Large classes of signals exhibit a very irregular behavior. In the
most complicated situations, this irregular behavior may follow
different regimes, and can switch from one regime to another almost
instantaneously. Such signals cannot be modelled by standard stationary
increments processes, such as Fractional Brownian Motions (or related
gaussian processes) for instance.
The techniques of multifractal signal analysis have been specifically
designed to analyze such behaviors. Initially developed in the mid 80's
in the context of turbulence analysis, they were applied successfully to a
large range of signals, including traffic data (cars and internet!),
stock market prices, speech signals, texture analysis of images, ...
We will give an overview of the mathematical tools that were
developed for that purpose (Holder regularity, Spectrum of singularities,
thermodynamic formalism, chirps and oscillating singularities), and we will
present some of the most successful applications.