Estimation of the multifractal spectrum using the crossing tree

The crossing tree of a signal has been recently used to construct monofractal and multifractal processes. In this paper, we use the crossing tree of a signal for the purpose of analysis. Given a realisation of a signal, its crossing tree is calculated, whose characteristics are used to estimate the multifractal spectrum. More precisely, a partition function is evaluated at each resolution to sum up the statistics of the crossing durations. For multifractal signals, the partition function is a power law of the resolution, whose exponent is linked to the spectrum. We illustrate numerically this approach for several classes of multifractal signals, including multiplicative cascades, fractional Brownian motion, and the Weierstrass function.