On Optimal Wavelet Bases for the Realization of Microcanonical Cascade Processes

Multiplicative cascades are often used to represent the structure of turbulence. Under the action of a multiplicative cascade, the relevant variables of the system can be understood as the result of a successive transfer of information in cascade from large to small scales. However, to make this cascade transfer explicit (i.e, being able to decompose each variable as the product of larger scale contributions) is only achieved when signals are represented in an optimal wavelet basis. Finding such a basis is a data-demanding, highly-complex task. In this paper we propose a formalism that allows to find the optimal wavelet of a signal in an efficient, little data-demanding way. We confirm the appropriateness of this approach by analyzing the results on synthetic signals constructed with prescribed optimal bases. We show the validity of our approach constrained to given families of wavelets, though it can be generalized for a continuous unconstrained search scheme.