Testing of fractional Brownian motion in a noisy environment

Abstract

Fractional Brownian motion (FBM) is the only Gaussian self-similar process with stationary increments. Its increment process, called fractional Gaussian noise, is ergodic and exhibits a property of power-like decaying autocorrelation function (ACF) which leads to the notion of long memory. These properties have made FBM important in modelling real-world data recorded in different experiments ranging from biology to telecommunication. These experiments are often disturbed by a noise which source can be just the instrument error. In this paper we propose a rigorous statistical test based on the ACF for FBM with added white Gaussian noise. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power and comparing with other tests existing in the literature. We also note that the introduced test procedure can be applied to an arbitrary Gaussian process.

Figures (6)

• Figure 1: Comparison of the imaginary (left panel) and real (right panel) parts of the empirical (dashed blue line) and analytical (dash-dotted red line) characteristic functions of the estimator $\hat{r}(3)$ for the FBM and noise. The analytical distribution of the estimator is given in terms of the generalized $\chi^2$ distribution. The top panel corresponds to a subdiffusion case with $H=0.3$ and the bottom panel to the superdiffusion with $H=0.7$. In both cases the magnitude of the additive noise is $\sigma=0.2$. The empirical characteristic function is calculated on the basis of $n=10^5$ trajectories of length $N = 2^7=128$.
• Figure 2: (Top panel) Critical surface $q(H, \sigma) = q(N=1000, a=0.05, H, \sigma)$ for parameters $H\in(0.1,0.9)$ and $\sigma\in[0,1]$. (Bottom panel) Heat maps for the lower (left panel) and upper (right panel) quantiles. To create the surface we simulated $10, 000$ replications of the generalized $\chi^2$ distribution.
• Figure 3: Cross-section of the critical surface presenting quantile lines of the estimator (\ref{['eq:rk']}) for three data lengths $N$, $H=0.3$ and various $\sigma$'s. Blue lines correspond to length $N=200$, red to $N=500$ and yellow to $N=1000$. In each pair of lines of the same colour the top line represents the quantile of order $1-a/2=0.975$ and the bottom the quantile of order $a/2=0.025$. To create the surfaces we simulated $10, 000$ replications of the generalized $\chi^2$ distribution.
• Figure 4: Cross-section of the critical surface presenting quantile lines of estimator (\ref{['eq:rk']}) for $\sigma=0.3$ and for three data lengths $N$. Blue line correspond to length $N=200$, red to $N=500$ and yellow to $N=1000$. In each pair of lines of the same colour the top line represents the quantile of order $1-\frac{a}{2}=0.975$, whereas the bottom the quantile of order $\frac{a}{2}=0.025$.
• Figure 5: Test's power for different null hypotheses, noise magnitudes and data lengths, for simulated FBM's. Top panel: dependence on data length $N$ and Hurst index $H$, assuming $\sigma_0=0.3$. Middle panel: dependence on magnitude of the noise $\sigma$ and $H$, assuming $N=200$. Bottom panel: dependence on lag $\tau$ in the ACVF and $H$, assuming $\sigma_0=0.3$.

Theorems & Definitions (3)

• Example 1
• Theorem 1
• proof