A microcanonical cascade formalism for multifractal systems and its application to data inference and forecasting

[eng] Complex systems are abundant in our natural environment. In linear systems, the equations of their dynamics can be very difficult to solve, but if they cannot be described with a single characteristic scale, at least they can be described by a set of few characteristic scales that are totally decoupled from each other. However, this takes on a completely different flavour in non-linear systems, where scales are coupled and appropriate multiscale analysis is in order. This is the case of complex systems and, more particularly, scale invariant systems. In these, the approach to their solution is different, and it usually involves a multiscale basis. In this context, wavelets are one of the most used representation paradigms. The research context of complex systems and, particularly, scale invariant systems and multifractals has been in constant evolution over the last few years. Theoretical advances, either statistical (stochastic processes and probability distributions) or geometrical (function analysis and measure theory), along with fancy signal-processing algorithms suited to scale invariant data (and additionally handling aliasing, discretization and other artefacts of experimental data), have originated new tools for multifractal characterization of systems. While ten years ago the only methods available were statistical, by the start of this thesis project, development of geometrical methods had begun (most notably, the microcanonical multifractal formalism (MMF)). Geometrical methods have a clear advantage over statistical methods: they characterize each point of the system and thus they permit new applications such as reconstruction and prediction of signals, i.e., not only statistical characterization. Additionally, geometrical methods provide statistical characterization with much less need of data than statistical methods. In the present thesis, we have worked on the generalization and improvement of MMF, as well as its applications to the inference and forecasting of systems that follow a cascade process. In particular, we have described applications to two very different systems: stock-market series and ocean turbulence. The representation of the signal as a microcanonical cascade plays a crucial role in these applications. This representation can be achieved with one particular wavelet called optimal wavelet. The most relevant theoretical achievements are the regularization of diverging multifractal measures, the establishment of the bridge between multiplicative variables in microcanonical cascade processes and local singularity exponents, and the design of accurate and robust measure of wavelet optimality for a given dataset. To achieve this, we have introduced a new formalism, that of microcanonical cascades, that marries the cascade formalisms with MMF. Regarding the developed applications, on stock-market time series, we have inferred the distribution of future returns conditioned by the cascade and we have shown that a prediction based on this inference improves that of an ARIMA model. From the distribution of future returns, future volatility and value-at-risk can be reliably forecasted. On ocean data we have characterized dynamical aspects from optimal wavelet cascade analysis. In particular, we have observed that anomalies in the cascade of sea surface temperature show particular points of heat transfer between structures at different scales in the zones of wind-driven currents, also in the gyres. Both understanding -- combined with appropriate modelling -- of dynamics and design of inference/forecasting algorithms have crucial importance for the anticipation of changes in natural phenomena. In this context, the chain formed by the three steps followed during the thesis, namely multifractal characterization first, then obtaining of the optimal wavelet and finally design of inference algorithms, summarizes the direction we have followed to tackle the study of econometric time series and ocean maps.[cat] Fenòmens naturals tant diversos com són la turbulència, les sèries economètriques i el camp magnètic heliocèntric tenen en comú el fet que són multifractals. La recent concepció de nous models que tenen en compte la presència de l'estructura multifractal estant permetent millorar la comprensió d'aquests fenòmens. D'igual manera, s'està explotant la capacitat d'aquests models en problemes com la codificació de dades amb mínim de redundància, la inferència de dades no disponibles i la previsió de l'evolució futura de la dinàmica del sistema. Malgrat aquests avenços, el disseny d'algorismes d'anàlisi multifractal de dades reals continua essent un repte important al qual fins ara s'ha donat resposta únicament de manera molt limitada. La presència de forats de dades, la discretització pròpia de les dades digitals, la presència de soroll i la de correlacions de llarg abast són dificultats comunes que necessiten ser tractades de forma curosa, mitjançant mètodes sofisticats, sobretot si es té en compte que les variables multifractals són intrínsecament irregulars a totes les escales i suavitzar-les implica modificar-ne les propietats. En aquesta tesi doctoral donem resposta als problemes esmentats per mitjà d'un formalisme multifractal basat en cascades multiplicatives microcanòniques. Mostrem que, si la base de representació de la cascada s'escull de forma adequada, les possibilitats d'inferència de la cascada milloren notablement i a més a més ho fan d'una forma robusta. En aquest sentit, mostrarem l'aplicació d'un model de cascada microcanònica per a la predicció de la distribució de cotització futura en sèries borsàries. Mostrarem també una altra aplicació d'un model de cascada microcanònica per a la detecció de transferències de calor entre escales a la superfície oceànica i com això permet identificar les zones de girs oceànics, les de ressorgiment d'aigües profundes i els corrents originats pels vents elisis