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The Statistical Significance of the Inclusion of Graph Neural Networks in the Financial Time Series Forecasting Problem

This paper introduces the Time-Geometric model, which integrates Graph Neural Networks to capture geometric patterns alongside temporal data, demonstrating through extensive empirical evaluation that this approach yields statistically significant improvements in financial univariate time series forecasting accuracy.

Original authors: Marco Gregnanin, Johannes De Smedt, Giorgio Gnecco, Maurizio Parton

Published 2026-05-21

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CC BY 4.0

Original authors: Marco Gregnanin, Johannes De Smedt, Giorgio Gnecco, Maurizio Parton

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

Imagine you are trying to predict the weather. Most forecasters look at the timeline of the past few days: "It rained yesterday, it was cloudy the day before, so it will probably rain today." They focus entirely on the order of events.

This paper argues that while looking at the timeline is good, it's not the whole story. The authors suggest that if you also look at the shape of the data—how the numbers rise and fall relative to each other, like a mountain range or a jagged coastline—you can make much better predictions.

Here is a breakdown of their research using simple analogies:

1. The Problem: Only Looking at the "When"

Financial markets (like stock prices) are messy and noisy. Traditional computer models (like RNNs, Transformers, and TCNs) are like runners who only look at the track in front of them. They are great at spotting patterns in time (e.g., "every Monday the price goes up"), but they often miss the bigger picture of how the data is structured.

The authors noticed that these models rarely check if their improvements are statistically real or just lucky guesses. They wanted to prove that adding a new "sense" to these models actually makes a measurable difference.

2. The Solution: The "Time-Geometric" Model

The authors built a new system called the Time-Geometric Model. Think of it as giving the computer a pair of 3D glasses.

The Time Component: This is the standard part. It looks at the data in a straight line (time), just like the old models. It asks, "What happened yesterday? What happened the day before?"
The Geometric Component (The Secret Sauce): This is the new part. The authors use a trick called a Visibility Graph.
- The Analogy: Imagine the stock prices are a series of vertical bars on a chart. The "Visibility Graph" asks: "If I stand on top of the bar for Tuesday, can I see the top of the bar for Friday without a taller bar blocking my view?"
- If you can see it, you draw a line connecting them.
- By doing this for every point, the straight line of time turns into a web or a spiderweb (a graph). This web captures the shape and structure of the data, revealing hidden patterns like fractals or random chaos that a straight line misses.

3. How They Tested It

They didn't just guess; they ran a massive experiment.

The Data: They used 90 different stocks from the S&P 100 (the biggest companies in the US).
The Setup: They took 8 different types of standard forecasting models (the "Time" runners) and gave them a partner: a Graph Neural Network (GNN). The GNN is a special type of AI designed to understand webs and connections.
The Goal: They wanted to see if the "Time + Geometry" team could predict the future price of a stock better than the "Time only" team.

4. The Results: Does the Shape Matter?

The answer is a resounding yes, but with a twist.

The Improvement: In many cases, adding the "Geometric" web helped the models predict prices more accurately. It was like giving a runner a map of the terrain, not just the track.
The Twist (Statistical Significance): The authors did something rare in this field: they ran strict statistical tests (like a referee checking the score with a stopwatch). They found that whether the improvement was "statistically significant" (meaning it wasn't just luck) depended on how you measured success.
- If you measured success by "how far off the prediction was" (RMSE), the geometric model won big for some types of AI.
- If you measured it differently (like percentage error), the results changed.
- Key Finding: The "Time-Geometric" model didn't win every single race, but when it did win, it was a real win, not a fluke. Specifically, it showed massive improvements when paired with certain models like Transformers and TCNs.

5. The Conclusion

The paper concludes that financial data isn't just a line; it has a shape. By using Graph Neural Networks to analyze that shape (the "geometry") alongside the timeline, we can significantly improve how well we forecast the future.

However, the authors warn that you can't just say "it's better" without checking the math. Depending on which "ruler" (metric) you use to measure the results, the winner might change. But overall, looking at the geometry of the data is a powerful new tool for understanding the financial markets.

Technical Summary: The Statistical Significance of the Inclusion of Graph Neural Networks in Financial Time Series Forecasting

Problem Statement

Univariate time series forecasting in financial markets is inherently challenging due to data noise and the stochastic nature of asset prices. While numerous statistical (e.g., ARIMA) and deep learning models (e.g., RNNs, Transformers, TCNs) exist, they predominantly focus on analyzing temporal patterns within the data. These models often neglect geometric patterns inherent in the time series structure. Furthermore, previous comparative studies frequently lack rigorous statistical analysis to validate performance improvements, often failing to determine if observed differences are statistically significant according to established criteria. This research addresses the gap by investigating whether the inclusion of geometric patterns, captured via Graph Neural Networks (GNNs), yields statistically significant improvements in forecasting accuracy for univariate financial time series.

Methodology

The Time-Geometric Framework

The authors propose the Time-Geometric model, a hybrid architecture designed to exploit both temporal and geometric patterns. The model consists of three primary components:

Visibility Graph Construction: The core innovation lies in transforming the univariate time series into a graph representation using the Visibility Graph algorithm. This method maps time series data points to nodes and establishes edges based on line-of-sight visibility between points. This transformation preserves structural properties, allowing the model to discern whether the underlying series is periodic, random, or fractal. The resulting graph is encoded as an adjacency matrix ( $A_t$ ).
Time Component: This component processes the feature matrix ( $X_t$ ) containing historical observations (e.g., Open, High, Low, Close, Volume) using standard neural network architectures. The study evaluates various baselines including LSTM, GRU, RNN, BiLSTM, BiGRU, Transformers, and TCNs.
Geometric Component: This component processes the adjacency matrix ( $A_t$ ) and the feature matrix ( $X_t$ ) using Graph Convolutional Networks (GCNs) to extract geometric patterns. The output of the GCN layers is further processed by an LSTM layer to capture temporal dependencies within the geometric patterns.
Fully Connected Component: The temporal patterns from the Time Component and the geometric patterns from the Geometric Component are summed (and optionally concatenated with the original features via a skip connection) and passed through a fully connected network to generate the final forecast.

Experimental Setup

Dataset: The study utilizes 90 univariate time series representing the constituents of the Standard & Poor 100 (S&P100) index. Data includes daily observations of Close, Open, High, Low, and Volume from January 2015 to June 2023.
Evaluation: The models are evaluated across three forecasting horizons: daily (1 day), weekly (5 days), and monthly (20 days).
Metrics: Performance is measured using Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Mean Absolute Scaled Error (MASE).
Statistical Analysis: To determine significance, the authors employ:
- Pairwise Comparisons: Paired t-tests, Wilcoxon signed-rank tests, and Sign tests comparing baseline models against their Time-Geometric counterparts.
- Multiple Comparisons: The Friedman test followed by the Nemenyi post-hoc test to rank all algorithms simultaneously and control for Type I errors.

Key Results

Numerical Performance

The Time-Geometric model generally demonstrates improved performance over baseline models, particularly when the baseline utilizes Transformer or TCN architectures. For 1-day predictions, the Time-Geometric model incorporating TCN (TG(TCN)) achieved the lowest average RMSE (1.1420) and MAPE (0.5658), significantly outperforming the baseline TCN. However, the magnitude of improvement varies by metric and model architecture; for instance, improvements were less pronounced or inconsistent for certain RNN-based baselines depending on the specific error metric used.

Statistical Significance

The statistical analysis reveals that the significance of including geometric patterns is metric-dependent and horizon-dependent:

Metric Dependence: In pairwise comparisons, models like $\Delta$ -Transformer and $\Delta$ -TCN showed consistent statistical significance (rejection of the null hypothesis) across all metrics for 1-day predictions. Conversely, models like $\Delta$ -RNN showed acceptance of the null hypothesis for several metrics, indicating that the geometric inclusion did not statistically improve performance for that specific baseline under those metrics.
Horizon Dependence: The statistical significance of the results shifts with the prediction horizon. For 5-day and 20-day predictions, the consistency of statistical rejection varies, suggesting that the utility of geometric patterns may change as the forecasting window extends.
Multiple Comparisons: The Friedman and Nemenyi tests confirmed that there are statistically significant differences among the set of all algorithms. The Time-Geometric models often achieved superior average ranks, particularly for the 5-day horizon where TG(BILSTM) and TG(GRU) performed notably well.

Claims and Significance

The paper claims to advance the domain of univariate time series prediction by:

Introducing the Time-Geometric Model: A novel framework that integrates dynamic GNNs with baseline time series neural networks via visibility graphs.
Providing Rigorous Statistical Validation: Unlike many prior studies that rely solely on average performance metrics, this research applies extensive statistical hypothesis testing (pairwise and multiple comparisons) across 90 financial datasets.
Demonstrating Conditional Significance: The authors conclude that leveraging geometric patterns yields statistically significant improvements in forecasting accuracy, but explicitly note that this significance is contingent upon the choice of evaluation metric and the forecasting horizon.

The paper does not claim that geometric patterns universally outperform temporal patterns in all scenarios. Instead, it posits that the inclusion of geometric information provides a statistically verifiable enhancement in specific contexts, particularly when combined with certain deep learning architectures like Transformers and TCNs. Future work is suggested to explore the behavior of evaluation metrics and the significance of geometric patterns across more diverse time series datasets beyond the financial domain.