Lf-Net:Generating Fractional Time-Series with Latent Fractional-Net

In this paper, we introduce a novel method for generating fractional time series through the utilization of neural networks. Although Neural Stochastic Differential Equations (Neural SDEs) have been presented as a method that combines Deep Neural Networks with numerical solvers of differential equations, these typically presume the noise structure of standard Brownian motion (Bm). Contrarily, numerous real-world time series data exhibit a fractal property, characterized by a Hurst index (H) that ranges from 0 to 1. This type of fractional time series pervades various domains including physics, biology, hydrology, network research, and financial mathematics. We propose a Latent Fractional Net (Lf-Net), devised to encapsulate both the long-range dependence (H > 1/2) and roughness (H < 1/2) intrinsic to fractional time series. This is accomplished by augmenting the noise term of the Neural SDEs using fractional Brownian motion (fBm) with an arbitrary Hurst index. We prove the existence and uniqueness of the solutions of the Lf-Net and theoretically show the convergence of the numerical solutions. We demonstrate the robustness of the Lf-Net under proper nonlinear transformations and construct a generative model for time-series data. The experiments show that the calibrated generator of the model can replicate the distributional properties of the original time series, especially the Hurst index. We conclude that our Lf-Net can effectively model the complex noise structure of real-world time series data and provide a promising direction for time series data generation.