Section: New Results
Semi and non-parametric methods
Modelling extremal events
Joint work with:Myriam Garrido (INRA Clermont-Ferrand), Armelle Guillou (Univ. Strasbourg), and Jean Diebolt (CNRS, Univ. Marne-la-vallée).
Our first achievement is the development of new estimators dedicated to Weibull-tail distributions ( 3 ): kernel estimators [Oops!] and bias correction through exponential regression [Oops!] , [Oops!] . Our second achievement is the construction of a goodness-of-fit test for the distribution tail. Usual tests are not adapted to this problem since they essentially check the adequation to the central part of the distribution. The proposed method [Oops!] is based on the comparison between two estimators of quantiles: classical parametric estimators and extreme-value statistics based quantiles.
Conditional extremal events
Joint work with:Cécile Amblard (TimB in TIMC laboratory, Univ. Grenoble 1).
The goal of the PhD thesis of Alexandre Lekina is to contribute to the development of theoretical and algorithmic models to tackle conditional extreme value analysis, ie the situation where some covariate information X is recorded simultaneously with a quantity of interest Y . In such a case, the tail heaviness of Y depends on X, and thus the tail index as well as the extreme quantiles are also functions of the covariate. We will investigate how to combine nonparametric smoothing techniques  with extreme-value methods in order to obtain efficient estimators of the conditional tail index and conditional extreme quantiles. 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 (the time). Bivariate examples include the study of extreme rainfalls as a function of the geographical location. Interaction between extreme-value statistics and environmental sciences has been discussed at the Statistical Extremes and Environmental Risk Workshop [Oops!] . The application part of the study will be joint work with the LTHE (Laboratoire d'étude des Transferts en Hydrologie et Environnement) located in Grenoble.
More future work will include the study of multivariate extreme values. To this aim, a research on some particular copulas  ,  has been initiated with Cécile Amblard, since they are the key tool for building multivariate distributions  .
Joint work with:Anatoli Iouditski (Univ. Joseph Fourier, Grenoble), Guillaume Bouchard (Xerox, Meylan), Pierre Jacob and Ludovic Menneteau (Univ. Montpellier II) and Alexandre Nazin (IPU, Moscow, Russia).
Two different and complementary approaches are developped.
Extreme quantiles approach.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 propose estimators based on projection as well as on kernel regression methods applied on the extreme values set [Oops!] , for particular set of points. Our work is to define similar methods based on wavelets expansions in order to estimate non-smooth boundaries, and on local polynomials estimators to get rid of boundary effects [Oops!] . Besides, we are also working on the extension of our results to more general sets of points. This work has been initiated in the PhD work of Laurent Gardes  , co-directed by Pierre Jacob and Stéphane Girard and in [Oops!] with the consideration of star-shaped supports.
Linear programming approach.The boundary of a set of points is defined has a closed curve bounding all the points and with smallest associate surface. It is thus natural to reformulate the boundary estimation method as a linear programming problem. The resulting estimate is parsimonious, it only relies on a small number of points. This method belongs to the Support Vector Machines (SVM) techniques. Their finite sample performances are very impressive but their asymptotic properties are not very well known, the difficulty being that there is no explicit formula of the estimator. However, such properties are of great interest, in particular to reduce the estimator bias.
Nuclear plants reliability
Joint work with:Nadia Perot, Nicolas Devictor and Michel Marquès (CEA).
One of the main activities of the LCFR (Laboratoire de Conduite et Fiabilité des Réacteurs), CEA Cadarache, concerns the probabilistic analysis of some processes using reliability and statistical methods. In this context, probabilistic modelling of steels tenacity in nuclear plants tanks has been developed. The databases under consideration include hundreds of data indexed by temperature, so that, reliable probabilistic models have been obtained for the central part of the distribution. However, in this reliability problem, the key point is to investigate the behaviour of the model in the distribution tail. In particular, we are mainly interested in studying the lowest tenacities when the temperature varies (Figure 4 ).
A postdoctoral position on this problem, supported by the CEA, has been opened. Laurent Gardes and Stéphane Girard will co-advise the student. We are currenlty investigating the possibility to sign a research contract on this topic involving mistis and the LCFR.
Quantifying uncertainties on extreme rainfall estimations
Joint work with:Gilles Molinié from Laboratoire d'Etude des Transferts en Hydrologie et Environnement (LTHE), France.
Extreme rainfalls are generally associated with two different precipitation regimes. Extreme cumulated rainfall over 24 hours results from stratiform clouds on which the relief forcing is of primary importance. Extreme rainfall rates are defined as rainfall rates with low probability of occurrence, typically with higher mean return-periods than the observed time period (data length). It is then of primary importance to study the sensitivity of the extreme rainfall estimation to the estimation method considered. A preliminary work on this topic is available in [Oops!] . mistis got a Ministry grant for a related ANR project (see Section 8.2 ).
Statistical methods for the analysis of complex remote sensing data
Joint work with:Sylvain Douté from Laboratoire de Planétologie de Grenoble, France.
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 the 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.
A new generation of imaging spectrometers is emerging with an additional angular dimension. The surface of the planets will now be observed from different view points on the satellite trajectory, corresponding to about ten different angles, instead of only one corresponding usually to the vertical (0 degree angle) view point. Multi-angle imaging spectrometers present several advantages: the influence of the atmosphere on the signal can be better identified and separated from the surface signal on focus, the shape and size of the surface components and the surfaces granularity can be better characterized.
However, this new generation of spectrometers also results in a significant increase in the size (several tera-bits expected) and complexity of the generated data. Consequently, HMA (Hyperspectral Multi Angular) data induce data manipulation and visualization problems due to its size and its 4 dimensionality.
We propose to investigate the use of statistical techniques to deal with these generic sources of complexity in data beyond the traditional tools in mainstream statistical packages. Our goal is twofold:
We first focus on developing or adapting dimension reduction methods, classification and segmentation methods for informative, useful visualization and representation of the data previous to its subsequent analysis.
We also address the problem of physical model inversion which is important to understand the complex underlying physics of the HMA signal formation. The models taking into account the angular dimension result in more complex treatments. We investigate the use of semi-parametric dimension reduction methods such as SIR (Sliced Inverse Regression,  ) to perform model inversion, in a reasonable computing time, when the number of input observations increases considerably. A preliminary version of this work is presented in [Oops!] .
mistis got a Ministry grant for a related ANR project (see Section 8.2 ).