Structural and Lagrangian properties of analogue ensembles to characterize multifractality of stochastic processes

Abstract

We present a framework for the scale-invariance characterization of stochastic processes in reconstructed finite-dimensional phase spaces. This framework analyses the structural and dynamical properties of the phase space and is based on a Takens embedding reconstruction followed by the definition of ensembles of analogue states. We define the analogues of a target state as its nearest neighbors. Then, we specify a collection of target states densely sampling the full phase space. For each target state, we search for the ensemble of its k-best analogues and we analyze its volume and dynamics. First, we study the probability distribution of the volumes and relate its mean and variance to the scale-invariance properties of the stochastic process. Second, we study the Lagrangian properties of the analogues by characterizing how they disperse in time. More particularly, we study the volume occupied by the analogue's successors in function of time and of their initial volume. We link these dynamical properties to the scale-invariance properties of the process. We analyze two types of stationary and dissipative 1-dimensional scale-invariant processes: regularized fractional Brownian motion and regularized multifractal random walk. For both processes, the structure and dynamics of the phase space are determined by their scale-invariant properties.