Multiple Testing and Variable Selection along Least Angle Regression's path

In this article, we investigate multiple testing and variable selection using Least Angle Regression (LARS) algorithm in high dimensions under the Gaussian noise assumption. LARS is known to produce a piecewise affine solutions path with change points referred to as knots of the LARS path. The cornerstone of the present work is the expression in closed form of the exact joint law of K-uplets of knots conditional on the variables selected by LARS, namely the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our finding. Our main contributions are three fold. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. This testing procedures are referred to as the Generalized t-Spacing tests (GtSt) and we prove that they have exact non-asymptotic level (i.e., Type I error is exactly controlled). In that way, we extend a work from (Taylor et al., 2014) where the Spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple false negatives test after model selection in the general design case when the noise level can be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Last, we give an exact control of the false discovery rate (FDR) under orthogonal design assumption. Monte-Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of LARS algorithm based on a recursive function.Comment: 62 pages; new: FDR control and power comparison between Knockoff, FCD, Slope and our proposed method; new: the introduction has been revised and now present a synthetic presentation of the main results. We believe that this introduction brings new insists compared to previous version

A Rice method proof of the Null-Space Property over the Grassmannian

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell\_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.Comment: 18 Pages, some Figure

The tail of the maximum of smooth Gaussian fields on fractal sets

We study the probability distribution of the maximum $M_S$ of a smooth stationary Gaussian field defined on a fractal subset $S$ of $\R^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $\P(M_S>u)$ as $u\rightarrow +\infty.$ The basic tool is Rice formula for the moments of the number of local maxima of a random field

Special session: Hot topics: Statistical test methods

International audienceThe process of testing Integrated Circuits involves a huge amount of data: electrical circuit measurements, information from wafer process monitors, spatial location of the dies, wafer lot numbers, etc. In addition, the relationships between faults, process variations and circuit performance are likely to be very complex and non-linear. Test (and its extension to diagnosis) should be considered as a challenging highly dimensional multivariate problem.Advanced statistical data processing offers a powerful set of tools, borrowed from the fields of data mining, machine learning or artificial intelligence, to get the most out of this data. Indeed, these mathematical tools have opened a number of novel and interesting research lines within the field of IC testing.In this special session, prominent researchers in this field will share their views on this topic and present some of their last findings. The first talk will discuss the interest of likelihood prevalence in random fault simulation. The second talk will show how statistical data analysis can help diagnosing test efficiency. The third talk will deal with the reliability of Alternate Test of AMS-RF circuits. The fourth and last talk will address the idea of mining the test data for improving design manufacturing and even test itself

Estimation of Piecewise-Deterministic Trajectories in a Quantum Optics Scenario

The manipulation of individual copies of quantum systems is one of the most groundbreaking experimental discoveries in the field of quantum physics. On both an experimental and a theoretical level, it has been shown that the dynamics of a single copy of an open quantum system is a trajectory of a piecewise-deterministic process. To the best of our knowledge, this application field has not been explored by the literature in applied mathematics, from both probabilistic and statistical perspectives. The objective of this chapter is to provide a self-contained presentation of this kind of model, as well as its specificities in terms of observations scheme of the system, and a first attempt to deal with a statistical issue that arises in the quantum world

Sign changes as a universal concept in first-passage-time calculations

First-passage-time problems are ubiquitous across many fields of study including transport processes in semiconductors and biological synapses, evolutionary game theory and percolation. Despite their prominence, first-passage-time calculations have proven to be particularly challenging. Analytical results to date have often been obtained under strong conditions, leaving most of the exploration of first-passage-time problems to direct numerical computations. Here we present an analytical approach that allows the derivation of first-passage-time distributions for the wide class of non-differentiable Gaussian processes. We demonstrate that the concept of sign changes naturally generalises the common practice of counting crossings to determine first-passage events. Our method works across a wide range of time-dependent boundaries and noise strengths thus alleviating common hurdles in first-passage-time calculations

Extremes of Gaussian random fields with regularly varying dependence structure

Let be a centered Gaussian random field with variance function sigma (2)(ai...) that attains its maximum at the unique point , and let . For a compact subset of a"e, the current literature explains the asymptotic tail behaviour of under some regularity conditions including that 1 - sigma(t) has a polynomial decrease to 0 as t -> t (0). In this contribution we consider more general case that 1 - sigma(t) is regularly varying at t (0). We extend our analysis to Gaussian random fields defined on some compact set , deriving the exact tail asymptotics of for the class of Gaussian random fields with variance and correlation functions being regularly varying at t (0). A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics

In Situ NMR Spectroscopy of Supercapacitors: Insight into the Charge Storage Mechanism

Electrochemical capacitors, commonly known as supercapacitors, are important energy storage devices with high power capabilities and long cycle lives. Here we report the development and application of in situ nuclear magnetic resonance(NMR) methodologies to study changes at the electrode−electrolyte interface in working devices as they charge and discharge. For a supercapacitor comprising activated carbon electrodes and an organic electrolyte, NMR experiments carried out at different charge states allow quantification of the number of charge storing species and show that there are at least two distinct charge storage regimes. At cell voltages below 0.75 V, electrolyte anions are increasingly desorbed from the carbon micropores at the negative electrode, while at the positive electrode there is little change in the number of anions that are adsorbed as the voltage is increased. However, above a cell voltage of 0.75 V, dramatic increases in the amount of adsorbed anions in the positive electrode are observed while anions continue to be desorbed at the negative electrode. NMR experiments with simultaneous cyclic voltammetry show that supercapacitor charging causes marked changes to the local environments of charge storing species, with periodic changes of their chemical shift observed. NMR calculations on a model carbon fragment show that the addition and removal of electrons from a delocalized system should lead to considerable increases in the nucleus-independent chemical shift of nearby species, in agreement with our experimental observations

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of ${\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \ \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} },\ \ u\to \infty ,$ where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena