⚡ electrical engineering

Least-squares-based deep learning for short-term ionospheric TEC prediction with uncertainty quantification and DIA quality control

This study proposes a least-squares-based deep learning framework that integrates stochastic weighting, uncertainty quantification, and detection–identification–adaptation quality control to deliver accurate, statistically interpretable short-term ionospheric TEC predictions with robust gross error handling.

Original authors: Qikang Chen, Pengyu Hou, Baocheng Zhang

Published 2026-09-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Qikang Chen, Pengyu Hou, Baocheng Zhang

Original paper licensed under CC BY 4.0 (https://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 sky above us is not empty; it is filled with a layer of charged gas called the ionosphere. This layer acts like a lens for the radio signals that travel between satellites and our phones or cars, guiding the Global Navigation Satellite System, or GNSS, that we rely on for timing and location. However, the ionosphere is restless. Driven by the sun and the Earth's magnetic field, the density of electrons within it shifts constantly, stretching and delaying signals in ways that can throw off precise positioning by meters or even more. To fix this, scientists create maps of the total electron content, known as TEC, which tell us how much charge is in the air at any given moment. But these maps are often just snapshots of the past. For navigation to work perfectly in real time, we need to know what the ionosphere will do in the next hour, not just what it did yesterday. Furthermore, knowing the prediction is not enough; we need to know how much we can trust it. If a navigation system is to be safe, it must be able to say, "I predict the delay will be this much, and I am 95 percent sure it will not be more than this much."

A team of researchers from the Chinese Academy of Sciences has developed a new way to make these short-term forecasts that does more than just guess the next value. They combined a powerful type of computer learning, known as deep learning, with the rigorous statistical rules used in geodesy, the science of measuring the Earth. Instead of treating all historical data as equally perfect, their method pays close attention to the quality of every single data point it learns from. They used a dataset from 2015 containing hourly measurements of electron density from 36 different locations stretching from the equator to the Arctic along a single line of longitude. By feeding the computer the last 24 hours of data, they asked it to predict the next hour. The result was a system that not only predicted the future state of the ionosphere with high accuracy but also calculated a reliable range of uncertainty for every prediction.

The researchers found that their new approach, which they call least-squares-based deep learning, performed just as well as, or better than, other popular forecasting methods like long short-term memory networks, or LSTM, and autoregressive models, or ARIMA. In tests across low, middle, and high latitudes, the average error of their predictions was about 1.14 units of electron content. This is a small margin of error for such a complex and shifting environment. What sets their work apart is how it handles the "noise" in the data. Real-world measurements are never perfect; some are clearer than others. The new system uses the known precision of each data point to weigh them correctly during training, giving more importance to the reliable measurements and less to the shaky ones. This allows the model to learn a clearer picture of the ionosphere's behavior without being misled by poor-quality data.

Beyond just making a prediction, the system provides a built-in measure of confidence. When the researchers tested the model, they found that their predicted ranges were remarkably accurate. When they said there was a 95 percent chance the real value would fall within a certain range, it actually did so 96.26 percent of the time. This kind of statistical honesty is rare in machine learning, where models often give a single number without explaining how sure they are. The researchers also tested how the system would react if the training data contained bad information, simulating a scenario where two to five percent of the data were corrupted by large errors. The system successfully spotted these bad data points and adjusted itself, either by ignoring them or by reducing their influence, effectively cleaning up its own learning process. This ability to detect and adapt to errors is crucial for safety-critical applications where a wrong guess could lead to a navigation failure.

One of the most surprising findings was the speed of the new method. While other deep learning models took over a hundred seconds to train on the same data, this system finished in less than one second. This dramatic difference comes from the way the math is structured: instead of adjusting every single part of the computer model at once through a slow, iterative process, the new method solves for the final layer of the model directly and quickly after each small adjustment to the earlier layers. This efficiency means the system could be updated frequently, perhaps even every hour, to keep pace with the rapidly changing ionosphere without requiring massive computing power. The study demonstrates that by blending the pattern-recognition power of modern artificial intelligence with the strict error-checking rules of traditional science, we can build tools that are not only smart but also trustworthy and fast.

The researchers are careful to note that their work is a step forward, not a final solution. The model was tested on a single line of longitude and did not include external factors like solar storms or geomagnetic activity, relying solely on the history of the electron density itself. While the results were strong, the uncertainty estimates were slightly conservative, meaning the predicted ranges were a bit wider than strictly necessary in some cases. The team plans to expand this work to cover larger areas of the globe and to incorporate more complex error structures in the future. For now, however, they have shown that it is possible to create a forecasting tool that understands not only what the ionosphere is likely to do next, but also how confident it should be in that guess, offering a new level of reliability for the navigation systems that guide our world.

Technical Summary: Least-Squares-Based Deep Learning for Short-Term Ionospheric TEC Prediction

Problem Statement
High-precision Global Navigation Satellite System (GNSS) applications require not only reliable short-term predictions of ionospheric Total Electron Content (TEC) but also statistically consistent uncertainty information and rigorous quality control. Conventional deep learning models often function as "black boxes," lacking explicit stochastic modeling, failing to incorporate the varying precision of training data (e.g., RMS values from Global Ionospheric Maps), and providing no mechanism for detecting anomalous training samples. Furthermore, existing forecasting studies typically report point-accuracy metrics (RMSE, MAE) without providing covariance information or predictive uncertainty intervals necessary for stochastic modeling in high-precision GNSS estimation. There is a need for a unified framework that integrates stochastic weighting, predictive uncertainty quantification, and geodetic quality control (specifically Detection–Identification–Adaptation, or DIA) into the TEC prediction process.

Methodology
This study proposes a Least-Squares-Based Deep Learning (LSBDL) framework tailored for short-term, one-hour-ahead TEC prediction. The methodology integrates neural network feature extraction with classical least-squares estimation theory.

Weighted LSBDL Formulation: The model employs a single-hidden-layer neural network where the hidden layer performs a nonlinear mapping of the input (a 24-hour sliding window of historical TEC). The output layer is treated as a linear parameter estimation problem solved via weighted least squares. Crucially, the relative precision of training observations is incorporated by constructing a cofactor matrix ( $Q_y$ ) using the squared RMS values provided with the TEC products. This ensures that observations with higher precision (lower RMS) receive greater weight during parameter estimation.
Joint First-Order Predictive Uncertainty: Unlike standard deep learning approaches that often ignore parameter uncertainty or only propagate output-layer variance, this study derives a joint first-order predictive uncertainty. The prediction function is linearized around the final estimates, and the covariance matrices of both the output-layer parameters ( $X$ ) and the hidden-layer weights ( $W$ ), along with their cross-covariances, are jointly propagated. This yields a prediction covariance matrix that accounts for the uncertainty in the entire model structure.
DIA Quality Control: The framework integrates the DIA procedure to handle contaminated training data. An overall model test (OMT) assesses the consistency of the training system. Standardized residuals are then used to identify candidate anomalies (gross errors). Two adaptation strategies are evaluated: "Remove" (deleting identified anomalies) and "Downweight" (reducing the cofactors of identified anomalies by a factor of 100). The model is retrained using these adapted datasets to restore consistency.
Experimental Design: Experiments utilized hourly vertical TEC and RMS data from CODE GIMs for the year 2015 along the 115°E meridian (0°–87.5°N, 36 grid points). A 24-hour sliding window was used to predict the subsequent hour. The final 240 hours served as the test set. Controlled gross-error scenarios (2% and 5% contamination with errors of 5 $\times$ RMS) were introduced to evaluate DIA performance. The LSBDL model was benchmarked against ARIMA, LSTM, and GRU.

Key Contributions

RMS-Driven Stochastic Modeling: The study develops a weighted LSBDL model that explicitly incorporates the relative precision information (RMS) of TEC products into the training stochastic model, addressing the issue of heteroscedastic data in ionospheric modeling.
Joint Uncertainty Propagation: It derives a formulation for joint first-order predictive uncertainty that simultaneously considers the output-layer parameters, hidden-layer weights, and their cross-covariances. This provides a standard deviation for predictions that reflects both model parameter uncertainty and the observation error scale.
Integrated DIA Quality Control: The study applies geodetic DIA procedures to the training residuals within the LSBDL framework. This transforms anomaly handling from empirical data cleaning into a statistically testable and traceable quality-control procedure, evaluating both detection performance and the efficacy of adaptation strategies.

Results

Prediction Accuracy: Under clean conditions, the LSBDL model achieved an average RMSE of 1.137 TECU across all 36 grid points. Performance varied by latitude, with higher RMSE at low latitudes (1.748 TECU) due to larger TEC dynamic ranges, and lower RMSE at middle (0.772 TECU) and high latitudes (0.812 TECU).
Benchmark Comparison: At representative latitudes (15°, 45°, 75°N), LSBDL demonstrated prediction accuracy comparable to or better than ARIMA, LSTM, and GRU. Notably, LSBDL required a mean training time of 0.749 seconds, which is approximately 145 times faster than LSTM (108.692 s) and 132 times faster than GRU (99.182 s) under the tested CPU environment.
Uncertainty Quantification: The pooled empirical coverage of the nominal 95% and 99% prediction intervals reached 96.26% and 98.80%, respectively, indicating well-calibrated uncertainty estimates. The standardized prediction errors had a mean close to zero and an RMS of 0.883, suggesting the intervals were slightly conservative. Analysis revealed that the total prediction uncertainty was dominated by the observation term (scaled test RMS) rather than the parameter propagation term (which accounted for only ~0.162% of the variance).
DIA Performance: In controlled gross-error scenarios, the DIA procedure achieved F1-scores of 0.988 (2% contamination) and 0.992 (5% contamination). Both "Remove" and "Downweight" strategies successfully restored the prediction accuracy and uncertainty calibration, preventing the inflation of the variance factor of unit weight and the subsequent widening of prediction intervals observed in the non-DIA baseline.

Significance
The paper claims that the LSBDL framework provides an effective solution for short-term TEC prediction by naturally integrating stochastic weighting, uncertainty quantification, and geodetic quality control within a unified least-squares framework. The significance lies in offering statistically interpretable TEC products that include reliability information (covariance and uncertainty intervals) and robustness against data anomalies. This approach bridges the gap between data-driven deep learning and classical geodetic adjustment theory, making the outputs suitable for subsequent GNSS ionospheric correction and stochastic modeling where reliability and traceability are critical. The method demonstrates that separating nonlinear feature learning from conditional least-squares estimation can yield high accuracy with significantly reduced computational costs compared to fully recurrent deep learning architectures.

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