Participants: A. Gégout-Petit

External collaborators: Y. Cao, S. Li, L. Guerin-Dubrana (Inra Bordeaux)

In the framework of a collaboration with INRA Bordeaux about the esca-illness of vines, Anne Gégout-Petit with Shuxian Li developed different spatial models and spatio-temporal models for different purposes: (1) study the distribution and the dynamics of esca vines in order to tackle the aggregation and the potential spread of the illness (2) propose a spatio-temporal model in order to capture the dynamics of cases and measure the effects of environmental covariates. For purpose (1), we propose different test based on the join count statistics, a paper is accepted for publication . We also developped a two-step centered autologistic model for the study of the dynamic of propagation. This work has been presented as invited paper in and is in preparation for publication.

Participant: S. Wantz-Mézières

External collaborator: J.-M. Moureaux, Y. Gaudeau, M. Ben Abdallah, M. Ouqamra (CRAN, Université de Lorraine), L. Taillandier, M. Blonski (CHU Nancy)

The collaboration with neurologists (CHU Nancy) and automaticians (CRAN) has carried on this year and led to the PhD presentation of M. Ben Abdallah, on December 12, 2016 , . We completed the modeling approach by a data analysis one. In the framework of a master 2 project, supervised and non supervised methods have confirmed our results on our local data base. This encourages us to continue our work in extending the data base via a collaboration with Montpellier CHU. Our perspectives are to validate multi-factor models, including biological and anatomopathological factors, and to design a decision-aid tool for praticians.

Participant: Céline Lacaux

External collaborator: Gennady Samorodnistky

In extreme value theory, one of the major topics is the study of the limiting behavior of the partial maxima of a stationary sequence. When this sequence is i.i.d., the unique limiting process is well-known and called the extremal process. Considering a long memory stable sequence, the limiting process is obtained as a simple power time change extremal process. Céline Lacaux and Gennady Samorodnistky have proved that this limiting process can also be interpreted as a restriction of a self-affine random sup measure. In addition, they have established that this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. Their results open the way to propose new self-similar processes with stationary max-increments.

Participant: Céline Lacaux

External collaborator: Hermine Biermé

Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest in the literature. Up to now, such models were only defined through stochastic integrals, without knowing explicitly their covariance functions. In link with this misunderstanding, one of the drawbacks is that no exact method of simulation has been proposed. In view to fill this lack, Hermine Biermé and Céline Lacaux have recently exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones. This allows them to propose a fast and exact method to synthetise an operator scaling Gaussian random fields with such a covariance function. Their algorithm is based on the famous circulant embedding matrix method. This is a first piece of work to popularized operator scaling Gaussian random field in anisotropic spatial data modeling.

Participants: P. Vallois

External collaborators: A. Lagnoux and S. Mercier (Toulouse)

In an article accepted at Bioinformatics, the goal is to illustrate different results on the local score distribution assuming an i.i.d. model, especially the one based on the pair (local score,length) and the one on the local score position. We measure with statistical tests how different approximations of the local score distribution fit to simulated sequences. In particular, our simulations show that the popular Karlin & Altschul approximation is not accurate in a wide range of situations. We add to the local score the length of the segment that realises it and we study the induced changes with numerical simulations. We also study specificity and sensitivity for the different methods. We introduce a new one dimensional statistic which is a function of $H_n^{*}$ and $L_n^{*}$ and we test its distribution. Finally, we estimate the probability that $H_n^{*}=H_n$ in different settings.

Participants: Romain Azaïs, Florian Bouguet, Anne Gégout-Petit, Florine Greciet, Aurélie Muller-Gueudin

External participants: Michel Benaïm (Université de Neuchâtel), Bertrand Cloez (Inra-SupAgro MISTEA), Alexandre Genadot (Inria CQFD, Université de Bordeaux)

A piecewise-deterministic Markov process is a stochastic process whose behavior is governed by an ordinary differential equation punctuated by random jumps occurring at random times. This class of stochastic processes offers a wide range of applications, especially in biology (kinetic diatery exposure model and growth of bacteria for example). BIGS' members mainly work on statistical inference techniques for these stochastic processes , , which is an essential step to build relevant application models. We also investigate the probabilistic properties of these processes , as well as the application in reliability to crack growth in some alloy in the industrial context of the PhD thesis of Florine Greciet with SAFRAN Aircraft Engines .

In a preprint recently accepted for publication in Electronic Journal of Statistics , we focus on the nonparametric estimation problem of the jump rate for piecewise-deterministic Markov processes observed within a long time interval under an ergodicity condition. More precisely, we introduce an uncountable class (indexed by the deterministic flow) of recursive kernel estimates of the jump rate and we establish their strong pointwise consistency as well as their asymptotic normality. In addition, we propose to choose among this class the estimator with the minimal variance, which is unfortunately unknown and thus remains to be estimated. We also discuss the choice of the bandwidth parameters by cross-validation methods. In , we state a new characterization of the jump rate when the transition kernel only charges a discrete subset of the state space. We deduce from this result a competitive nonparametric technique for estimating this feature of interest. We state the uniform convergence in probability of the estimator. Both the methodologies have been illustrated on numerical examples and real data.

The article deals with a class of conservative growth-fragmentation equations with a deterministic viewpoint. With the help of Foster-Lyapunov criteria, we study the long-time behavior of some associated piecewise-deterministic Markov process, which represents a typical individual following the dynamics of the equation. If the growth and the fragmentation are balanced, it is possible to provide existence and unicity for the stationary distribution on the process, as well as precise bounds for its tails of distributions in the neighborhoods of both 0 and $+\infty$. Our probabilistic results are systematically compared to estimates already obtained with deterministic methods.

In , we are interested by the long-time behavior of inhomogeneous-time Markov chains. We put forward an original and unified approach to relate some of their asymptotic properties (stationary distribution, speed of convergence, ...) to the ones of an auxiliary homogeneous-time Markov process. Such results are close to traditional functional limit theorems, but our method differs from the standard “Tightness/Identification” argument; it is based on the notion of asymptotic pseudotrajectories on the space of probability measures. We recover classical results, such as normalized bandit algorithms converging to a piecewise-deterministic Markov process, or weighted random walks or decreasing step Euler schemes approximated with solutions of stochastic differential equations.

Participant: Romain Azaïs

External participants: Bernard Delyon (Université Rennes 1), François Portier (Télécom ParisTech)

Suppose that a mobile sensor describes a Markovian trajectory in the ambient space. At each time the sensor measures an attribute of interest, e.g., the temperature. Using only the location history of the sensor and the associated measurements, the aim of the paper is to estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g., Monte Carlo, the proposed approach does not require any knowledge on the distribution of the sensor trajectory. Probabilistic bounds on the convergence rates of the estimator are established. These rates are better than the traditional “root $n$”-rate, where $n$ is the sample size, attached to other probabilistic integration methods. For finite sample sizes, the good behaviour of the procedure is demonstrated through simulations and an application to the evaluation of the average temperature of oceans is considered.

Participant: T. Bastogne

Photodynamic therapy (PDT) is an alternative treatment for cancer that involves the administration of a photosensitizing agent, which is activated by light at a specific wavelength. This illumination causes after a sequence of photoreactions, the production of reactive oxygen species responsible for the death of the tumor cells but also the degradation of the photosensitizing agent, which then loose the fluorescence properties. The phenomenon is commonly known as photobleaching process and can be considered as a therapy efficiency indicator. In , we present the design and validation of a real time controller able to track a preset photobleaching trajectory by modulating the light impulses width during the treatment sessions. This innovative solution was validated by in vivo experiments that have shown a significantly improvement of reproducibility of the inter-individual photobleaching kinetic. We believe that this approach could lead to personalized photodynamic therapy modalities in the near future.

Participant: T. Bastogne

The increase of computational environments dedicated to the simulation of nanoparticles (NP)-X-Rays interactions has opened new perspectives in computer-aided-design of nanostructured materials for biomedical applications. Several published studies have shown a crucial need of standardization of these numerical simulations . That is why, we proposed to perform a robustness multivariate analysis in . A gold nanoparticle (GNP) of 100 nm diameter was selected as a standard nano-system activated by a X-ray source placed just below the NP. Two response variables were examined: the dose enhancement in seven different spatial regions of interest around the NP and the duration of the experiments. 9 factors were pre-identified as potentially critical. A Plackett-Burman design of numerical experiments was applied to estimate and test the effects of each simulation factors on the examined responses. Four factors: the working volume, the spatial resolution, the spatial cutoff and the computational mode (parallelization) do not significantly affect the dose deposition results and none except the last one may reduce the computational duration. The energy cutoff may cause significant variations of the dose enhancement in some specific regions of interest: the higher the cutoff, the closer the secondary particles will stop from the GNP. By contrast, the Auger effect as well as the choice of the physical medium and the fluence level clearly appear as critical simulation parameters. Consequently, these four factors may be compulsory examined before comparing and interpreting any simulation results coming from different simulation sessions.

In , we address the prediction issue of organometallic nanoparticles (NPs)-based radiosensitization enhancement. The goal was to carry out computational experiments to quickly identify efficient nanostructures and then to preferentially select the most promising ones for the subsequent in vivo studies. To this aim, this interdisciplinary article introduces a new theoretical Monte Carlo computational ranking method and tests it using 3 different organometallic NPs in terms of size and composition. While the ranking predicted in a classical theoretical scenario did not fit the reference results at all, in contrast, we showed for the first time how our accelerated in silico virtual screening method, based on basic in vitro experimental data (which takes into account the NPs cell biodistribution), was able to predict a relevant ranking in accordance with in vitro clonogenic efficiency. This corroborates the pertinence of such a prior ranking method that could speed up the preclinical development of NPs in radiation therapy.

This in-silico approach was tested in  to screen radiosensitizing nanoparticles and the results have been validated by in vitro assays.

Participant: Bruno Scherrer

Given a Markov Decision Process (MDP) with $n$ states and a total number $m$ of actions, we study in the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal $\gamma$-discounted policy. We consider two variations of PI: Howard's PI that changes the actions in all states with a positive advantage, and Simplex-PI that only changes the action in the state with maximal advantage. We show that Howard's PI terminates after at most $O\left(\frac{m}{1-\gamma }log\left(\frac{1}{1-\gamma }\right)\right)$ iterations, improving by a factor $O(logn)$ a result by Hansen et al., while Simplex-PI terminates after at most $O\left(\frac{nm}{1-\gamma }log\left(\frac{1}{1-\gamma }\right)\right)$ iterations, improving by a factor $O(logn)$ a result by Ye. Under some structural properties of the MDP, we then consider bounds that are independent of the discount factor $\gamma$: quantities of interest are bounds ${\tau }_t$ and ${\tau }_r$—uniform on all states and policies—respectively on the expected time spent in transient states and the inverse of the frequency of visits in recurrent states given that the process starts from the uniform distribution. Indeed, we show that Simplex-PI terminates after at most $\tilde{O}\left(n^3{m}^2{\tau }_t{\tau }_r\right)$ iterations. This extends a recent result for deterministic MDPs by Post & Ye, in which ${\tau }_t\le 1$ and ${\tau }_r\le n$; in particular it shows that Simplex-PI is strongly polynomial for a much larger class of MDPs. We explain why similar results seem hard to derive for Howard's PI. Finally, under the additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient and recurrent for all policies, we show that both Howard's PI and Simplex-PI terminate after at most $\tilde{O}\left(m(n^2{\tau }_t+n{\tau }_r)\right)$ iterations.

Participant: Bruno Scherrer

We have made two contributions to the analysis of Approximate Dynamic Programming algorithms for Markov Games.

First, we extend in several non-stationary Reinforcement Learning (RL) algorithms and their theoretical guarantees to the case of discounted zero-sum Markov Games (MGs). As in the case of Markov Decision Processes (MDPs), non-stationary algorithms are shown to exhibit better performance bounds compared to their stationary counterparts. The obtained bounds are generically composed of three terms: 1) a dependency over gamma (discount factor), 2) a concentrability coefficient and 3) a propagation error term. This error, depending on the algorithm, can be caused by a regression step, a policy evaluation step or a best-response evaluation step. As a second contribution, we empirically demonstrate, on generic MGs (called Garnets), that non-stationary algorithms outperform their stationary counterparts. In addition, it is shown that their performance mostly depends on the nature of the propagation error. Indeed, algorithms where the error is due to the evaluation of a best-response are penalized (even if they exhibit better concentrability coefficients and dependencies on gamma) compared to those suffering from a regression error.

Furthermore, we report in theoretical and empirical investigations on the use of quasi-Newton methods to minimize the Optimal Bellman Residual (OBR) of zero-sum two-player Markov Games. First, it reveals that state-of-the-art algorithms can be derived by the direct application of Newton's method to different norms of the OBR. More precisely, when applied to the norm of the OBR, Newton's method results in the Bellman Residual Minimization Policy Iteration (BRMPI) and, when applied to the norm of the Projected OBR (POBR), it results into the standard Least Squares Policy Iteration (LSPI) algorithm. Consequently, new algorithms are proposed, making use of quasi-Newton methods to minimize the OBR and the POBR so as to take benefit of enhanced empirical performances at low cost. Indeed, using a quasi-Newton method approach introduces slight modifications in term of coding of LSPI and BRMPI but improves significantly both the stability and the performance of those algorithms. These phenomena are illustrated on an experiment conducted on artificially constructed games called Garnets.

Participants: Aurélie Muller-Gueudin

We relate here a collaboration with researchers in Automatic in Nancy (CRAN).

We consider here networks, modeled as a graph with nodes and edges representing the agents and their interconnections, respectively. The connectivity of the network, persistence of links and interactions reciprocity influence the convergence speed towards a consensus.

The problem of consensus or synchronization is motivated by different applications as communication networks, power and transport grids, decentralized computing networks, and social or biological networks.

We then consider networks of interconnected dynamical systems, called agents, that are partitioned into several clusters. Most of the agents can only update their state in a continuous way using only inner-cluster agent states. On top of this, few agents also have the peculiarity to rarely update their states in a discrete way by reseting it using states from agents outside their clusters. In social networks, the opinion of each individual evolves by taking into account the opinions of the members belonging to its community. Nevertheless, one or several individuals can change its opinion by interacting with individuals outside its community. These inter-cluster interactions can be seen as resets of the opinions. This leads us to a network dynamics that is expressed in term of reset systems. We suppose that the reset instants arrive stochastically following a Poisson renewal process.

We have an accepted paper in the journal IEEE Transactions on Automatic Control .

Participant: Romain Azaïs

External participants: Jean-Baptiste Durand (ENSIMAG, Inria MISTIS), Christophe Godin (Inria Virtual Plants), Benoît Henry (Inria TOSCA puis Madynes), Alexandre Genadot (Université de Bordeaux, Inria CQFD)

Tree-structured data naturally appear in various fields, particularly in biology where plants and blood vessels may be described by trees, but also in computer science because XML documents form a tree structure. Among trees, the class of self-nested trees presents remarkable compression properties because of the systematic repetition of subtrees in their structure. In a recent preprint , we provide a better combinatorial characterization of this specific family of trees. We show that self-nested trees may be considered as a good approximation class of unordered trees. In addition, we compare our approximation algorithms with a competitive approach of the literature on a simulated dataset. On the other hand, the paper is devoted to the estimation of the relative scale of ordered trees that share the same layout. The theoretical study is achieved for the stochastic model of conditioned Galton-Watson trees. New estimators are introduced and their consistency is stated. A comparison is made with an existing approach of the literature. A simulation study shows the good behavior of our procedure on finite-sample sizes. An application to the analysis of revisions of Wikipedia articles is also considered through real data.

Participants: A. Gégout-Petit, A. Muller-Gueudin, Y. Shi

External collaborators: B. Bastien (Transgene, Strasbourg)

In the purpose to select factors linked to the efficiency of a treatment in the context of high dimension (about 100.000 covariates), we have developed a new methodology to select and rank covariates associated to a variable of interest in a context of high-dimensional data under dependence but few observations. The methodology imbricates successively rough selection, clustering of variables, decorrelation of variables using Factor Latent Analysis, selection using aggregation of adapted methods and finally ranking through bootstrap replications. Simulations study shows the interest of the decorrelation inside the different clusters of covariates. The methodology was applied to select covariates among genomics, proteomics covariates linked to the success of a immunotherapy treatment for the lung cancer. A paper on the subject is in preparation.

Participant: J.-M. Monnez

External collaborator: O. Collignon (LIH Luxembourg)

In supervised learning the number of values of a response variable to predict can be very high. Grouping these values in a few clusters can be useful to perform accurate supervised classification analyses. On the other hand selecting relevant covariates is a crucial step to build robust and efficient prediction models. We propose in this paper an algorithm that simultaneously groups the values of a response variable into a limited number of clusters and selects stepwise the best covariates that discriminate this clustering. These objectives are achieved by alternate optimization of a user-defined selection criterion. This process extends a former version of the algorithm to a more general framework. Moreover possible further developments are discussed in detail .

Participant: J.-M. Monnez, K. Duarte

External collaborator: E. Albuisson (CHU, Nancy)

The purpose of this study was to define a short term event (death or hospitalization) score for heart failure patients based on the observation of biological, clinical and medical historical variables. Some of them were transformed or winsorized. Two methods of statistical learning were performed, logistic regression and linear discriminant analysis, different variable selection methods were used, on bootstrap samples. Aggregation of classifiers and out-of-bag validation were used. Finally a score taking values between 0 and 100 was established and an odds-ratio was defined in order to support medical decision (writing in progress).

Participant: J.-M. Monnez, K. Duarte

External collaborator: E. Albuisson

We consider the problem of sequential least square multidimensional linear regression using a stochastic approximation process. The choice of the stepsize may be crucial in this type of process. In order to avoid the risk of numerical explosion which can be encountered, we define three processes with a variable or a constant stepsize and establish their convergence. Finally these processes are compared to classic processes on 11 datasets, 6 with a continuous output and 5 with a binary output, for a fixed total number of observations used and then for a fixed processing time. It appears that the third-defined process with a very simple choice of the stepsize gives usually the best results (paper to be submitted).

Participants: T. Bastogne, L. Batista

System identification is a data-driven modeling approach more and more used in biology and biomedicine . In this application context, each assay is always repeated to estimate the response variability. The inference of the modeling conclusions to the whole population requires to account for the inter-individual variability within the modeling procedure. One solution consists in using mixed effects models but up to now no similar approach exists in the field of dynamical system identification. In , we propose a new solution based on an ARX (Auto Regressive model with eXternal inputs) structure using the EM (Expectation-Maximisation) algorithm for the estimation of the model parameters. Simulations show the relevance of this solution compared with a classical procedure of system identification repeated for each subject.

In , we propose a solution to firstly estimate the Fisher information matrix using the Louis' method and secondly to determine the parameters confidence intervals of an ARX model structure. We show relevance of the proposed solution in simulation and using real in-vitro data coming from realtime cell impedance measurements.

In parallel, we applied the mixed-effect modeling approach to the analysis in vivo responses in order to identify pronostic biomarkers of tumor regrowth after photodynamic therapy . This application corroborated the practical relevance of our model-based approach.

Participants: S.Ferrigno, A. Muller-Gueudin, M. Maumy-Bertrand (IRMA, Strasbourg)

In this work, we study the conditional cumulative distribution function and a nonparametric estimator associated to this function. The conditional cumulative distribution function has the advantages of completely characterizing the law of the random considered variable, allowing to obtain the regression function, the density function, the moments and the conditional quantile function. As a nonparametric estimator of this function, we focus on local polynomial techniques described in Fan and Gijbels [ref]. In particular, we use the local linear estimation of the conditional cumulative distribution function.

The objective of this work is to establish uniform asymptotic certainty bands for the conditional cumulative distribution function. To this aim, we give exact rate of strong uniform consistency for the local linear estimator of this function (writing in progress).

Participants: R.Azaïs, S.Ferrigno, M-J Martinez Marcoux (LJK, Grenoble)

The aim of this collaboration begins is to compare, through simulations, several methods to test the validity of a regression model. These tests can be "directional" in that they are designed to detect departures from mainly one given assumption of the model (for example the regression function, the variance or the error) or global (for example the conditional distribution function). The establishment of such statistical tests require the use of nonparametric estimators various functions (regression, variance, cumulative distribution function). The idea would then be able to build a tool (package R) that allows a user to test the validity of the model it uses through different methods and varying parameters associated with modeling. This work is currently in progress.