Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases

We investigate the optimality for model selection of the so-called slope heuristics, $V$-fold cross-validation and $V$-fold penalization in a heteroscedastic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics---when the optimal penalty shape is known---and $V$ -fold penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for a fixed value of $V$ since it recovers asymptotically the oracle learned from a sample size equal to $1-V^{-1}$ of the original amount of data. Our results are based on genuine concentration inequalities for the true and empirical excess risks that are of independent interest. We show in our experiments the good behavior of the slope heuristics for the selection of linear wavelet models. Furthermore, $V$-fold cross-validation and $V$-fold penalization have comparable efficiency

Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model

We observe $n$ heteroscedastic stochastic processes $\{Y_v(t)\}_{v}$, where for any $v\in\{1,\ldots,n\}$ and $t \in [0,1]$, $Y_v(t)$ is the convolution product of an unknown function $f$ and a known blurring function $g_v$ corrupted by Gaussian noise. Under an ordinary smoothness assumption on $g_1,\ldots,g_n$, our goal is to estimate the $d$-th derivatives (in weak sense) of $f$ from the observations. We propose an adaptive estimator based on wavelet block thresholding, namely the "BlockJS estimator". Taking the mean integrated squared error (MISE), our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax in the least favorable situation. We also report a comprehensive suite of numerical simulations to support our theoretical findings. The practical performance of our block estimator compares very favorably to existing methods of the literature on a large set of test functions

On the estimation of density-weighted average derivative by wavelet methods under various dependence structures

International audienceThe problem of estimating the density-weighted average derivative of a regression function is considered. We present a new consistent estimator based on a plug-in approach and wavelet projections. Its performances are explored under various dependence structures on the observations: the independent case, the $\rho$-mixing case and the $\alpha$-mixing case. More precisely, denoting $n$ the number of observations, in the independent case, we prove that it attains $1/n$ under the mean squared error, in the $\rho$-mixing case, $1/\sqrt{n}$ under the mean absolute error, and, in the $\alpha$-mixing case, $\sqrt{\ln n /n}$ under the mean absolute error. A short simulation study illustrates the theory

Dynamic Reconfiguration for Software and Hardware Heterogeneous Real-time WSN

International audienceWireless Sensor Network (WSN) technology has imposed itself in civilian and industrial applications as a promising technology for wireless monitoring due to its wireless connectivity, removing many hardware constraints. Initially used in low frequency sampling applications, the increasing performances of electronic circuits has driven WSNs to integrate more powerful computation units, paving the way for a new generation of applications based on distributed computation. These new applications (process control, active control, visual surveillance, multimedia streaming) involving medium to heavy computation present real-time requirements at node level where reactivity becomes a primary concern as well as at the network level where latency must be bounded. In this paper, we present the implementation of a high-level language MinTax coupled with an in-situ compilation solution for real time Operating Systems enabling energy-aware dynamic reconfiguration while supporting hardware heterogeneity in Wireless Sensor Networks

Heterogeneous Wireless Sensor Network Simulation

International audienceBased on our previous work on the development of a Wireless Sensor Network (WSN) simulation platform, we present here its ability to run simulations on heterogeneous nodes. This platform allows system-level simulations with low level accurate models, with graphical inputs and outputs to easily simulate such distributed systems. In the testbed we consider, the well known IEEE 802.15.4 standard is used, and different microcontrollers units (MCU) and radiofrequency transceivers compose the heterogeneous nodes. It is also possible to simulate complex networks or interacting networks; that is a more realistic case, as more and more hardware devices exist and standards permit their interoperability. This simulation platform can be used to explore design space in order to find the hardware devices and IEEE 802.15.4 algorithm that best fit a given application. Packet Delivery Rate (PDR) and packet latency can be evaluated, as other network simulators do. Energy consumption of sensor nodes is detailed with a very fine granularity: partitioning over and into hardware devices that compose the node is studied

On adaptive wavelet estimation of a class of weighted densities

We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations

Nonparametric estimation in a regression model with additive and multiplicative noise

In this paper, we consider an unknown functional estimation problem in a general nonparametric regression model with the characteristic of having both multiplicative and additive noise. We propose two wavelet estimators, which, to our knowledge, are new in this general context. We prove that they achieve fast convergence rates under the mean integrated square error over Besov spaces. The rates obtained have the particularity of being established under weak conditions on the model. A numerical study in a context comparable to stochastic frontier estimation (with the difference that the boundary is not necessarily a production function) supports the theory