Optimization methods for the analysis of scale invariant processes
Scale invariance relies on the intuition that temporal dynamics are not driven by one (or a few) characteristic scale(s). This property is massively used in the modeling and analysis of univariate data stemming from real-world applications. However, its use in practice encounters two difficulties when dealing with modern applications: scaling properties are not necessarily homogenous in time or space ; the multivariate nature of data leads to the minimization of highly non-linear and non-convex functionals in order to estimate the scaling parameters.The first originality of this work is to investigate the study of non-homogenous scale invariance as a joint problem of detection/segmentation and estimation, and to propose its formulation by the minimization of vectorial functionals constructed around a total variation penalization, in order to estimate both the boundaries delimiting the changes and the scaling properties within each region.The second originality lies in the design of a branch and bound minimization procedure of non-convex functional for the full identification of the bivariate extension of fractional Brownian motion, considered as the reference for modeling univariate scale invariance. Such procedure is applied in practice on Internet traffic data in the context of anomaly detection.Thirdly, we propose some contributions specific to total variation denoising: Poisson data-fidelity model related to a state detection problem in intermittent fluorescence ; automatic selection of the regularization parameter.