Figure 1. Tests of exponent estimation for the DLFNN using N = 10⁴ simulated fBm trajectories.

(a) Plots showing the Hurst exponent estimates of fBm trajectories with $n={10}^2$ data points by a triangular DLFNN with three hidden layers compared with conventional methods. Plots are vertically grouped by Hurst exponent estimation method: (left to right) rescaled range, MSD, sequential range and DLFNN. ${\sigma }_H$ values are shown in the title. Top row: Scatter plots of estimated Hurst exponents $H_{est}$ and the true value of Hurst exponents from simulation $H_{sim}$. The red line shows perfect estimation. Second row: Due to the density of points, a Gaussian kernel density estimation was made of the plots in the top row (see Materials and methods). Third row: Scatter plots of the difference between the true value of Hurst exponents from simulation and estimated Hurst exponent $\mathrm{\Delta }H=H_{sim}-H_{est}$. Last row: Gaussian kernel density estimation of the plots in the third row. (b) ${\sigma }_H$ as a function of the number of consecutive fBm trajectory data points $n$ for different methods of exponent estimation. Example structures for two hidden layers and $n=5$ time series input points of the anti-triangular, rectangular and triangular DLFNN are shown in (c, d and e), respectively. (f) ${\sigma }_H$ as a function of the number of hidden layers in the DLFNN for triangular, rectangular and anti-triangular structures. (g) ${\sigma }_H$ as a function of the number of randomly sampled fBm trajectory data points $n_{rand}$ with different number of hidden layers in the DLFNN shown in the legend. (h) ${\sigma }_H$ as a function of the noise-to-signal ratio ($\frac{Noise}{Signal}$) (NSR) from Gaussian random numbers added to all $n={10}^2$ data points in simulated fBm trajectories. (i) Plots of bias $b(H_{sim})$, variance $\text{Var}(H_{sim})$ and mean square error (MSE) as functions of $H_{sim}$. For each value of $H_{sim}$, fBm trajectories with $n=100$ points were simulated and estimated by a triangular DLFNN.