Joint work with Nathalie Peyrard (INRA, Avignon), Chris Fraley and Adrian Raftery (Statistics Department, University of Washington, Seattle).

The project started as a collaboration between the University of Washington Department of Statistics, the University of Washington Breast Imaging Center, and Toshiba America MRI, with the latter two collaborating to acquire the data. In a first version of this work only three patients were analysed. We then had to extend the analysis and analyze data from patients that we had not analyzed for the first version. Changes in some of the participants affiliation, together with human subjects constraints, meant that it was logistically complicated for us to get permission to analyze the new data and then to recover them. In the end we did succeed to extend our analyis to 19 patients, and the results were very good .

Outside simple cases, the EM algorithm is seldom tractable analytically. In practice, difficulties arise due to the dependence structure in the models and approximations are required. A heuristic solution using mean field approximation principle has been proposed in . Using ideas from this principle, we proposed a class of EM-like algorithms generalizing . The mean field approach consists of neglecting fluctuations from the mean in the environment of each variable. More generally, we talk about mean field-like approximations when the value at node i does not depend on the values at other nodes which are all set to constants (not necessarily the means) independently of the value at node i (). The following computation then reduces to dealing with systems of independent variables, which is much simpler.

This approach is very flexible in that many ways to set the neighboring nodes are possible and lead to as many different algorithms. We investigated some of these choices which led to promising procedures. Their behavior is satisfying in practice but no theoretical study as regards convergence properties is available yet.

To investigate such convergence properties, we propose to consider a particular way to set the neighbors which induces the increase of a function of interest. The function is chosen so as to facilitate the the convergence study of the subsequent algorithm. After implementing and assessing the performance of this algorithm in practice, a second step is to consider techniques developped in to link the properties of the algorithm to the other algorithms originally developped in .

The purpose of our work is the development of dynamic factor models for multivariate financial time series, and the incorporation of stochastic volatility components for latent factor processes. The models are direct generalizations of univariate stochastic volatility models, and represent specific varieties of models recently discussed in the growing multivariate stochastic volatility literature.

We investigated a part of the exploratory analysis of bacterial genomes, beyond gene detection. We aim at detecting relationships among genes based on different kinds of information: nucleotide sequence, gene position, functional annotation,... The ideal goal is to link proximities among genes on the chromosome with genetic mechanisms of the cell. In fact, the cell machinery is thought to be coded inside the genome. We reviewed the main work in progress on the subject in order to suggest an appropriate formalism. We focused on the notion of neighborhood, related to intrinsic properties among entities (genes) considered. Neighborhood must be understood in a broad sense which leads to some specific mathematical tools and processes. Our investigation is based on tools from mixture models and markovian models. We consider various classification methods.

We present a new probabilistic framework for recognizing textures in images. Images are described by local affine-invariant descriptors and by spatial relationships between these descriptors. A graph is associated to an image with the nodes representing feature vectors describing image regions and the edges joining spatially related regions. Incorporating information about the spatial organization of the descriptors leads to better recognition results. Current approaches consist in augmenting the data with information coming from the spatial relationships, for instance by using co-occurence statistics, but without modeling explicitly the dependencies between neighboring descriptors. In such approaches the underlying model is one where the descriptors are statistically independent variables. Our claim is that recognition results can be further improved by considering that descriptors are statistically dependent. We propose to introduce in texture recognition the use of statistical parametric models of the dependence between descriptors. In this work, we chose Hidden Markov Models (HMM) which are both well statistically-based and appropriate models for such a task. They are parametric models and their use requires non trivial parameter estimation. We propose to use recent estimation procedures based on the mean field principle of statistical physics. Using sample images, textures are then learned as HMM's and a set of estimated parameters is associated to each texture. At recognition time, another HMM is used to compute, for each feature vector, the membership probabilities to the different texture classes. Preliminary experiments show promising results .

Joint work with Mhamed El Aroui (ISG, Tunis), Myriam Garrido (ENAC, Université Toulouse 3, Jean Diebolt (CNRS)).

The first part of our work is to propose new estimates of the extremal index. This parameter is important in practice since it drives the behaviour of the distribution tail. The second part is then to deduce estimates for extreme quantiles.

In ,, we investigate the asymptotical behaviour of two new estimates based on double threshold methods.

We also introduce a quasi-conjugate Bayes approach for estimating Generalized Pareto Distribution (GPD) parameters, distribution tails and extreme quantiles within the Peaks-Over-Threshold framework . Bayes credibility intervals are defined, they provide assessment of the quality of the extreme events estimates. Posterior estimates are computed by Gibbs samplers with Hastings-Metropolis steps. Even if non-informative priors are used in this work, the suggested approach could incorporate informative priors. It brings solutions to the problem of estimating extreme events when data are scarce but expert opinion is available.

Finally, we introduce estimates dedicated to the important case of Weibull tail-distributions  which includes for instance Gaussian, gamma, and Weibull distributions.

Joint work with Anatoli Iouditski (Univ. Joseph Fourier, Grenoble), Pierre Jacob, Ludovic Menneteau (Univ. Montpellier) and Alexandre Nazin (IPU, Moscow, Russia).

The first part of our work consists in building nonparametric estimates of the boundary of some support based on the extreme values of the sample , . These estimates require to select which extreme values are to be used. This problem is difficult in practice. To overcome this limitation, estimates based on a linear programming formulation are defined. In this case, the important points of the sample are selected automatically by solving a linear optimization problem . Our current work consists in building an optimization problem leading to an optimal estimate for the L₁-distance. We refer to  and  for similar studies on other estimates.

Joint work with Nicolas Devictor (CEA - Cadarache).

The first motivation of J. Jacques thesis was to take into account model uncertainty in sensitivity analysis. Two types of uncertainty have been studied: uncertainty due to the use of a simplified model and uncertainty du to a mutation of the model. A second motivation was exhibited during the first thesis year: the problem of sensitivity analysis of models with correlated inputs.

This last year of thesis has been devoted to the formalisation of the proposed solutions and to several applications in nuclear engineering.

This thesis work has been presented at the fourth international conference on Sensitivity Analysis of Model Output, and at two others French conferences. A paper has been accepted in the journal Reliability Engineering and System Safety.

Joint work with Serge Iovleff (Université Lille 3) and Cordelia Schmid (Lear, Inria).

In the first part of this work, we focus on nonlinear PCA based on manifold approximation of the set of points introduced in . This method proves especially useful when the observations are images  and thus located in high dimensional spaces. The joint work with Serge Iovleff consists in defining a probabilistic framework for nonlinear PCA permitting new extensions of this dimension-reduction method .

The second part of our work is to propose new methods combining dimension-reduction with a classification step. This is the context of the PhD thesis of Charles Bouveyron which takes place in collaboration with C. Schmid (Lear) in the ACI Movistar in the ``Masse de données'' program. A new method of discriminant analysis, called High Dimensional Discriminant Analysis (HHDA) is introduced. Our approach is based on the assumption that high dimensional data live in different subspaces with low dimensionality. Thus, HDDA reduces the dimension for each class independently and regularizes class conditional covariance matrices in order to adapt the Gaussian framework to high dimensional data. This regularization is achieved by assuming that classes are spherical in their eigenspace.

Joint work with P. Borgnat (Inria post-doctoral fellowship).

This ongoing work, initiated with P. Borgnat during his post-doctoral stay at IST-ISR (sept. 2003 – sept. 2004), aims at recovering a signal from the sparse set of local maxima coefficients of its wavelet decomposition. Starting with the conjugate gradient algorithm proposed by Mallat and Zhong to pseudo-inverse the transform, we adapted it to complex wavelets. There are two main advantages in using complex wavelets for this purpose:

With J. Lewalle (Univ. of Syracuse, New York, USA), we are now tackling the continuous wavelet inversion problem from the point of view of its diffusion formulation (PDE).

This topic is at the core of J. Gosme (Univ. Tech. Troyes) Ph.D. thesis (to be defended on December 20, 2004) advised by C. Richard (Univ. Tech. Troyes) and co-advised by P. Gonçalvès (INRIA).

Our aim is to propose a totally adaptive (signal driven) smoothing of time-frequency representations, relying on non linear anisotropic diffusion schemes inspired from the heat equation. We derived a set of partial differential equations applied to standard time-frequency representations (e.g. Wigner-Ville distribution) to locally adapt the amount of smoothing to the local (time-frequency) characteristics of the signal. The outcomes are for instance interference free representations with sharp localization properties, but the versatility of this approach allows for enhancing any other desired feature of the distributions, defining a corresponding diffusion control strategy (conductance function). An important achievement this year, was to derive an equivalent diffusion process that preserves covariances with respect to time shifts and scale changes, opening up in this way the scope of adaptive smoothing to the affine class of time-scale representations.

This topic is the main line of our scientific collaboration with Ecole Normale Superieure de Lyon (France). P. Flandrin and P. Goncalvès are co-advising the PhD thesis of G. Rilling (starting date, Sept. 2004) on ``Empirical Mode Decomposition" (EMD).

We now briefly describe the EMD technique. This entirely data-driven algorithm introduced by N. E. Huang decomposes iteratively a complex signal (i.e. with several characteristic time scales coexisting) into elementary AM-FM type components (Intrinsic Mode Functions). The rationale of this decomposition is to locally identify in the signal the most rapid oscillations, defined as the waveform interpolating interwoven local maxima and minima. To do so, local maxima points (respectively local minima points) are interpolated with a cubic spline, to yield the upper (resp. lower) envelope. The mean envelope (half sum of upper and lower envelopes) is then subtracted from the initial signal, and the same interpolation scheme is re-iterated on the remainder. The so-called sifting process stops when the mean envelope is reasonably zero everywhere, and the resulting signal is designated the first Intrinsic Mode Function. The higher order IMFs are iteratively extracted applying the same procedure to the initial signal after the previous IMFs have been removed.

With P. Flandrin (ENS-Lyon, France) and G. Rilling (ENS-Lyon, France), we are pursuing the qualitative study of EMD as an adaptive dyadic filter bank. In the course of this analysis we have also proposed several modifications of this decompositions, that significantly improved its performances (cf. corresponding publications).

With S. Bausson (IST-ISR) and P. de Oliveira (marinha & IST-ISR), we are continuing a work that P. Goncalvès had initiated at Inria with B. Esterni, a post-graduate student from Ensimag (France). We endeavored to transpose the EMD to 2D signals, and more specifically to quadratic time-frequency representations of 1D signals. The idea is to use EMD to separate signal components (low pass structures) from cross-components (high pass oscillating terms).

In parallel to this, we are investigating several different approaches to the 2D-EMD, including for instance a row-wise/column-wise decomposition, in the spirit of the so-called non-standard wavelet transform. This is also a joint work with J.C. Nunes (Université de Créteil, France).

Joint work with P. Borgnat. This research topic was prompted by the tight connection between the work of P. Borgnat developed during his PhD thesis (ENS-Lyon, Nov. 2002) and the current activities on local stationarity of Professors I. Lourtie (IST-ISR) and F. Garcia (IST-ISR). For timetable issues, the achievement of this work has been delayed, but should remain the backbone of a collaboration between INRIA, IST-ISR and Ecole Normale Supérieure de Lyon (France).

The proposed work deals with 2D statistical fields, for instance images but also other random fields coming from other domains (e.g., in physics, the turbulent velocity fields, or a representation of a 1D signal on a time-frequency plane). Knowing how to define the symmetries of one image is a classical way to describe textures (leaving out the study of shapes for now).

Among the interesting symmetries, the scale invariance property has a special relevance both for images (to deal with multi-scale structures) and physical fields. The first part of this work was to define what are the possible choices of symmetries for images, especially in the case of scale invariance (or self-similarity for random fields). Using preliminary work on plane transformations, we have studied how one can use a stationarization of those invariances to prescribe the statistical properties of the random fields. Stationarization is a method that studies a signal or field that has some invariance by means of a stationary generator. Namely, one tries to find a stationary generator Y(t) that can be warped by some warping t = f(u) in the original field X(u) = Y(f(u)) that has a different invariance. This method was introduced in geostatistics and used in some problems of imaging. We develop this approach for self-similarity of images.

A first point was to describe possible warping functions and the kinds of self-similarity that can be targeted this way. The correlation structure is then controlled by the invariance. We have studied how using the stationary generator (and thus, means to synthesize this field Y using this stationarity – spectral or parametric methods) induces an efficient method for the synthesis of self-similar random fields. A second point is the question of analysis: is it possible to recover the stationarizing warping from one realization of the random field ? Drawing on the method proposed by Perrin and Senoussi (1999) based on the variogram, and on the work of Clerc and Mallat (2000) on wavelet decompositions, we address the problem of scale invariant fields. Preliminary results show that it is possible in this case to recover the warping but a more robust method should be designed. An insight would be to adapt results about local stationarity (work of F. Garcia and I. Lourtie at the ISR) to cross-check the stationarity of the unwarped process locally, during the estimation of the inverse warping.

This work was presented in a workshop at INRIA Rocquencourt in December 2003 (journées Thalweg).

Joint work with G. Celeux and N. Bousquet. In the reliability context, we are interested in lifetime data analysis. We have especially examined a simple competing risk model that may be viewed as a possible alternative to the standard Weibull model. In particular our model enables to take into account both accidental causes of failure and aging. The estimation of parameters is made by Maximum Likelihood and Bayesian inference. Moreover in order to discriminate between our model and Weibull (or exponential) models, a test procedure has been proposed. Finally different applications have been presented.