Technical Summary: A Generative Adversarial Graph Neural Network for Synthetic Time Series Data

Problem Statement

The generation of synthetic financial time series data is critical for addressing data scarcity, privacy concerns, and the need for diverse training sets to improve model generalization. However, this task is hindered by the non-stationary and stochastic nature of financial data. Traditional statistical models (e.g., ARIMA, Black-Scholes) often rely on assumptions of weak stationarity and homoskedasticity, which constrain their effectiveness in capturing evolving volatility and complex market dynamics. While data-driven deep learning approaches, particularly Generative Adversarial Networks (GANs), have shown promise, existing models primarily focus on the autoregressive nature of time series. They frequently neglect the underlying geometric patterns and non-linear dependencies inherent in financial data, which are crucial for accurately modeling risk and volatility.

Methodology: The Sig-Graph GAN Framework

The authors propose Sig-Graph GAN, a novel architecture that integrates three distinct components to capture both temporal and geometric properties of time series:

1. Graph Representation via Visibility Graph:
   To address the non-stationarity and capture geometric structures, the model transforms univariate time series into a graph representation using the Visibility Graph algorithm. In this approach, data points become nodes, and edges are formed if a clear line of sight exists between them. This transformation preserves structural properties (e.g., periodicity, randomness, fractal characteristics) and allows the model to operate in a non-Euclidean space without assuming stationarity.

2. Hybrid Neural Architecture:
   The core of the generator and discriminator consists of three interconnected blocks:

   • Recurrent Block: Utilizes Long Short-Term Memory (LSTM) networks to capture the inherent autoregressive structure and temporal dependencies of the data.
   • Geometric Block: Employs Graph Neural Networks (GNNs), specifically Graph Convolutional Networks (GCNs), to process the adjacency matrix derived from the visibility graph. This block learns local structural information and geometric patterns by aggregating information from neighboring nodes.
   • Feedforward Block: Combines the temporal and geometric feature maps. It sums the outputs of the recurrent and geometric blocks (optionally concatenating with the original input via a skip connection) and passes them through fully connected layers with Parametric Rectified Linear Unit (PReLU) activations to produce the final prediction.

3. Signature-Based Loss Functions:
   To evaluate the quality of generated data beyond simple pointwise errors, the authors introduce custom loss functions based on the Time Series Signature (a concept from path theory). The raw outputs are transformed via a lead-lag transformation, and their truncated signatures (up to degree 5) and cumulative truncated signatures are computed. Two loss metrics are defined:

   • Mean Square Error (MSE): Measures pointwise discrepancies between the signatures of real and generated data.
   • Kullback-Leibler Divergence (KLD): Measures the disparity between the probability distributions of the signatures.
     These losses aim to capture both local path attributes and global distributional characteristics.

Key Contributions

The paper outlines three primary contributions:

1. Novel Architecture: The introduction of a GAN model that dynamically integrates GNNs with LSTM networks, specifically designed to leverage both autoregressive properties and geometric structures for synthetic data generation.
2. Custom Loss Functions: The proposal of signature-based loss functions (MSE and KLD) tailored for the Sig-Graph GAN, enabling the model to capture discrepancies in both pointwise values and probability distribution functions.
3. Comprehensive Evaluation: A rigorous evaluation across three major stock exchanges (S&P 500, Nasdaq Composite, and Nikkei 225), demonstrating performance improvements over established benchmarks.

Experimental Results

The model was evaluated on daily closing prices from 2010 to 2019 for the S&P 500, Nasdaq (IXIC), and Nikkei 225 (N225). The data was pre-processed to compute logarithmic returns and induce fat-tailed distributions.

• Performance Metrics: The study utilized the Earth Mover's Distance (EMD) to measure distributional similarity, Root Mean Square Error (RMSE) on signatures, and a Leverage Effect score.
• Comparative Analysis: Sig-Graph GAN consistently outperformed baseline models, including Quant GAN (using TCNs), GARCH(1,1), and Monte Carlo simulations based on Geometric Brownian Motion.
  • The model achieved lower EMD scores, indicating a closer match to the distribution of real logarithmic returns.
  • It demonstrated superior capability in replicating the leverage effect compared to most baselines.
  • The choice of loss function (MSE vs. KLD) influenced performance depending on the specific dataset (e.g., KLD favored N225 and IXIC, while MSE favored S&P 500).
• Ablation Study: Removing the Geometric Block resulted in the most significant performance degradation, confirming the importance of capturing geometric patterns. The impact of the Geometric Block was particularly pronounced when using the KLD loss function.

Significance and Claims

The paper claims that Sig-Graph GAN advances the field of time series generation by successfully integrating autoregressive modeling with geometric structure analysis. By utilizing the visibility graph to represent time series in a non-Euclidean space and employing GNNs to learn these structures, the model overcomes the limitations of stationarity assumptions found in traditional statistical models.

The authors assert that this approach effectively captures both the temporal evolution and the intrinsic geometric patterns (such as fractal characteristics) of financial data. Consequently, the model generates synthetic data that more accurately replicates the statistical characteristics, temporal dependencies, and distributional properties (including fat tails and volatility clustering) of real financial time series. The work suggests that incorporating geometric patterns is a pivotal step toward more robust synthetic data generation for financial applications, though the authors note that future work could explore replacing the LSTM component with more advanced architectures like Transformers.