The following result has been obtained by J. Anselmi (Inria CQFD), F. Dufour (Inria CQFD) and T. Prieto-Rumeau.

We propose an approach for approximating the value function and computing an ε-optimal policy of a continuous-time Markov decision processes with Borel state and action spaces, with possibly unbounded cost and transition rates, under the total expected discounted cost optimality criterion. Under the assumptions that the controlled process satisfies a Lyapunov type condition and the transition rate has a density function with respect to a reference measure, together with piecewise Lipschitz continuity of the elements of the control model, one can approximate the original controlled process by a sequence of models that are computationally solvable. Convergence of the approximations takes place at an exponential rate in probability.

The following result has been obtained by J. Anselmi (Inria CQFD) and N. Walton.

Load balancing is a powerful technique commonly used in communication and computer networks to improve system performance, robustness and fairness. In this paper, we consider a general model capturing the performance of communication and computer networks, and on top of it we propose a decentralized algorithm for balancing load among multiple network paths. The proposed algorithm is inspired by the modus operandi of the processor-sharing queue and on each network entry point operates as follows: every time a unit of load completes its service on a path, it increases by one unit the load of that path and decreases by one unit the load of a path selected at random with probability proportional to the amount of load on each of the available paths. We develop a dynamical system to argue that our load-balancer achieves a desirable network-wide utility optimization.

The following result has been obtained by O. Costa, F. Dufour (Inria CQFD), and A. B. Piunovskiy.

The main goal of this paper is to study the infinite-horizon expected discounted continuous-time optimal control problem of piecewise deterministic Markov processes with the control acting continuously on the jump intensity $\lambda$ and on the transition measure $Q$ of the process but not on the deterministic flow $\phi$. The contributions of the paper are for the unconstrained as well as the constrained cases. The set of admissible control strategies is assumed to be formed by policies, possibly randomized and depending on the history of the process, taking values in a set valued action space. For the unconstrained case we provide sufficient conditions based on the three local characteristics of the process $\phi$, $\lambda$, $Q$ and the semicontinuity properties of the set valued action space, to guarantee the existence and uniqueness of the integro-differential optimality equation (the so-called Bellman–Hamilton–Jacobi equation) as well as the existence of an optimal (and $\delta$-optimal, as well) deterministic stationary control strategy for the problem. For the constrained case we show that the values of the constrained control problem and an associated infinite dimensional linear programming (LP) problem are the same, and moreover we provide sufficient conditions for the solvability of the LP problem as well as for the existence of an optimal feasible randomized stationary control strategy for the constrained problem.

The following result has been obtained by B. Saporta and E. F. Costa.

The aim of this paper is to propose a new numerical approximation of the Kalman-Bucy filter for semi-Markov jump linear systems. This approximation is based on the selection of typical trajectories of the driving semi-Markov chain of the process by using an optimal quantization technique. The main advantage of this approach is that it makes pre-computations possible. We derive a Lipschitz property for the solution of the Riccati equation and a general result on the convergence of perturbed solutions of semi-Markov switching Riccati equations when the perturbation comes from the driving semi-Markov chain. Based on these results, we prove the convergence of our approximation scheme in a general infinite countable state space framework and derive an error bound in terms of the quantization error and time discretization step. We employ the proposed filter in a magnetic levitation example with markovian failures and compare its performance with both the Kalman-Bucy filter and the Markovian linear minimum mean squares estimator.

The following result has been obtained by B. Saporta in collaboration with B. Delyon, N. Krell and Lydia Robert.

The data we analyze derives from the observation of numerous cells of the bacterium Escherichia coli (E. coli) growing and dividing. Single cells grow and divide to give birth to two daughter cells, that in turn grow and divide. Thus, a colony of cells from a single ancestor is structured as a binary genealogical tree. At each node the measured data is the growth rate of the bacterium. In this paper, we study two different data sets. One set corresponds to small complete trees, whereas the other one corresponds to long specific sub-trees. Our aim is to compare both sets. This paper is accessible to post graduate students and readers with advanced knowledge in statistics.

The following result has been obtained by F. Dufour (Inria CQFD) and A. B. Piunovskiy.

In this paper, we investigate an optimization problem for continuous-time Markov decision processes with both impulsive and continuous controls. We consider the so-called constrained problem where the objective of the controller is to minimize a total expected discounted optimality criterion associated with a cost rate function while keeping other performance criteria of the same form, but associated with different cost rate functions, below some given bounds. Our model allows multiple impulses at the same time moment. The main objective of this work is to study the associated linear program defined on a space of measures including the occupation measures of the controlled process and to provide sufficient conditions to ensure the existence of an optimal control.

The following result has been obtained by F. Dufour (Inria CQFD) and T. Prieto-Rumeau.

We consider a discrete-time constrained discounted Markov decision process (MDP) with Borel state and action spaces, compact action sets, and lower semi-continuous cost functions. We introduce a set of hypotheses related to a positive weight function which allow us to consider cost functions that might not be bounded below by a constant, and which imply the solvability of the linear programming formulation of the constrained MDP. In particular, we establish the existence of a constrained optimal stationary policy. Our results are illustrated with an application to a fishery management problem.

The following result has been obtained by A. Genadot (Inria CQFD).

We obtain a limit theorem endowed with quantitative estimates for a general class of infinite dimensional hybrid processes with intrinsically two different time scales and including a population. As an application, we consider a large class of conductance-based neuron models describing the nerve impulse propagation along a neural cell at the scales of ion channels.

The following result has been obtained by Pierrick Legrand (Inria CQFD) in collaboration with Y. Martinez, E. Naredo, L. Trujillo, U. Lopez.

The canonical approach towards fitness evaluation in Genetic Programming (GP) is to use a static training set to determine fitness, based on a cost function averaged over all fitness-cases. However, motivated by different goals, researchers have recently proposed several techniques that focus selective pressure on a subset of fitness-cases at each generation. These approaches can be described as fitness-case sampling techniques, where the training set is sampled, in some way, to determine fitness. This paper shows a comprehensive evaluation of some of the most recent sampling methods, using benchmark and real-world problems for symbolic regression. The algorithms considered here are Interleaved Sampling, Random Interleaved Sampling, Lexicase Selection and a new sampling technique is proposed called Keep-Worst Interleaved Sampling (KW-IS). The algorithms are extensively evaluated based on test performance, overfitting and bloat. Results suggest that sampling techniques can improve performance compared with standard GP. While on synthetic benchmarks the difference is slight or none at all, on real-world problems the differences are substantial. Some of the best results were achieved by Lexicase Selection and KeepWorse-Interleaved Sampling. Results also show that on real-world problems overfitting correlates strongly with bloating. Furthermore, the sampling techniques provide efficiency, since they reduce the number of fitness-case evaluations required over an entire run.

The following result has been obtained by Pierrick Legrand (Inria CQFD) in collaboration with Y. Martínez, L. Trujillo and E. Galván-López.

The estimation of problem difficulty is an open issue in genetic programming (GP). The goal of this work is to generate models that predict the expected performance of a GP-based classifier when it is applied to an unseen task. Classification problems are described using domain-specific features, some of which are proposed in this work, and these features are given as input to the predictive models. These models are referred to as predictors of expected performance. We extend this approach by using an ensemble of specialized predictors (SPEP), dividing classification problems into groups and choosing the corresponding SPEP. The proposed predictors are trained using 2D synthetic classification problems with balanced datasets. The models are then used to predict the performance of the GP classifier on unseen real-world datasets that are multidimensional and imbalanced. This work is the first to provide a performance prediction of a GP system on test data, while previous works focused on predicting training performance. Accurate predictive models are generated by posing a symbolic regression task and solving it with GP. These results are achieved by using highly descriptive features and including a dimensionality reduction stage that simplifies the learning and testing process. The proposed approach could be extended to other classification algorithms and used as the basis of an expert system for algorithm selection.

The following result has been obtained by Pierrick Legrand (Inria CQFD) in collaboration with E. Naredo, L. Trujillo, S. Silvac andLuis Muñoza.

Novelty Search (NS) is a unique approach towards search and optimization, where an explicit objective function is replaced by a measure of solution novelty. However, NS has been mostly used in evolutionary robotics while its usefulness in classic machine learning problems has not been explored. This work presents a NS-based genetic programming (GP) algorithm for supervised classification. Results show that NS can solve real-world classification tasks, the algorithm is validated on real-world benchmarks for binary and multiclass problems. These results are made possible by using a domain-specific behavior descriptor. Moreover, two new versions of the NS algorithm are proposed, Probabilistic NS (PNS) and a variant of Minimal Criteria NS (MCNS). The former models the behavior of each solution as a random vector and eliminates all of the original NS parameters while reducing the computational overhead of the NS algorithm. The latter uses a standard objective function to constrain and bias the search towards high performance solutions. The paper also discusses the effects of NS on GP search dynamics and code growth. Results show that NS can be used as a realistic alternative for supervised classification, and specifically for binary problems the NS algorithm exhibits an implicit bloat control ability.

The following result has been obtained by Pierrick Legrand (Inria CQFD) in collaboration with E. Z-Floresa, L. Trujillo, A. Soteloa and L. N. Coriaa.

The neurological disorder known as epilepsy is characterized by involuntary recurrent seizures that diminish a patient's quality of life. Automatic seizure detection can help improve a patient's interaction with her/his environment, and while many approaches have been proposed the problem is still not trivially solved.

In this work, we present a novel methodology for feature extraction on EEG signals that allows us to perform a highly accurate classification of epileptic states. Specifically, Hölderian regularity and the Matching Pursuit algorithm are used as the main feature extraction techniques, and are combined with basic statistical features to construct the final feature sets. These sets are then delivered to a Random Forests classification algorithm to differentiate between epileptic and non-epileptic readings.

Several versions of the basic problem are tested and statistically validated producing perfect accuracy in most problems and 97.6% accuracy on the most difficult case. Comparison with existing methods: A comparison with recent literature, using a well known database, reveals that our proposal achieves state-of-the-art performance. The experimental results show that epileptic states can be accurately detected by combining features extracted through regularity analysis, the Matching Pursuit algorithm and simple time-domain statistical analysis. Therefore, the proposed method should be considered as a promising approach for automatic EEG analysis.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with C. Verge, J. Morio and J.C Dolado Perez.

Collision between satellites and space debris seldom happens, but the loss of a satellite by collision may have catastrophic consequences both for the satellite mission and for the space environment. To support the decision to trigger off a collision avoidance manoeuver, an adapted tool is the determination of the collision probability between debris and satellite. This probability estimation can be performed with rare event simulation techniques when Monte Carlo techniques are not enough accurate. In this chapter, we focus on analyzing the influence of different simulation parameters (such as the drag coefficient) that are set for to simplify the simulation, on the collision probability estimation. A bad estimation of these simulation parameters can strongly modify rare event probability estimations. We design here a new island particle Markov chain Monte Carlo algorithm to determine the parameters that, in case of bad estimation, tend to increase the collision probability value. This algorithm also gives an estimate of the collision probability maximum taking into account the likelihood of the parameters. The principles of this statistical technique are described throughout this chapter.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with J. Houssineau.

In the last decade, the area of multiple target tracking has witnessed the introduction of important concepts and methods, aiming at establishing principled approaches for dealing with the estimation of multiple objects in an efficient way. One of the most successful classes of multi-object filters that have been derived out of these new grounds includes all the variants of the Probability Hypothesis Density (phd) filter. In spite of the attention that these methods have attracted, their theoretical performances are still not fully understood. In this chapter, we first focus on the different ways of establishing the equations of the phd filter, using a consistent set of notations. The objective is then to introduce the idea of observation path, upon which association measures are defined. We will see how these concepts highlight the structure of the first moment of the multi-object distributions in time, and how they allow for devising solutions to practical estimation problems.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with D. Villemonais.

We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of Rd with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion semigroup; when a particle is absorbed, another selected particle splits into two offsprings. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with R. Kohn and F. Patras.

This result analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with Feynman-Kac models. This perspective sheds new light on the foundations and the mathematical analysis of this class of methods. A key consequence is their equivalence with the Gibbs sampling of a (many-body) Feynman-Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques to derive non-asymptotic Taylor type series for the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler. These results provide sharp quantitative estimates of the convergence rate of the models with respect to the time horizon and the size of the systems. We illustrate the direct implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of the corresponding samplers, and explicit and non-asymptotic L p -mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called SMC${}^2$ ) type particle methodologies. We illustrate this general framework and results in the context of nonlinear filtering, hidden Markov chain problems with fixed unknown parameters, and Feynman-Kac path- integration models arising in computational physics and chemistry.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with L. Murray.

We propose sequential Monte Carlo (SMC) methods for sampling the posterior distribution of state-space models under highly informative observation regimes, a situation in which standard SMC methods can perform poorly. A special case is simulating bridges between given initial and final values. The basic idea is to introduce a schedule of intermediate weighting and resampling times between observation times, which guide particles towards the final state. This can always be done for continuous-time models, and may be done for discrete-time models under sparse observation regimes; our main focus is on continuous-time diffusion processes. The methods are broadly applicable in that they support multivariate models with partial observation, do not require simulation of the backward transition (which is often unavailable), and, where possible, avoid pointwise evaluation of the forward transition. When simulating bridges, the last cannot be avoided entirely without concessions, and we suggest an epsilon-ball approach (reminiscent of Approximate Bayesian Computation) as a workaround. Compared to the bootstrap particle filter, the new methods deliver substantially reduced mean squared error in normalising constant estimates, even after accounting for execution time. The methods are demonstrated for state estimation with two toy examples, and for parameter estimation (within a particle marginal Metropolis–Hastings sampler) with three applied examples in econometrics, epidemiology and marine biogeochemistry.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with R. Kohn and F. Patras.

This result presents a new duality formula between genetic type genealogical tree based particle models and Feynman–Kac measures on path spaces. Among others, this formula allows us to design reversible Gibbs–Glauber Markov chains for Feynman–Kac integration on path spaces. Our approach yields new Taylor series expansions of the particle Gibbs–Glauber semigroup around its equilibrium measure w.r.t. the size of the particle system, generalizing the recent work of Andrieu, Doucet, and Holenstein [1]. We analyze the rate of convergence to equilibrium in terms of the ratio of the length of the trajectories to the number of particles. The analysis relies on a tree-based functional and combinatorial representation of a class of Feynman–Kac particle models with a frozen ancestral line. We illustrate the impact of these results in the context of Quantum and Diffusion Monte Carlo methods.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with F. Giraud.

Sequential and quantum Monte Carlo methods, as well as genetic type search algorithms can be interpreted as a mean field and interacting particle approximations of Feynman-Kac models in distribution spaces. The performance of these population Monte Carlo algorithms is strongly related to the stability properties of nonlinear Feynman-Kac semigroups. In this paper, we analyze these models in terms of Dobrushin ergodic coefficients of the reference Markov transitions and the oscillations of the potential functions. Sufficient conditions for uniform concentration inequalities w.r.t. time are expressed explicitly in terms of these two quantities. We provide an original perturbation analysis that applies to annealed and adaptive Feynman-Kac models, yielding what seems to be the first results of this kind for these types of models. Special attention is devoted to the particular case of Boltzmann-Gibbs measures' sampling. In this context, we design an explicit way of tuning the number of Markov chain Monte Carlo iterations with temperature schedule. We also design an alternative interacting particle method based on an adaptive strategy to define the temperature increments. The theoretical analysis of the performance of this adaptive model is much more involved as both the potential functions and the reference Markov transitions now depend on the random evolution on the particle model. The nonasymptotic analysis of these complex adaptive models is an open research problem. We initiate this study with the concentration analysis of a simplified adaptive models based on reference Markov transitions that coincide with the limiting quantities, as the number of particles tends to infinity.

The following result has been obtained by P. Del Moral (Inria CQFD) in collaboration with A. Doucet and S.S. Singh.

Particle methods, also known as Sequential Monte Carlo methods, are a principled set of algorithms used to approximate numerically the optimal filter in nonlinear non-Gaussian state-space models. However, when performing maximum likelihood parameter inference in state-space models, it is also necessary to approximate the derivative of the optimal filter with respect to the parameter of the model. References [G. Poyiadjis, A. Doucet, and S. S. Singh, Particle methods for optimal filter derivative: Application to parameter estimation, in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 5, Philadelphia, 2005, pp. 925–928 and G. Poyiadjis, A. Doucet, and S. S. Singh, Biometrika, 98 (2011), pp. 65–80] present an original particle method to approximate this derivative, and it was shown in numerical examples to be numerically stable in the sense that it did not deteriorate over time. In this paper we theoretically substantiate this claim. ${𝕃}_p$ bounds and a central limit theorem for this particle approximation are presented. Under mixing conditions these ${𝕃}_p$ bounds and the asymptotic variance are uniformly bounded with respect to the time index.

The following result has been obtained by M. Chavent, and J. Saracco in collaboration with R. Genuer.

High-dimensional data classification is a challenging problem. A standard approach to tackle this problem is to perform variables selection, e.g. using stepwise procedures or LASSO approches. Another standard way is to perform dimension reduction, e.g. by Principal Component Analysis (PCA) or Partial Least Square (PLS) procedures. The approach proposed in this paper combines both dimension reduction and variables selection. First, a procedure of clustering of variables (CoV) is used to built groups of correlated variables in order to reduce the redundancy of information. This dimension reduction step relies on the R package ClustOfVar which can deal with both numerical and categorical variables. Secondly, the most relevant synthetic variables (which are numerical variables summarizing the groups obtained in the first step) are selected with a procedure of variable selection using random forests (VSURF), implemented in the R package VSURF. Numerical performances of the proposed methodology called CoV/VSURF are compared with direct applications of VSURF or random forests (RF) on the original $p$ variables. Improvements obtained with the CoV/VSURF procedure are illustrated on two simulated mixed datasets (cases $n>p$ and $n<<p$) and on a real proteomic dataset.

The following result has been obtained by M. Chavent and J. Saracco in collaboration with M.P. Ellies.

This work sets out a methodological approach to assess how to simultaneously control together animal performances, nutritional value, sensory quality of meat. Seventy-one young bulls were characterized by 97 variables. Variables of each element were arranged into either 5 homogeneous Intermediate Scores (IS) or 2 Global Indices (GI) via a clustering of variables and analysed together by Principal Component Analysis (PCA). These 3 pools of 5 IS (or 2 GI) were analysed together by PCA to established the links existing among the triptych. Classification on IS showed no opposition between animal performances and nutritional value of meat, as it seemed possible to identify animals with a high butcher value and intramuscular fat relatively rich in polyunsaturated fatty acids. Concerning GI, the classification indicated that animal performances were negatively correlated with sensory quality. This method appeared to be a useful contribution to the management of animal breeding for an optimal trade-off between the three elements of the triptych.

The following result has been obtained by J. Saracco (Inria CQFD) in collaboration with B. Liquet.

In a massive data setting, we focus on a semiparametric regression model involving a real dependent variable $Y$ and a $p$-dimensional covariable $X$. This model includes a dimension reduction of $X$ via an index $X^{\text{'}}\beta$. The Effective Dimension Reduction (EDR) direction $\beta$ cannot be directly estimated by the Sliced Inverse Regression (SIR) method due to the large volume of the data. To deal with the main challenges of analysing massive datasets which are the storage and computational efficiency, we propose a new SIR estimator of the EDR direction by following the “divide and conquer” strategy. The data is divided into subsets. EDR directions are estimated in each subset which is a small dataset. The recombination step is based on the optimisation of a criterion which assesses the proximity between the EDR directions of each subset. Computations are run in parallel with no communication among them. The consistency of our estimator is established and its asymptotic distribution is given. Exten- sions to multiple indices models, $q$-dimensional response variable and/or SIR $\alpha$-based methods are also discussed. A simulation study using our edrGraphicalTools R package shows that our approach enables us to reduce the computation time and conquer the memory constraint problem posed by massive datasets. A combination of foreach and bigmemory R packages are exploited to offer efficiency of execution in both speed and memory. Finally, results are visualised using the bin-summarise-smooth approach through the bigvis R package