Finding optimal wavelet bases of cascade processes

Multiplicative cascade processes are found in a wide range of different physical systems. In these systems, the energy or an analogous quantity is transferred from large to small scales through an independent scale-invariant factor, thus conferring a statistically self-similar behaviour to this quantity. Such a behaviour is usually referred to as multifractality, because it is related to the presence of a multifractal structure (a hierarchical combination of fractal sets), which is a very general case of self-similarity. This way, the presence of multifractality allows to recognize the cascade process, either as a real mechanism or an effective one. An example of multiplicative cascade process is the case of Fully Developed Turbulence (FDT), where the cascade transfers the energy from large to small scales (where it is finally dissipated) giving rise to its multifractal structure, but such a behaviour is quite ubiquitous in nature and in fact has been observed in systems as diverse as stock market series, natural images, the heliospheric magnetic field, human gait, heartbeat dynamics, network traffic, fractures, fire plumes, as well as many other complex systems. While studying the cascade process is known to be a good strategy to obtain the global descriptors of a system (such as its multifractal characterization), it is possible to also achieve a local dynamical description, thanks to the optimal wavelet of the system. The cascade process with the optimal wavelet describes a local effective dynamics that can be used in reconstruction of gaps or lost information, data compression and time-series forecast. Wavelet transforms are integral transforms that allow to separate the details of a signal that are relevant at different scale levels. In other words, this means that wavelet transforms are precisely tuned to an adjustable scale and, for this reason, they are a powerful strategy to represent cascade processes. In addition, wavelets are Hilbert bases, i.e., the wavelet transform is invertible, so that the signal can be completely represented from the cascade process. Given a signal s(t), the wavelet transform at scale r is defined as αr(t) = s ⊗ Ψr, where Ψr(t) = Ψ( t r ) and Ψ is a certain function called wavelet. In a cascade process, the wavelet transform follows a multiplicative relation, i.e., two different scales r, L with r < L are related through a multiplicative variable: