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

This paper proposes a machine learning framework that utilizes activation functions to convert predicted signal quality scores into optimal weights for Weighted Least Squares GNSS positioning, significantly improving accuracy in urban environments by effectively mitigating non-line-of-sight and multipath errors.

Original authors: Pin-Hsun Lee, Harry Leib

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pin-Hsun Lee, Harry Leib

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Problem: Getting Lost in the "Concrete Canyon"

Imagine you are trying to find your way in a city filled with towering skyscrapers and narrow streets. This is what experts call an "urban canyon." When you use a GPS in these areas, the signal from the satellites has to bounce off buildings before it reaches your phone.

Think of it like shouting in a canyon. Sometimes you hear your own voice directly (a clear signal). Other times, you hear an echo bouncing off a wall (a "multipath" signal) or a voice coming from a different direction entirely (a "non-line-of-sight" signal). These echoes and wrong directions confuse your GPS, making it think you are somewhere you aren't. In extreme cases, the error can be as large as a kilometer!

The Old Way: Throwing Out the Bad Guys

Traditionally, to fix this, engineers tried to identify the "bad" signals (the echoes) and simply throw them away. They would say, "This signal looks suspicious, ignore it."

However, there's a catch. If you throw away too many signals, you don't have enough "voices" left to triangulate your position. It's like trying to solve a puzzle but throwing away half the pieces; you might have a clean picture, but you can't finish the puzzle at all.

The New Solution: A Smart Weighting System

This paper proposes a smarter way to handle the chaos. Instead of just throwing signals away, the authors built a Machine Learning Framework that acts like a sophisticated "Quality Control Manager."

Here is how it works, step-by-step:

1. The Detective Work (Machine Learning)

The system uses a team of digital detectives (called Ensemble Learning algorithms like Random Forest and AdaBoost). These detectives look at several clues about each satellite signal:

How high is it in the sky? (Low angles are more likely to bounce off buildings).
How loud is the signal? (Weak signals are suspicious).
Does the math add up? (If the signal says you are 10 miles away, but your clock says you are 2 miles away, something is wrong).

Based on these clues, the detectives give every signal a Quality Score from 0 to 1. A score of 1 means "Trust me completely," and a score of 0 means "I'm lying."

2. The Translator (Activation Functions)

This is the paper's biggest innovation. The Machine Learning gives a score, but the GPS calculator needs a weight (a number that tells it how much to listen to that signal).

The authors tested different "translators" (called Activation Functions) to turn the Quality Score into a Weight.

The "All or Nothing" Translator: This is like a strict bouncer. If a signal is below a certain score, it gets kicked out (weight = 0). If it's above, it gets full trust (weight = 1). The paper found this is too harsh; it throws away too many signals, ruining the puzzle.
The "Linear" Translator: This just passes the score along directly. The problem here is that most signals end up with very similar scores, so the GPS treats them all the same, which doesn't help much.
The "Sigmoid" Translator (The Winner): This is the star of the show. Think of this as a dimmer switch rather than an on/off switch.
- If a signal is terrible, the dimmer turns it almost all the way down (near zero weight).
- If a signal is great, the dimmer turns it all the way up.
- If a signal is "okay," it gives it a medium weight.

The paper found that this Sigmoid approach is the best because it doesn't just delete bad signals; it gently lowers their volume so they don't ruin the calculation, while keeping enough signals to solve the puzzle.

The Results: Testing in Real Cities

The authors tested this system using real data from two very busy cities: Hong Kong and Tokyo.

The Test: They trained the system on data from one part of the city and tested it on another. They even tried training on data from Hong Kong and testing it in Tokyo.
The Outcome:
- The new system significantly reduced positioning errors compared to standard methods.
- The Sigmoid function consistently outperformed the others.
- Geographic Transferability: This is a key finding. The system trained on the dense, tall buildings of Hong Kong worked almost just as well when tested in the dense, tall buildings of Tokyo. This means you don't need to re-train the system for every single city, as long as the "urban density" is similar.

The Bottom Line

The paper claims that by using Machine Learning to grade signal quality and a specific mathematical "dimmer switch" (the Sigmoid function) to adjust how much the GPS listens to each signal, we can navigate urban canyons much more accurately. It's not about ignoring the bad signals; it's about knowing exactly how much to trust them.

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

Problem Statement
Global Navigation Satellite Systems (GNSS) are critical for positioning, velocity, and timing (PVT) in applications ranging from transportation to intelligent agriculture. However, in urban environments characterized by high-rise buildings and narrow streets ("urban canyons"), GNSS signals suffer from obstruction, non-line-of-sight (NLOS) reception, and multipath effects. These phenomena introduce significant errors in pseudorange measurements, often exceeding one kilometer. While multi-constellation GNSS increases the number of available satellites, the inclusion of degraded signals can exacerbate positioning errors. Traditional methods that simply exclude degraded signals can adversely affect satellite geometry, leading to higher Dilution of Precision (DOP). Conversely, standard Weighted Least Squares (WLS) methods often rely on simplistic weighting schemes (e.g., elevation angle, Carrier-to-Noise density ratio $C/N_0$ ) that fail to capture the complex relationships between signal features and measurement quality.

Methodology
The authors propose a novel framework that integrates ensemble machine learning with a specific activation function mechanism to enhance WLS positioning. The methodology proceeds through the following stages:

Feature Extraction: Six signal quality indicators are extracted for each epoch to serve as input features for machine learning:
- Elevation Angle
- Carrier-to-Noise Density ( $C/N_0$ )
- Pseudorange Residual
- GDOP Contribution (measuring the impact of a specific satellite on geometry)
- Pseudorange Rate Consistency (derived from Doppler shifts)
- Estimated Receiver Clock Offset
  These features are normalized using Z-score standardization on an epoch-by-epoch basis.
Machine Learning Training: Ensemble learning algorithms—specifically Random Forest, AdaBoost, and Gradient Boosting—are trained to classify signal quality.
- Labeling: A supervised learning approach is used where the "best" subset of satellites (the combination yielding the minimum 3D positioning error via Least Squares) is labeled as 1 (good quality), and all other signals are labeled as 0.
- Training Data: The models are trained on data from Hong Kong (deep and harsh urban environments) and Tokyo (Shinjuku and Odaiba), with specific configurations used to test generalization across different geographical locations.
Activation Function Framework: The core innovation lies in mapping the machine learning's predicted probability scores (ranging from 0 to 1) into weights for the WLS algorithm using activation functions. Several functions were evaluated:
- Constant: Serves as a baseline (equivalent to Ordinary Least Squares).
- Linear: Directly maps scores to weights.
- Unit Step: Hard thresholding (excludes signals below a threshold).
- ReLU: Assigns zero weight to low scores and linearly maps the rest.
- Sigmoid: A soft decision function that gradually reduces weights for poor-quality signals while amplifying weights for high-quality ones, preserving satellite geometry by avoiding the hard exclusion of marginal signals.
Positioning Calculation: The transformed weights are applied to the WLS algorithm to compute the final position solution.

Key Contributions
The paper outlines three primary contributions:

Signal Quality Assessment: The employment of three ensemble learning techniques to assess GNSS signal quality using a comprehensive set of indicators reflecting NLOS/multipath errors and satellite geometry.
Activation Function Framework: The introduction of a novel framework that utilizes activation functions to convert machine learning quality scores into WLS weights. The study systematically evaluates various functions to identify the optimal one for minimizing 3D positioning errors.
Geographical Transferability Validation: The validation of the algorithm using real-world datasets from Hong Kong and Tokyo, demonstrating effectiveness across single- and multi-constellation scenarios and assessing the model's ability to generalize across different urban environments.

Experimental Results
Experiments were conducted using the UrbanNav open-source dataset, comparing GPS-only, BeiDou-only, and GPS+BeiDou configurations.

Algorithm Performance: Among the machine learning models, AdaBoost consistently demonstrated the best generalization, achieving the highest testing accuracy with minimal overfitting compared to Random Forest and Gradient Boosting.
Activation Function Efficacy: The Sigmoid function consistently outperformed other activation functions (including ReLU and Unit Step). It provided the greatest improvement in positioning accuracy by effectively suppressing poor-quality signals while maintaining a diverse set of satellites to preserve geometry.
Positioning Accuracy (Hong Kong Data):
- For GPS-only positioning, the "AdaBoost + Sigmoid" approach reduced the 3D Root Mean Square Error (RMSE) by 75.52% relative to the baseline (Constant function), bringing the error within 24.48% of the theoretical "best set" (the optimal subset of satellites).
- For GPS+BeiDou positioning, the same approach improved the 3D RMSE by 53.78% over the baseline, reducing the gap to the best set to 46.22%.
Geographical Transferability:
- When models trained on Odaiba (suburban/open) data were tested on Shinjuku (dense urban) data, performance improvements were modest (62.02% away from the best set) due to the environmental mismatch.
- However, when models trained on Hong Kong (dense urban) data were tested on Shinjuku data, performance improved significantly (54.21% away from the best set). This indicates that the algorithm exhibits strong geographical transferability provided the training and testing environments share similar levels of urbanization.

Significance and Claims
The paper claims that the proposed framework offers a robust solution for GNSS positioning in challenging urban environments without requiring the exclusion of satellites, which often degrades geometry. The study emphasizes that:

Soft Weighting is Superior: The use of sigmoid activation functions allows for a "soft" rejection of degraded signals, which is more effective than hard exclusion methods (like Unit Step or ReLU) in maintaining satellite geometry while still mitigating errors.
Generalization via Urban Similarity: The algorithm does not require location-specific training data for every new site. Instead, a model trained on data from an environment with a similar level of urbanization (e.g., Hong Kong to Tokyo) can be effectively applied to other locations, reducing the need for extensive, site-specific dataset collection.
Multi-Constellation Synergy: While multi-constellation GNSS increases signal availability, it does not automatically guarantee higher accuracy if degraded signals are included. The proposed ML-weighted approach is necessary to realize the benefits of multi-constellation systems in urban canyons.

The authors conclude that while the current results are substantial, future work could involve integrating this framework with Inertial Navigation Systems (INS) and exploring adaptive selection of activation function parameters based on real-time signal quality.