Description, modelling and forecasting of data with optimal wavelets

Cascade processes have been used to model many different self-similar systems, as they are able to accurately describe most of their global statistical properties. The so-called optimal wavelet basis allows to achieve a geometrical representation of the cascade process-named microcanonical cascade-that describes the behavior of local quantities and thus it helps to reveal the underlying dynamics of the system. In this context, we study the benefits of using the optimal wavelet in contrast to other wavelets when used to define cascade variables, and we provide an optimality degree estimator that is appropriate to determine the closest-to-optimal wavelet in real data. Particularizing the analysis to stock market series, we show that they can be represented by microcanonical cascades in both the logarithm of the price and the volatility. Also, as a promising application in forecasting, we derive the distribution of the value of next point of the series conditioned to the knowledge of past points and the cascade structure, i.e., the stochastic kernel of the cascade process.