning converts GNSS signal quality scores into weights that cut urban positioning errors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework that trains ensemble learning models on signal quality indicators from GNSS receivers to assign quality scores to satellite signals. These scores are then passed through activation functions to generate weights for a weighted least squares positioning solution. Experiments on real urban data from Hong Kong and Tokyo show lower positioning errors for both single and multiple satellite constellations. The approach also works when the model is trained in one city and tested in another with similar urban density. A sympathetic reader would care because accurate positioning matters for navigation in dense urban environments where traditional methods struggle with blocked and reflected signals.

Core claim

The central claim is that incorporating machine learning predicted quality scores transformed by activation functions into the weighted least squares algorithm substantially reduces positioning errors in urban GNSS scenarios and demonstrates strong geographical transferability between similar urban areas.

What carries the argument

An ensemble learning model that predicts signal quality scores from indicators, combined with activation functions such as the sigmoid to map those scores into weights for the WLS solver.

If this is right

  • The method yields lower positioning errors than standard WLS in both single-constellation and multi-constellation cases.
  • Sigmoid activation functions produce the largest improvements among those tested.
  • Performance remains high when the model trained on data from one urban region is applied to another.
  • Different machine learning algorithms can be used within the framework with consistent benefits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This weighting approach might extend to other positioning technologies like INS or visual odometry in challenging environments.
  • Future work could test the framework in rural or indoor settings to see if the transferability holds.
  • The activation function choice could be learned jointly with the model rather than selected separately.

Load-bearing premise

The assumption that signal quality indicators from real urban GNSS data can reliably predict which signals contribute most to positioning errors, allowing the activation functions to produce weights that genuinely improve the solution.

What would settle it

Collecting new urban GNSS data from a different city, training the model on one dataset, and checking whether the weighted least squares positioning error is lower than the unweighted version on the held-out data.

Figures

Figures reproduced from arXiv: 2605.21461 by Harry Leib, Pin-Hsun Lee.

Figure 1
Figure 1. Figure 1: Flowchart of the proposed algorithm. B. Pseudorange Model The pseudorange ρ [32] is modeled by: ρ model i,j = q (xi,j − xu) 2 + (yi,j − yu) 2 + (zi,j − zu) 2 + δ i ρ (1) where (x, y, z) is the 3-dimensional (3D) position in earth-centered earth-fixed (ECEF) coordinates, subscript u indicates user, subscripts (i, j) represent j-th satellite in i-th con￾stellation and δ i ρ is the geometric distance equivale… view at source ↗
Figure 2
Figure 2. Figure 2: Decision Tree. Adaptive boosting, abbreviated as AdaBoost, is a boosting algorithm developed by Freund and Schapire [44]. The algorithm trains individual weak learners using identical data sets but applies different sample weights. Consider training data S with N data points S = {(d1, l1),(d2, l2), ...,(dN , lN )} (22) where d denotes feature vector and l is the corresponding label. The first weak learner … view at source ↗
Figure 3
Figure 3. Figure 3: Hidden Layer. IV. Experimental Results This section presents results obtained when applying our framework to real life measure￾ments. A. Experimental Setup This study employs the UrbanNav open-source dataset, which provides GNSS observation data and ground truth information for datasets collected in Hong Kong and Tokyo urban areas [49]. The GNSS observation data are written in Receiver Independent Exchange… view at source ↗
Figure 4
Figure 4. Figure 4: Activation Functions. (RINEX) format. The ground truth data are written in txt files for Hong Kong data and csv files for Tokyo data. GPS L1 C/A and BeiDou B1I signals are used in this study. An elevation mask that excludes signals with elevation angle below 15◦ is applied to avoid low-quality signals. 1) Hong Kong Dataset: For Hong Kong dataset, the GNSS observation data were recorded using a low-cost u-b… view at source ↗
Figure 5
Figure 5. Figure 5: Ground truth of testing data (a) medium urban data fro [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Google Earth 3D view of the open sky ground truth traje [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Weight distribution of GPS data with activation func [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 3D RMSE of GPS-only positioning using Medium Urban da [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 3D errors of GPS-only positioning using Medium Urban [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 3D RMSE of GPS+BeiDou positioning using Medium Urba [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: 3D errors of GPS+Beidou positioning using Medium Ur [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Single and multi-constellation 3D positioning err [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: 3D RMSE of GPS+BeiDou positioning using Shinjuku da [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: 3D errors of GPS-only positioning using Shinjuku da [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: 3D RMSE of GPS+BeiDou positioning using Shinjuku da [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: 3D errors of GPS-only positioning using Shinjuku da [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
read the original abstract

Global Navigation Satellite Systems (GNSS) are widely used to provide position, velocity, and timing (PVT) information for various applications, including transportation, location-based communication services, and intelligent agriculture. In urban canyons, high-rise buildings and narrow streets can cause signal obstruction, non-line-of-sight (NLOS) reception, and multipath effects that introduce errors in GNSS pseudorange measurements. Although multi-constellations GNSS effectively increase the number of available satellites, the inclusion of degraded signals can lead to severe positioning errors. This study proposes a machine learning framework for the weighted least squares (WLS) algorithm incorporating activation functions to enhance positioning accuracy. Several signal quality indicators are employed as training features for ensemble learning algorithms to identify poor quality signals by providing quality scores. Then, activation functions are employed to transform the machine learning predicted scores to appropriate weights for WLS positioning. To evaluate the performance of our approach, experiments are conducted using real-world datasets from Hong Kong and Tokyo urban areas. Comparative analysis of activation functions reveals that sigmoid functions consistently yield the greatest improvements with different machine learning algorithms and GNSS constellation configurations. The proposed algorithm demonstrates substantial reductions in positioning errors for both single- and multiconstellation scenarios. Furthermore, our results indicate that the proposed algorithm exhibits strong geographical transferability. The proposed algorithm maintains comparable level of performance when trained on data from other regions with similar levels of urbanization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a machine learning framework to enhance weighted least squares (WLS) GNSS positioning in urban environments. Ensemble models are trained on signal quality indicators (C/N0, elevation, pseudorange residuals) to output quality scores per satellite; these scores are mapped through activation functions (sigmoid performing best) to produce weights for the WLS solution. Real-world datasets from Hong Kong and Tokyo are used to demonstrate substantial positioning error reductions in single- and multi-constellation cases, plus geographical transferability when models trained on one city are applied to the other.

Significance. If the empirical gains prove robust, the approach could supply a practical, data-driven improvement to GNSS accuracy in dense urban settings by down-weighting degraded signals more effectively than conventional heuristics. The integration of activation functions for weight mapping and the reported cross-city transferability are potentially useful contributions for real-world deployment.

major comments (2)
  1. [Abstract and Evaluation] Abstract and results sections: the central claims of 'substantial reductions in positioning errors' and 'strong geographical transferability' are presented without quantitative error statistics, error bars, baseline definitions, or details on data exclusion/cross-validation. This information is load-bearing for assessing whether the reported improvements exceed standard elevation/SNR weighting.
  2. [Methodology and Results] Methodology and results: no direct diagnostics are reported on the correlation between ML-predicted quality scores and ground-truth pseudorange errors or residuals. Without such analysis it remains unclear whether the activation-mapped weights meaningfully reflect per-satellite error contributions rather than simply reweighting in a training-distribution-dependent manner.
minor comments (2)
  1. [Abstract] Abstract: 'comparative analysis of activation functions' is mentioned but the set of functions tested (beyond sigmoid) is not enumerated.
  2. [Throughout] Notation: ensure consistent terminology for 'quality scores' versus 'weights' across sections to prevent reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We have addressed each of the major comments in detail below and made revisions to the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract and Evaluation] Abstract and results sections: the central claims of 'substantial reductions in positioning errors' and 'strong geographical transferability' are presented without quantitative error statistics, error bars, baseline definitions, or details on data exclusion/cross-validation. This information is load-bearing for assessing whether the reported improvements exceed standard elevation/SNR weighting.

    Authors: We agree that the abstract would benefit from more specific quantitative details to support the claims. The results section already provides quantitative error statistics, including positioning RMSE values for various scenarios and comparisons against baseline weighting methods such as elevation-only and C/N0-based weighting. We have revised the abstract to include specific error reduction figures (e.g., average improvements observed) and added details on the cross-validation procedure and data exclusion criteria used in the experiments. Additionally, we have included error bars in the revised figures and text to indicate variability. These changes ensure the claims are better supported and allow direct comparison to standard methods. revision: yes

  2. Referee: [Methodology and Results] Methodology and results: no direct diagnostics are reported on the correlation between ML-predicted quality scores and ground-truth pseudorange errors or residuals. Without such analysis it remains unclear whether the activation-mapped weights meaningfully reflect per-satellite error contributions rather than simply reweighting in a training-distribution-dependent manner.

    Authors: This is a valid point. To address it, we have added a new analysis in the results section showing the correlation between the machine learning predicted quality scores and the ground-truth pseudorange residuals. This includes computed correlation coefficients and visualizations demonstrating that lower quality scores correspond to larger residuals, validating that the weights reflect per-satellite error contributions. The cross-city transferability results further support that the approach is not merely overfitting to the training distribution but generalizes to similar urban environments. revision: yes

Circularity Check

0 steps flagged

No significant circularity: empirical results rest on external GNSS datasets

full rationale

The paper trains ensemble models on real-world signal quality indicators (C/N0, elevation, residuals) from Hong Kong and Tokyo urban recordings to output quality scores, maps them through activation functions (sigmoid reported best) to WLS weights, and evaluates positioning error reductions plus cross-regional transferability directly against measured ground-truth errors in held-out or alternate-region data. No equations, fitted parameters, or self-citations reduce the reported accuracy gains or transferability to quantities defined inside the training loop itself; the central claims are falsifiable against independent positioning benchmarks outside the model's own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The framework rests on standard GNSS measurement models and machine-learning assumptions about feature relevance; no new physical entities are introduced.

free parameters (2)
  • Ensemble learning hyperparameters
    Tuned on training data to optimize quality score prediction.
  • Activation function parameters
    Sigmoid and alternatives selected and compared for best empirical performance.
axioms (1)
  • domain assumption Signal quality indicators are informative predictors of pseudorange error severity in urban environments
    Used as input features for the machine learning models.

pith-pipeline@v0.9.0 · 5777 in / 1363 out tokens · 42990 ms · 2026-05-21T05:25:39.331657+00:00 · methodology

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