BLANC - Blanc

Stationnarité relative et approches connexes – StaRAC

Submission summary

The word « stationarity » is ubiquitous in signal processing and data analysis but, often used in a loose sense, it may correspond to different qualities that are not necessarily captured by what is referred to as « stationarity » in textbooks. Classically, « stationarity » refers to stochastic processes and is defined as the invariance in time of statistical properties or, in other words, as the independence of those properties with respect to some absolute time. In practice, however, stationarity is commonly advocated in rather different contexts and/or with additional features that may enter the picture. First, an observation scale is commonly and implicitly used, which should enter the definition (a very same signal can be considered as, e.g., « short-term stationary » and « long-term nonstationary »). Second, whereas the classical definition is given in a stochastic framework, stationarity is also often used for deterministic signals exhibiting periodicity properties. Third, though classical stationarity corresponds to an invariance with respect to shifts (in time or space), other forms of invariance or symmetries are known that can be interpreted as a generalized form of stationarity: for instance self-similarity is an invariance with respect to dilations, or isotropy with respect to rotations. These considerations serve as motivations for the scientific objectives of StaRAC: propose and develop operational (i.e., interpretable, relative and testable) approaches to the concept of stationarity, with the purpose of filling existing gaps between theory and practice; extend the concept of stationarity to general groups of transformations; develop new methods to measure, test and model departures from stationarity in the two above contexts. The project is primarily of a theoretical and methodological nature, but its implications are obvious in terms of applications too, especially for real signals with meaningful temporal evolution (such as sound) or coming from systems with specific symmetries (as is the case for experiments in physics). Concerning methods, signal processing is equipped with many powerful algorithms devoted to stationary processes, whose applicability should therefore be first assessed prior using them, and whose modeling can pave the road for adaptive extensions. Turning to interpretation, rejecting stationarity is of primary importance in numerous domains ranging from exploratory data analysis to diagnosis or surveillance. As far as generalized forms of stationarity are concerned, the interest is twofold. On the one hand, stationarizing processes obeying other symmetries makes them enter standard frameworks and eases their processing. On the other hand, new tools specifically adapted to given forms of nonstationarities, or of other kinds of symmetries, may result from a proper de-stationarization of the classical stationary tools. The key point of the approaches to be developed in StaRAC is to make use of adapted representation spaces. A time-frequency framework (at large) will allow to address the three issues: i) defining stationarity from its interpretation as a property relative to a measurement scale; ii) providing tools to estimate the time-frequency features needed to assess the stationarity of a signal; iii) designing, from these features, methods to quantify nonstationarities and their statistical relevance. Major new insights on this issue will emerge from the combination of several ideas: the comparison of local vs. global analyses, the use of surrogate data as a stationary learning set, and the use of statistical learning theory to go beyond the basics of testing for stationarity. The existing interplay between kernel methods and time-frequency analysis will be used both for this last aspect of statistical testing, for the design of adapted time-frequency schemes and associated features, and for effective adaptive modeling. Generalizations of stationarity find their formal background in a seminal work by Hannan, but they have rarely been declined as practical tools, excepted in specific cases such as the Lamperti transformation (dedicated to self-similarity) and some extensions. A contribution of the project will be to construct warpings from shifts to new symmetry operators, even in cases where the warping is not an invertible function any more. Based on adaptive and/or non-uniform sampling theories, practical schemes will be be developed. Multidimensional generalizations with richer symmetries are envisioned too, as well as situations where « broken » symmetries are encountered. Finally, it has to be remarked that all the above-mentioned issues are not independent, in the sense that, e.g., stationarity tests may be ultimately considered in a generalized sense.

Project coordination

Patrick FLANDRIN (Organisme de recherche)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

Help of the ANR 225,365 euros
Beginning and duration of the scientific project: - 36 Months

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