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Personne# Lionel Coulot

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Thierry Blu, Lionel Coulot, Pier Luigi Dragotti, Pina Marziliano, Martin Vetterli

Sparse sampling of continuous-time sparse signals is addressed. In particular, it is shown that sampling at the rate of innovation is possible, in some sense applying Occam's razor to the sampling of sparse signals. The noisy case is analyzed and solved, proposing methods reaching the optimal performance given by the Cramer-Rao bounds. Finally, a number of applications have been discussed where sparsity can be taken advantage of. The comprehensive coverage given in this article should lead to further research in sparse sampling, as well as new applications. One main application to use the theory presented in this article is ultra-wide band (UWB) communications.

We introduce in this thesis the idea of a variable lookback model, i.e., a model whose predictions are based on a variable portion of the information set. We verify the intuition of this model in the context of experimental finance. We also propose a novel algorithm to estimate it, the variable lookback algorithm, and apply the latter to build investment strategies. Financial markets under information asymmetry are characterized by the presence of better-informed investors, also called insiders. The literature in finance has so far concentrated on theoretical models describing such markets, in particular on the role played by the price in conveying information from informed to uninformed investors. However, the implications of these theories have not yet been incorporated into processing methods to extract information from past prices and this is the aim of this thesis. More specifically, the presence of a time-varying number of insiders induces a time-varying predictability in the price process, which calls for models that use a variable lookback window. Moreover, although our initial motivation comes from the study of markets under information asymmetry, the problem is more general, as it touches several issues in statistical modeling. The first one concerns the structure of the model. Existing methods use a fixed model structure despite evidences from data, which support an adaptive one. The second one concerns the improper handling of the nonstationarity in data. The stationarity assumption facilitates the mathematical treatment. Hence, existing methods relies on some form of stationarity, for example, by assuming local stationary, as in the windowing approach, or by modeling the underlying switching process, for example, with a Markov chain of order 1. However, these suffer from certain limitations and more advanced methods that take explicitly into account the nonstationariry of the signal are desirable. In summary, there is a need to develop a method that constantly monitors what is the appropriate structure, when a certain model works and when not or when are the underlying assumptions of the model violated. We verify our initial intuition in the context of experimental finance. In particular, we highlight the diffusion of information in the market. We give a precise definition to the notion of the time of maximally informative price and verify, in line with existing theories, that the time of maximally informative price is inversely proportional to the number of insiders in the market. This supports the idea of a variable lookback model. Then, we develop an estimation algorithm that selects simultaneously the order of the process and the lookback window based on the minimum description length principle. The algorithm maintains a series of estimators, each based on a different order and/or information set. The selection is based on an information theoretic criterion, that accounts for the ability of the model to fit the data, penalized by the model complexity and the amount of switching between models. Finally, we put the algorithm at work and build investment strategies. We devise a method to draw dynamically the trend line for the time-series of log-prices and propose an adaptive version of the well-known momentum strategy. The latter outperforms standard benchmarks, in particular during the 2009 momentum crash.