# Team : mistis

## Section: New Results

### Semi and non parametric methods

#### Modelling extremal events

Participants : Stéphane Girard, Laurent Gardes.

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 ,[12], 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 [24]. 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 [9][13] which includes for instance Gaussian, gamma, and Weibull distributions.

#### Boundary estimation

Participants : Stéphane Girard, Laurent Gardes.

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 [30][29][27],
[17][16].
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 [1][11].
Our current work consists in building an optimization problem
leading to an optimal estimate for the L_{1}-distance.
We refer to [15] and [14] for similar studies
on other estimates.

#### Sensitivity analysis and model uncertainty

Participants : Julien Jacques, Christian Lavergne.

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*[18].

#### Dimension reduction for image processing

Participants : Stéphane Girard, Charles Bouveyron.

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 [8]. This method proves especially useful when the observations are images [4] 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 [28].

The second part of our work is to propose new methods combining dimension-reduction with a classification step[20]. 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.

#### Sparse Continuous Wavelet Transform Inversion

Participant : Paulo Gonçalves.

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:

the number of local maxima is considerably reduced when considering the magnitude of the complex wavelet transform field, as compared to its real part.

Although the reconstruction error is slightly smaller with real wavelets, in most case, it decreases faster with complex wavelets.

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).

#### Diffusion of time-frequency representations

Participant : Paulo Gonçalvès.

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.

#### Empirical Mode Decomposition

Participant : Paulo Gonçalvès.

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).

#### Statistical Modelling of Image Symmetries and Stationarization

Participant : Paulo Gonçalvès.

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*).