Section: Overall Objectives

Overall objectives

singularity exponent

A measure of the unpredictability around a point in a complex signal. Based on local reconstruction around a point, singularity exponents can be evaluated in different ways and in different contexts (e.g. non-localized, through the consideration of moments and structure fonctions, leading to singularity spectra). In GEOSTAT we study approaches corresponding to far from equilibrium hypothesis (e.g. microcanonical) leading to geometrically localized singulariy exponents.


Local Predictability Exponent: another name for singularity exponents, that better underlines the relation with predictability.

Framework of reconstructible systems

Complex systems whose acquisitions can be reconstructed from the knowledge of the geometrical sets that maximize statistical information content. Study of complex signals' compact representations associated to unpredictability.


Microcanonical Multiscale Formalism.

Sparse representation

The representation of a signal as a linear combination of elements taken in a dictionary, with the aim of finding the sparset possible one.

Optimal wavelet

(OW). Wavelets whose associated multiresolution analysis optimizes inference along the scales in complex systems.

GEOSTAT is a research project in nonlinear digital signal processing, with the fundamental distinction that it considers the signals as the realizations of complex dynamic systems. The research in GEOSTAT encompasses nonlinear signal processing and the study of emergence in complex systems, with a strong emphasis on geometric approaches to complexity. Consequently, research in GEOSTAT is oriented towards the determination, in real signals, of quantities or phenomena that are known to play an important role both in the evolution of dynamical systems whose acquisitions are the signals under study, and in the compact representations of the signals themselves. Hence we first mention:

  • Singularity exponents, also called Local Predictability Exponents or LPEs,

  • how singularity exponents can be related to sparse representations with reconstruction formulae,

  • comparison with embedding techniques, such as the one provided by the classical theorem of Takens [39] , [31] .

  • Lyapunov exponents, how they are related to intermittency, large deviations and singularity exponents,

  • various forms of entropies,

  • multiresolution analysis, specifically when performed on the singularity exponents,

  • the cascading properties of associated random variables,

  • persistence along the scales, optimal wavelets,

  • the determination of subsets where statistical information is maximized, their relation to reconstruction and compact representation,

and, above all, the ways that lead to effective numerical and high precision determination of nonlinear characteristics in real signals. The MMF (Multiscale Microcanonical Formalism) is one of the ways to partly unlock this type of analysis, most notably w.r.t. LPEs and reconstructible systems [8] . We presently concentrate our efforts on it, but GEOSTAT is intended to explore other ways [27] . Presently GEOSTAT explores new methods for analyzing and understanding complex signals in different applicative domains through the theoretical advances of the MMF, and the framework of reconstructible systems [40] . Derived from ideas in Statistical Physics, the methods developped in GEOSTAT provide new ways to relate and evaluate quantitatively the local irregularity in complex signals and systems, the statistical concepts of information content and most informative subset. That latter notion is developed through the notion of transition front and Most Singular Manifold. As a result, GEOSTAT is aimed at providing radically new approaches to the study of signals acquired from different complex systems (their analysis, their classification, the study of their dynamical properties etc.). The common characteristic of these signals, as required by universality classes [35] [36] [33] , being the existence of a multiscale organization of the systems. For instance, the classical notion of edge or border, which is of multiscale nature, and whose importance is well known in Computer Vision and Image Processing, receives profound and rigorous new definitions, in relation with the more physical notion of transition and fits adequately to the case of chaotic data. The description is analogous to the modeling of states far from equilibrium, that is to say, there is no stationarity assumption. From this formalism we derive methods able to determine geometrically the most informative part in a signal, which also defines its global properties and allows for compact representation in the wake of known problematics addressed, for instance, in time-frequency analysis. In this way, the MMF allows the reconstruction, at any prescribed quality threshold, of a signal from its most informative subset, and is able to quantitatively evaluate key features in complex signals (unavailable with classical methods in Image or Signal Processing). It appears that the notion of transition front in a signal is much more complex than previously expected and, most importantly, related to multiscale notions encountered in the study of nonlinearity [37] . For instance, we give new insights to the computation of dynamical properties in complex signals, in particular in signals for which the classical tools for analyzing dynamics give poor results (such as, for example, correlation methods or optical flow for determining motion in turbulent datasets). The problematics in GEOSTAT can be summarized in the following items:

  • the accurate determination in any n-dimensional complex signal of LPEs at every point in the signal domain  [41] [13] .

  • The geometrical determination and organization of singular manifolds associated to various transition fronts in complex signals, the study of their geometrical arrangement, and the relation of that arrangement with statistical properties or other global quantities associated to the signal, e.g. cascading properties [9] .

  • The study of the relationships between the dynamics in the signal and the distributions of LPEs [42] [9] .

  • Multiresolution analysis and inference along the scales [9] , [2] .

  • The study of the relationships between the distributions of LPEs and other formalisms associated to predictability in complex signals and systems, such as cascading variables, large deviations and Lyapunov exponents.

  • The ability to compute optimal wavelets and relate such wavelets to the geometric arrangement of singular manifolds and cascading properties[3] .

  • The translation of recognition, analysis and classification problems in complex signals to simpler and more accurate determinations involving new operators acting on singular manifolds using the framework of reconstructible systems.