Context Tree Selection: A Unifying View

The present paper investigates non-asymptotic properties of two popular procedures of context tree (or Variable Length Markov Chains) estimation: Rissanen's algorithm Context and the Penalized Maximum Likelihood criterion. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning overestimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The underestimation properties rely on loss-of-memory and separation conditions of the process. These results improve and generalize the bounds obtained previously. Context tree models have been introduced by Rissanen as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics

A past discharges assimilation system for ensemble streamflow forecasts over France – Part 1: Description and validation of the assimilation system

International audienceTwo Ensemble Streamflow Prediction Systems (ESPSs) have been set up at M´et´eo-France. They are based on the French SIM distributed hydrometeorological model. A deterministic analysis run of SIM is used to initialize the two ESPSs. In order to obtain a better initial state, a past discharges assimilation system has been implemented into this analysis SIM run, using the Best Linear Unbiased Estimator (BLUE). Its role is to improve the model soil moisture by using streamflow observations in order to better simulate streamflow. The skills of the assimilation system were assessed for a 569-day period on six different configurations, including two different physics schemes of the model (the use of an exponential profile of hydraulic conductivity or not) and, for each one, three different ways of considering the model soil moisture in the BLUE state variables. Respect of the linearity hypothesis of the BLUE was verified by assessing of the impact of iterations of the BLUE. The configuration including the use of the exponential profile of hydraulic conductivity and the combination of the moisture of the two soil layers in the state variable showed a significant improvement of streamflow simulations. It led to a significantly better simulation than the reference one, and the lowest soil moisture corrections. These results were confirmed by the study of the impacts of the past discharge assimilation system on a set of 49 independent stations

Aubry sets vs Mather sets in two degrees of freedom

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide.Comment: Revised and expanded version. New proof of Lemma 2.3 (formerly Lemma 14

A past discharge assimilation system for ensemble streamflow forecasts over France – Part 2: Impact on the ensemble streamflow forecasts

International audienceThe use of ensemble streamflow forecasts is developing in the international flood forecasting services. Ensemble streamflow forecast systems can provide more accurate forecasts and useful information about the uncertainty of the forecasts, thus improving the assessment of risks. Nevertheless, these systems, like all hydrological forecasts, suffer from errors on initialization or on meteorological data, which lead to hydrological prediction errors. This article, which is the second part of a 2-part article, concerns the impacts of initial states, improved by a streamflow assimilation system, on an ensemble streamflow prediction system over France. An assimilation system was implemented to improve the streamflow analysis of the SAFRAN-ISBAMODCOU (SIM) hydro-meteorological suite, which initializes the ensemble streamflow forecasts at M´et´eo-France. This assimilation system, using the Best Linear Unbiased Estimator (BLUE) and modifying the initial soil moisture states, showed an improvement of the streamflow analysis with low soil moisture increments. The final states of this suite were used to initialize the ensemble streamflow forecasts of M´et´eo-France, which are based on the SIM model and use the European Centre for Medium-range Weather Forecasts (ECMWF) 10-day Ensemble Prediction System (EPS). Two different configurations of the assimilation system were used in this study: the first with the classical SIM model and the second using improved soil physics in ISBA. The effects of the assimilation system on the ensemble streamflow forecasts were assessed for these two configurations, and a comparison was made with the original (i.e. without data assimilation and without the improved physics) ensemble streamflow forecasts. It is shown that the assimilation system improved most of the statistical scores usually computed for the validation of ensemble predictions (RMSE, Brier Skill Score and its decomposition, Ranked Probability Skill Score, False Alarm Rate, etc.), especially for the first few days of the time range. The assimilation was slightly more efficient for small basins than for large ones

Assimilation of atmospheric methane products into the MACC-II system: From SCIAMACHY to TANSO and IASI

The Monitoring Atmospheric Composition and Climate Interim Implementation (MACC-II) delayed-mode (DM) system has been producing an atmospheric methane (CH4) analysis 6 months behind real time since June 2009. This analysis used to rely on the assimilation of the CH4 product from the SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY) instrument onboard Envisat. Recently the Laboratoire de Météorologie Dynamique (LMD) CH4 products from the Infrared Atmospheric Sounding Interferometer (IASI) and the SRON Netherlands Institute for Space Research CH4 products from the Thermal And Near-infrared Sensor for carbon Observation (TANSO) were added to the DM system. With the loss of Envisat in April 2012, the DM system now has to rely on the assimilation of methane data from TANSO and IASI. This paper documents the impact of this change in the observing system on the methane tropospheric analysis. It is based on four experiments: one free run and three analyses from respectively the assimilation of SCIAMACHY, TANSO and a combination of TANSO and IASI CH4 products in the MACC-II system. The period between December 2010 and April 2012 is studied. The SCIAMACHY experiment globally underestimates the tropospheric methane by 35 part per billion (ppb) compared to the HIAPER Pole-to-Pole Observations (HIPPO) data and by 28 ppb compared the Total Carbon Column Observing Network (TCCON) data, while the free run presents an underestimation of 5 ppb and 1 ppb against the same HIPPO and TCCON data, respectively. The assimilated TANSO product changed in October 2011 from version v.1 to version v.2.0. The analysis of version v.1 globally underestimates the tropospheric methane by 18 ppb compared to the HIPPO data and by 15 ppb compared to the TCCON data. In contrast, the analysis of version v.2.0 globally overestimates the column by 3 ppb. When the high density IASI data are added in the tropical region between 30° N and 30° S, their impact is mainly positive but more pronounced and effective when combined with version v.2.0 of the TANSO products. The resulting analysis globally underestimates the column-averaged dry-air mole fractions of methane (xCH4) just under 1 ppb on average compared to the TCCON data, whereas in the tropics it overestimates xCH4 by about 3 ppb. The random error is estimated to be less than 7 ppb when compared to TCCON data

Sparsity and Incoherence in Compressive Sampling

We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds $$m\geq \mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n,$$ where $S$ is the number of nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples