Section: New Results
Semi and non-parametric methods
Modelling extremal events
Participants : Stéphane Girard, El-Hadji Deme.
Joint work with: L. Gardes (Univ. Strasbourg) and E. Deme (Univ. Gaston Berger, Sénégal)
We are working on the estimation of the second
order parameter
In addition to this work, we have established a review on the Weibull-tail distributions [29] .
Conditional extremal events
Participants : Stéphane Girard, Gildas Mazo, Jonathan El-Methni.
Joint work with: L. Gardes (Univ. Strasbourg) and A. Daouia (Univ. Toulouse I and Univ. Catholique de Louvain)
The goal of the PhD thesis of Alexandre Lekina was to contribute to
the development of theoretical and algorithmic models to tackle
conditional extreme value analysis, ie the situation where
some covariate information
Conditional extremes are studied in climatology where one is interested in how climate change over years might affect extreme temperatures or rainfalls. In this case, the covariate is univariate (time). Bivariate examples include the study of extreme rainfalls as a function of the geographical location. The application part of the study is joint work with the LTHE (Laboratoire d'étude des Transferts en Hydrologie et Environnement) located in Grenoble.
Estimation of extreme risk measures
Participants : Stéphane Girard, Jonathan El-Methni, El-Hadji Deme.
Joint work with: L. Gardes and A. Guillou (Univ. Strasbourg)
One of the most popular risk measures is the Value-at-Risk (VaR) introduced in the 1990's.
In statistical terms,
the VaR at level
Multivariate extremal events
Participants : Stéphane Girard, Gildas Mazo, Florence Forbes, Van Trung Pham.
Joint work with: C. Amblard (TimB in TIMC laboratory, Univ. Grenoble I) and L. Menneteau (Univ. Montpellier II)
Copulas are a useful tool to model multivariate distributions [72] . At first, we developed an extension of some particular copulas [1] . It followed a new class of bivariate copulas defined on matrices [56] and some analogies have been shown between matrix and copula properties.
However, while there exist various families of bivariate copulas, much fewer has been done when the dimension is higher. To this aim an interesting class of copulas based on products of transformed copulas has been proposed in the literature. The use of this class for practical high dimensional problems remains challenging. Constraints on the parameters and the product form render inference, and in particular the likelihood computation, difficult. We proposed a new class of high dimensional copulas based on a product of transformed bivariate copulas [61] . No constraints on the parameters refrain the applicability of the proposed class which is well suited for applications in high dimension. Furthermore the analytic forms of the copulas within this class allow to associate a natural graphical structure which helps to visualize the dependencies and to compute the likelihood efficiently even in high dimension. The extreme properties of the copulas are also derived and an R package has been developed.
As an alternative, we also proposed a new class of copulas constructed by introducing a latent factor. Conditional independence with respect to this factor and the use of a nonparametric class of bivariate copulas lead to interesting properties like explicitness, flexibility and parsimony. In particular, various tail behaviours are exhibited, making possible the modeling of various extreme situations. A pairwise moment-based inference procedure has also been proposed and the asymptotic normality of the corresponding estimator has been established [53] .
Level sets estimation
Participant : Stéphane Girard.
Joint work with: A. Guillou and L. Gardes (Univ. Strasbourg), G. Stupfler (Univ. Aix-Marseille) and A. Daouia (Univ. Toulouse I and Univ. Catholique de Louvain)
The boundary bounding the set of points is viewed as the larger level set of the points distribution. This is then an extreme quantile curve estimation problem. We proposed estimators based on projection as well as on kernel regression methods applied on the extreme values set, for particular set of points [10] . We also investigate the asymptotic properties of existing estimators when used in extreme situations. For instance, we have established in collaboration with G. Stupfler that the so-called geometric quantiles have very counter-intuitive properties in such situations [60] and thus should not be used to detect outliers.
In collaboration with A. Daouia, we investigate the application of such methods in econometrics [17] : A new characterization of partial boundaries of a free disposal multivariate support is introduced by making use of large quantiles of a simple transformation of the underlying multivariate distribution. Pointwise empirical and smoothed estimators of the full and partial support curves are built as extreme sample and smoothed quantiles. The extreme-value theory holds then automatically for the empirical frontiers and we show that some fundamental properties of extreme order statistics carry over to Nadaraya's estimates of upper quantile-based frontiers.
In collaboration with G. Stupfler and A. Guillou, new estimators of the boundary are introduced. The regression is performed on the whole set of points, the selection of the “highest” points being automatically performed by the introduction of high order moments [26] , [27] .
Retrieval of Mars surface physical properties from OMEGA hyperspectral images.
Participants : Stéphane Girard, Alessandro Chiancone.
Joint work with: S. Douté from Laboratoire de Planétologie de Grenoble, J. Chanussot (Gipsa-lab and Grenoble-INP) and J. Saracco (Univ. Bordeaux).
Visible and near infrared imaging spectroscopy is
one of the key techniques
to detect, to map and to characterize mineral and volatile (eg.
water-ice)
species existing at
the surface of planets. Indeed the chemical composition,
granularity, texture, physical state, etc. of the materials
determine the existence and morphology of the absorption bands.
The resulting spectra contain therefore very useful information.
Current imaging spectrometers provide data organized as three
dimensional hyperspectral images: two spatial dimensions and one
spectral dimension. Our goal is to estimate the functional
relationship
High-dimensional change-point detection with sparse alternatives
Participant : Farida Enikeeva.
Joint work with: Zaid Harchaoui from LEAR team Inria Grenoble
The change-point problem is a classical problem of statistics that arises in various applications as signal processing, bioinformatics, financial market analysis. The goal of change-point problems is to make an inference about the moment of a change in the distribution of the observed data. We consider the problem of detection of a simultaneous change in mean in a sequence of Gaussian vectors.
The state-of-the-art approach to the change-point detection/estimation is based on the assumption of growing number of observations and fixed dimension of the signal. We work in high-dimensional setting assuming that the vector dimension tends to infinity and the length of the sequence grows slower than the dimension of the signal. Assuming that the change occurs only in a subset of the vector components of unknown cardinality we can reduce our problem to the problem of testing non-zero components in a sequence of sparse Gaussian vectors. We construct a testing procedure that is adaptive to the number of components with a change. This testing procedure is based on combination of two chi-squared type test statistics. This combined test provides an optimal performance of the test both in the cases of high and moderate sparsity. We obtain the detection boundary of the test and show its rate-optimality in minimax sense.
The results of the paper [59] were presented at
-
NIPS 2013, Workshop on Modern Nonparametric Methods in Machine Learning (Dec. 2013)
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Conference on Structural Inference in Statistics, Potsdam, Germany (Sept. 2013)
Yield Improvement by the Redundancy Method for Component Calibration
Participant : Farida Enikeeva.
Joint work with: Dominique Morche (CEA-LETI) and Alp Oguz (CEA-LETI)
This work [23] was done in the framework of the Optimyst II project of MINALOGIC in collaboration with CEA-LETI and LJK-UJF. In this project we explore the benefits of the redundant channels methodology for the calibration of electronic components.
The demand for high data rate in communication puts stringent requirements on components' dynamic range. However, the extreme size reduction in advanced technology results inadvertently in increased process variability, which inherently limits the performances. The redundancy approach is based on the idea of dividing an elementary component (capacitor, resistor, transistor) into several subsets and then choosing an optimal combination of such subsets to provide the production of a component with very precise characteristics. For several years, the redundancy method has been identified as complementary to digital calibration to improve the performances. On practice, it is hard for a designer to select an optimal number of redundant components to provide the desired production yield and to minimize the area occupied by the components. The usual way to solve this problem is to resort to statistical simulations which are time consuming and sometimes misleading. We propose a normal approximation of the yield in order to estimate the number of redundant components needed to provide a minimal area occupied by the components.