Modified Multi-fractional Gaussian Noise and Its Application to Traffic
Ming Li · 2022
This chapter discusses a novel multi-fractional noise called modified multi-fractional Gaussian noise. It is the conventional fractional Gaussian noise equipped with the time-varying Hurst parameter H ( t ). The discussed theory of modified multi-fractional Gaussian noise contains four parts. The first part consists of the representations of its autocorrelation function, power spectrum density, and approximation of autocorrelation function. The second part is about its statistical dependences and the condition of multi-fractional Gaussian noise to be 1/ f noise. The third part is about its stationarity behavior. The analysis exhibits that multi-fractional Gaussian noise is stationary when H ( t ) 0.70. A set of stationary ranges regarding H ( t ) > 0.70 are given. The last part is about an application case of using H ( t ) to describe the stationarity of real traffic and introduce the concept of the stationarity scale of traffic, which implies that the stationarity of traffic is observation-scale-dependent. Generally, it is stationary at small time scales and non-stationary at large time scales.