The Quadratic Time-Varying Hausdorff and Large Deviation Multifractal Spectrum of Stochastic Fractal Signal

Although multifractal describes the spectrum distribution of Singularity Exponent (SE), it loses the temporal information, and it’s hard to describe the dynamics evolving process of non-stationary system. The time-varying singularity distribution indicates the spatial dynamics character of system. Therefore, the time-varying quadratic multifractal spectrum is proposed. Similar to the Wigner-Ville time-frequency analysis, the time-delayed conjugation of analyzed signal is selected as the windows function, and the quadratic time-singularity exponent distribution of the instantaneous self-correlation is deduced based on the short-time multifractal analysis, i.e. quadratic time-singularity multifractal distribution, which includes Hausdorff Measure, time-varying singular spectrum distribution, time-varying large deviation multifractal spectrum, which exhibits the singular exponent distribution of signal at arbitrary time.