Many applications hinge on modelling phenomena observed or sampled through signal sequences or time realizations whose dynamics come from heterogeneous sources of uncertainty. This requirement applies to many scientific contexts, and results particularly challenging when a low signal-to-noise ratio exists, due to structural or experimental conditions, and the information core appears dispersed in a wide spectrum of frequency bands or resolution levels. The methodological and empirical work that is here presented aims to design ad hoc approximation instruments dealing with a particularly complex class of random processes that generates financial returns, or their aggregates in the form of index returns. It is important to note that the underlying volatility function appears to be subject to the signature of noise and the masking effects of non-stationary superimposed dynamics, together with multi-scaling regimes. Due to the unobservability of the volatility function, its recovery represents an inverse problem that can be cast in a latent variable model designed to account for both switching multi-scaling regime and cascade system dynamics. Together with emphasizing the role of independent component analysis for achieving dimensionality reduction of the addressed inverse problem, we also stress the role of atomic functional dictionaries in improving the volatility feature detection power, and show the performance of greedy approximation algorithms in delivering sparse representations and coherent decompositions of the return sequences.
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