Search

International audience; Sensitivity indices are commonly used to quantity the relative inuence of any specic group of input variables on the output of a computer code. In this paper, we focus both on computer codes the output of which is a cumulative distribution function and on stochastic computer codes. We propose a way to perform a global sensitivity analysis for these kinds of computer codes. In the rst setting, we dene two indices: the rst one is based on Wasserstein Fréchet means while the second one is based on the Hoeding decomposition of the indicators of Wasserstein balls. Further, when dealing with the stochastic computer codes, we dene an ideal version of the stochastic computer code thats ts into the frame of the rst setting. Finally, we deduce a procedure to realize a second level global sensitivity analysis, namely when one is interested in the sensitivity related to the input distributions rather than in the sensitivity related to the inputs themselves. Several numerical studies are proposed as illustrations in the dierent settings.

International audience; In this article we study the natural nonparametric estimator of a Wasserstein type cost between two distinct continuous distributions Fand G on R. The estimator is based on the order statistics of a sample having marginals F, G and any joint distribution. We prove a central limit theorem under general conditions relating the tails and the cost function. In particular, these conditions are satisfied by Wasserstein distances of order p>1and compatible classical probability distributions.; Dans cet article nous étudions l’estimateur non paramétrique naturel d’un coût de type Wasserstein entre deux lois Fet G distinctes et continues sur R. Cet estimateur est construit à partir des statistiques d’ordre d’un échantillon d’un couple quelconque de lois marginales F et G. Nous démontrons un théorème limite central sous des conditions générales reliant les queues de distribution à la fonction de coût. En particulier, ces conditions sont satisfaites par les distances de Wasserstein d’ordre p>1 et les lois classiques compatibles.

We study the Wasserstein distance $W_{2}$ for Gaussian samples. We establish the exact rate of convergence $\sqrt{\log \log n/n} $ of the expected value of the $W_{2}$ distance between the empirical and true $c.d.f.$’s for the normal distribution. We also show that the rate of weak convergence is unexpectedly $1/\sqrt{n} $ in the case of two correlated Gaussian samples.

The purpose of a Satellite Based Augmentation System, such as EGNOS or WAAS, is to decompose all range error sources and to distribute them to the civil aviation user community with reliable navigation services for different flight phases. Integrity refers to the notion of trust that the user may have in the positioning. Integrity includes the ability of the system to provide confidence thresholds as well as Alarms in case of anomalies. Considering satellite integrity, two ways are possible for an SBAS. The first is to broadcast per satellite the same UDRE value applicable for every user located inside the service area defined by the message type 27 content. As these UDREs are dimensioned to protect the worst user location – generally at the border zone – it penalizes automatically the users having lower residual errors. The second is to broadcast per satellite a covariance matrix of residual errors through the message type 28. This matrix contains all the structure of the orbitography and synchronization residual errors. It can be seen as a protective ellipsoid around the computed satellite position and clock containing the true – but unknown – satellite position and clock at a certain level of confidence. Using the MT28 message each user is able to reconstruct the integrity value by projecting this protective ellipsoid along its line of sight. EGNOS operational system is implementing the MT27 solution whilst the WAAS is based on the MT28 one. The MT28 approach is the current baseline for EGNOS V3. Starting from recent R&D activities, Thales Alenia Space together with CNES (French Space Agency) has developed a dedicated module providing the message type 28 based on orbitography and fast synchronization variance-covariance matrices combined with satellites residual measurement errors. The MT28 formulation has been designed in collaboration with Statistics and Probabilities laboratories (Paris Descartes and Toulouse Paul Sabatier) in France. The algorithm contains also a mechanism that reacts immediately to orbit or clock satellite feared events such as a clock jump. This new concept for EGNOS provides a drastic improvement with respect to the first studies provided during EGNOS V3 phase A & B and HISTB V2 by the Thales consortium. The new MT28 module is integrated into a SPEED platform, the SBAS Operational Test-bed that fully represents EGNOS performances in terms of accuracy, continuity, availability and integrity for Safety Of Life services. The performance evaluation shows a very good level of EGNOS LPV200 availability with respect to the MT27 current approach. The integrity is constantly maintained on the Geostationary Broadcast Area with a good level of integrity margin. This paper provides a high level architecture description of this new EGNOS algorithm as well as a set of performance figures showing the achieved improvements.

We propose a new algorithm for vector quantization:Average Competitive Learning Vector Quantization (ACLVQ). It is a rather simple modification of the classical Competitive Learning Vector Quantization (CLVQ). This new formulation give us similar results for the quantization error to those obtained by the CLVQ and reduce considerably the computation time to achieve the optimal quantizer. We establish the convergence of the method via the Kushner–Clark approach, and compare the two algorithms via the Central Limit Theorem. A simulation study is carried out showing the good performance of our proposal.

We propose to estimate the Hurst parameter involved in fractional processes via a method based on the Karhunen–Loeve expansion of a Gaussian process. We specifically investigate the cases of the fractional Brownian motion, the fractional Ornstein–Uhlenbeck family and the fractional Brownian bridge. We numerically compare our results with the ones obtained by the maximum-likelihood method, which show the validity of our proposal.

Brain atrophy measured by magnetic resonance structural imaging has been proposed as a surrogate marker for the early diagnosis of Alzheimer's disease. Studies on large samples are still required to determine its practical interest at the individual level, especially with regards to the capacity of anatomical magnetic resonance imaging to disentangle the confounding role of the cognitive reserve in the early diagnosis of Alzheimer's disease. One hundred and thirty healthy controls, 122 subjects with mild cognitive impairment of the amnestic type and 130 Alzheimer's disease patients were included from the ADNI database and followed up for 24 months. After 24 months, 72 amnestic mild cognitive impairment had converted to Alzheimer's disease (referred to as progressive mild cognitive impairment, as opposed to stable mild cognitive impairment). For each subject, cortical thickness was measured on the baseline magnetic resonance imaging volume. The resulting cortical thickness map was parcellated into 22 regions and a normalized thickness index was computed using the subset of regions (right medial temporal, left lateral temporal, right posterior cingulate) that optimally distinguished stable mild cognitive impairment from progressive mild cognitive impairment. We tested the ability of baseline normalized thickness index to predict evolution from amnestic mild cognitive impairment to Alzheimer's disease and compared it to the predictive values of the main cognitive scores at baseline. In addition, we studied the relationship between the normalized thickness index, the education level and the timeline of conversion to Alzheimer's disease. Normalized thickness index at baseline differed significantly among all the four diagnosis groups (P < 0.001) and correctly distinguished Alzheimer's disease patients from healthy controls with an 85% cross-validated accuracy. Normalized thickness index also correctly predicted evolution to Alzheimer's disease for 76% of amnestic mild cognitive impairment subjects after cross-validation, thus showing an advantage over cognitive scores (range 63–72%). Moreover, progressive mild cognitive impairment subjects, who converted later than 1 year after baseline, showed a significantly higher education level than those who converted earlier than 1 year after baseline. Using a normalized thickness index-based criterion may help with early diagnosis of Alzheimer's disease at the individual level, especially for highly educated subjects, up to 24 months before clinical criteria for Alzheimer's disease diagnosis are met.

We consider one dimensional probability distributions μ having a continuous and positive probability density function. We find the asymptotic of the size and the mass of the Voronoi cells and we prove that the local distortion associated with stationary or optimal quantizers is asymptotically uniform. Numerical simulations and computations illustrate the theoretical results and lead to the design of some good-fit test for the stationary equilibria.

We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non-log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew & Wise Theorem and finally the asymptotic size of the Voronoi tessels.

We first derive from abstract results on Feller transition kernels that, under some mild assumptions, a Markov stochastic algorithm with constant step size $\varepsilon$ usually has a tight family of invariant distributions $\nu^{\varepsilon}$, $\varepsilon \in(0,\varepsilon_0]$, whose weak limiting distributions as $\varepsilon\downarrow 0$ are all flow-invariant for its ODE. Then the main part of the paper deals with a kind of converse: what are the possible limiting distributions among all flow-invariant distributions of the ODE? We first show that no repulsive invariant (thin) set can belong to their supports. When the ODE is a stochastic pseudogradient descent, these supports cannot contain saddle or spurious equilibrium points either, so that they are eventually supported by the set of local minima of their potential. Such results require only the random perturbation to lie in L2. Various examples are treated, showing that these results yield some weak convergence results for the $\nu^{\varepsilon}$'s, sometimes toward a saddle point when the algorithm is not a pseudogradient.

We show in a very general framework the a.s. convergence of the one-dimensional Kohonen algorithm–after self-organization–to its unique equilibrium when the learning rate decreases to 0 in a suitable way. The main requirement is a log-concavity assumption on the stimuli distribution which includes all the usual (truncated) probability distributions (uniform, exponential, gamma distribution with parameter ≥ 1, etc.). For the constant step algorithm, the weak convergence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are the Poincaré-Hopf Theorem and a result of Hirsch on the convergence of cooperative dynamical systems.

We prove the a.s. convergence of the one dimensional Kohonen algorithm after selforganization and with decreasing step. Under some mild asumption of log-concavity on the input probability distribution, we first show the uniqueness of the limit and then the a.s. convergence. The proof relies on the index formula for a vector field on a manifold and a result by M. Hirsh about the convergence of cooperative dynamical systems applied to the ODE.

International audience; In this paper, we develop new algorithms for parameter estimation in the case of models type Input/Output in order to represent and to characterize a phenomenon Y. From experimental data Y_{1},...,Y_{n} supposed to be i.i.d from Y, we prove risk bounds qualifying the proposed procedures in terms of the number of experimental data n, computing budget m and model complexity. The methods we present are general enough which should cover a wide range of applications.

We consider a functional linear model where the explicative variables are known stochastic processes taking values in a Hilbert space, the main example is given by Gaussian processes in L 2 ( [ 0 , 1 ] ) . We propose estimators of the Sobol indices in this functional linear model. Our estimators are based on U-statistics. We prove the asymptotic normality and the efficiency of our estimators and we compare them from a theoretical and practical point of view with classical estimators of Sobol indices.

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.

This note shows that some assumption on small balls probability, frequently used in the domain of functional statistics, implies that the considered functional space is of finite dimension. To complete this result an example of L2 process is given that does not fulfill this assumption., 4 pages

International audience; Inverse problem is a current practice in engineering where the goal is to identify parameters from observed data through numerical models. These numerical models, also called Simulators, are built to represent the phenomenon making possible the inference. However, such representation can include some part of variability or commonly called uncertainty (see [4]), arising from some variables of the model. The phenomenon we study is the fuel mass needed to link two given countries with a commercial aircraft, where we only consider the Cruise phase . From a data base of fuel mass consumptions during the cruise phase, we aim at identifying the Speci c Fuel Consumption (SFC) in a robust way, given the uncertainty of the cruise speed V and the lift-to-drag ratio F. In this paper, we present an estimation procedure based on Maximum-Likelihood estimation, taking into account this uncertainty.

Coding of odour quality in the first two neuronal layers of olfactory systems is studied, taking especially insects as reference. First, odour quality is defined by K independent molecular properties that activate differentially the chemosensitive receptors of first-order neurons. Each odour quality discriminated is assumed to be coded by a distinct spatial activity pattern of the principal neurons (or equivalently glomeruli) of the second layer. Second, computer simulations show that the differential projections into glomeruli of several types of first-order neurons (convergence) is the main factor responsible for the generation of activity maps. These maps give a complete and unbiased representation of all odour qualities which make them suitable as an internal code. Third, lateral inhibition mediated by local neurons can significantly increase the difference of activity between the least and most active glomeruli for each odour, and consequently the total number of discriminated odours. Fourth, the mos...

In this paper, we propose an innovative classification method dedicated to hyperspectral images which uses both spectral information (Principal Component Analysis bands, Minimum Noise Fraction bands) and spatial information (textural features and segmentation). The process includes a segmentation as a pre-processing step, a spatial/spectral features calculation step and finally an area-wise classification. The segmentation, a region growing method, is processed according to a criterion called J-image which avoids the risks of over-segmentation by considering the homogeneity of an area at a textural level as well as a spectral level. Then several textural and spectral features are calculated for each area of the segmentation map and these areas are classified with a hierarchical ascendant classification. The method has been applied on several data sets and compared to the Gaussian Mixture Model classification. The JSEG classification process finally appeared to gives equivalent, and most of the time more accurate classification results.

This article is devoted to a survey on generic stochastic gradient methods. It provides the main results on stochastic gradient approximations such as the almost sure convergence, the central limit theorem, and the almost sure central limit theorem. The theory is illustrated by a wide range of applications going from least-square approximation to quantile estimation and quantization. Keywords: stochastic approximation; ordinary differential equation method; almost sure convergence; central limit theorem

Several spatial features are compared for the spatial/spectral classification of hyperspectral data. These features are extracted from texture spectra, co-occurrence matrices and morphological profiles. First, a PCA (Principal Components Analysis) is carried out on the hyperspectral image and textural features are computed on the first principal components. These textural features are concatenated together with spectral features (the principal components previously used) and the resulting image vector is then classified using SVM (Support Vector Machines) and a gaussian mixture algorithm. In the latter case, a hierarchical classification is used as a post-processing in order to reach a desired number of classes.

SummaryConsidering the problem of predicting the whole annual load curve of some Electricité de France customers from easily available explanatory variables, we derive simultaneous confidence bands for this prediction. The methodology that is developed, which can be applied to numerous problems, uses results on the maximum of Gaussian sequences. The paper ends with the application to Electricité de France’s problem.

grey matter (GM) probability in the MTL were obtained. Neuropsychological testing was performed at baseline for all subjects, and during follow-up between 1-3.5 years in MCI patients. Conversion to AD was defined based upon clinical and neuropsychological evaluation and change in Clinical Dementia Rating (CDR) scale score from 0.5 to 1.0. Subjects were reclassified to four cognitive status levels: 1 control, 2 MCI-Nonconverter, 3 MCI-Converter, and 4 AD. An ordinal regression model was created with MTL structural and PMC fMRI measures, age and MMSE as the independent variables. Results: Both imaging and follow-up data were available in 32 MCI subjects, 10 of whom converted to AD (total sample 28 controls, 22 MCI-Nonconverters, 10 MCI-Converters, and 13 AD). In our regression model, the main effects of all four factors were statistically significant (p 0.05), independent of one another. Contributions to model variance (highest to lowest) were as follows: MTL GM probability, MMSE, Age, and PMC signal change. Conclusions: PMC deactivation and MTL GM probability measures contribute independently to diagnostic classification, including prognosis in MCI patients. Combined use of structural and functional MRI information may serve as adjuncts to neuropsychological testing in early identification and monitoring of disease progression.

19 pages; International audience; We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments for powers of martingales. Several statistical applications on autoregressive and branching processes are also provided.

Kohonen self-organisation maps are a well know classification tool, commonly used in a wide variety of problems, but with limited applications in time series forecasting context. In this paper, we propose a forecasting method specifically designed for multi-dimensional long-term trends prediction, with a double application of the Kohonen algorithm. Practical applications of the method are also presented., \`{a} la suite de la conf\'{e}rence ANNPR, Florence 2003

Hyperspectral imagery provides detailed spectral information on the observed scene which enhances detection possibility, in particular for subpixel targets. In this context, we have developed and compared several anomaly detection algorithms based on a projection pursuit approach. The projection pursuit is performed either on the ACP or on the MNF (Minimum Noise Fraction) components. Depending on the method, the best axes of the eigenvectors basis are directly selected, or a genetic algorithm is used in order to optimize the projections. Two projection index (PI) have been tested: the kurtosis and the skewness. These different approaches have been tested on Aviris and Hymap hyperspectral images, in which subpixel targets have been included by simulation. The proportion of target in pixels varies from 50% to 10% of the surface. The results are presented and discussed. The performance of our detection algorithm is very satisfactory for target surfaces until 10% of the pixel.

A la suite de la conférence WSOM 03 à Kitakiushu; International audience; The Kohonen self-organization map is usually considered as a classification or clustering tool, with only a few applications in time series prediction. In this paper, a particular time series forecasting method based on Kohonen maps is described. This method has been specifically designed for the prediction of long-term trends. The proof of the stability of the method for long-term forecasting is given, as well as illustrations of the utilization of the method both in the scalar and vectorial cases.

Shows that the main proofs of the above paper (Yu et al., Trans Neural Networks, vol. 4, no. 2, p. 207-220, 1993) are incomplete and not correct: in fact, the self-organization cannot be achieved if the adaptation parameter satisfies the classical Robins-Monro conditions and Proposition 2 is erroneous. On the other hand, the two-dimensional extension (Theorem 3) is not proved. The main point is that the four classes that the authors consider as stable classes are not stable at all. Some references are finally given. >

The Kohonen algorithm (SOM) was originally designed as a stochastic algorithm which works in an on-line way and which was designed to model some adaptative features of the human brain. In fact it is nowadays extensively used for data mining, data visualization, and exploratory data analysis. Some users are tempted to use the batch version of the Kohonen algorithm since it is a deterministic algorithm which can be convenient if one needs to get reproducible results and which can go faster in some cases. In this paper, we try to elucidate the mathematical nature of this batch variant and give some elements of comparison of both algorithms. Then we compare both versions on a real data set.