Sampling per mode for rare event simulation in switching diffusions

An interacting particle system (IPS) approach is virtually applicable to estimate rare event for switching diffusions, since these processes own the strong Markov property. Nevertheless, in practice the straightforward application of this approach to switching diffusions fails to produce reasonable estimates within a reasonable amount of simulation time. This happens because each resampling step tends to sample more "heavy" particles from modes with higher probabilities, thus "light" particles in the modes with small probability tend to be discarded. To avoid this, a conditional "sampling per mode" algorithm has been proposed by Krystul (2006): instead of starting the algorithm with particles randomly distributed, we draw in each mode a fixed number of particles and at each resampling step, the same number of particles is sampled for each visited mode. In this paper, we establish a law of large numbers theorem as well as a central limit theorem (CLT) for the estimate of the rare event probability.Un algorithme de branchement multi-niveaux peut en principe être utilisé pour estimer des évènements rares dans des diffusions à paramètre markovien, puisque ces processus vérifient la propriété de Markov forte. En pratique, l'application directe de cet algorithme peut échouer à produire une estimation raisonnable en un temps de simulation raisonnable. En effet, chaque étape de ré-échantillonnage tend à favoriser les particules dans les modes de forte probabilité, de sorte que les particules dans les modes de faible probabilité tendent à être éliminées. Pour éviter cet écueil, un algorithme de ré-échantillonnage par mode a été proposé par Krystul (2006) : au lieu de démarrer l'algorithme avec des particules distribuées aléatoirement, on génère dans chaque mode un nombre fixé de particules, et à chaque étape de ré-échantillonnage, le même nombre de particules est généré dans chaque mode visité. Dans cet article, on établit une loi des grands nombres et un théorème central limite (TCL) pour l'estimation de la probabilité de l'évènement rare

Sampling per mode simulation for switching diffusions

We consider the problem of rare event estimation in switching diffusions using an Interacting Particle Systems (IPS) based Monte Carlo simulation approach \cite{DelMoral}. While in theory the IPS approach is virtually applicable to any strong Markov process, in practice the straightforward application of this approach to switching diffusions may fail to produce reasonable estimates within a reasonable amount of simulation time. The reason is that there may be few if no particles in modes with small probabilities (i.e.\ "light" modes). This happens because each resampling step tends to sample more "heavy" particles from modes with higher probabilities, thus, "light" particles in the "light" modes tend to be discarded. This badly affects IPS estimation performance. By increasing the number of particles the IPS estimates should improve but only at the cost of substantially increased simulation time which makes the performance of IPS approach in switching diffusions similar to one of the standard Monte Carlo. To avoid this, a conditional "sampling per mode" algorithm has been proposed in \cite{Krystul}; instead of starting the algorithm with N particles randomly distributed, we draw in each mode j, a fixed number Nj particles and at each resampling step, the same number of particles is sampled for each visited mode. Using the techniques introduced in \cite{LeGland}, we recently established a Law of Large Number theorem as well as a Central Limit Theorem for the estimate of the rare event probability

Chernoff and Berry–Esséen inequalities for Markov processes

In this paper, we develop bounds on the distribution function of the empirical mean for general ergodic Markov processes having a spectral gap. Our approach is based on the perturbation theory for linear operators, following the technique introduced by Gillman

Chernoff and Berry–Esseen inequalities for Markov processes

Abstract. In this paper, we develop bounds on the distribution function of the empirical mean for general ergodic Markov processes having a spectral gap. Our approach is based on the perturbation theory for linear operators, following the technique introduced by Gillman. Mathematics Subject Classification. 60F10

Étude quantitative des chaînes de Markov par perturbation de leur noyau.

The development of the modelling of the random phenomena using Markov chains raises the problem of the control of convergence of the algorithms of simulation. The methods of simulations by ergodic Markov chains is based on the law of large numbers, which stipulates that for any initial distribution and any function f, the empirical average converges to the average of f, calculated with the unique invariant probability of the chain. It is then advisable, to determine a sufficient number of step of simulation in order to approximate, in a relatively precise way, the average of a f by its empirical average. Several works studied the speed of convergence of the chain towards its steady state. However even in steady state, the problem of the control remains, as we want to obtain a confidence interval for a fixed level. Two approaches exist to determine a sufficient number of steps. Either by bounding directly the probability of deviation between the empirical average and the average of f under the invariant probability, or by using the CLT. The starting point of this thesis was the inequality of Gillman and more particularly the method used: namely tools presented in the book of Kato on the theory of the perturbation of the linear operators.Le développement de la modélisation des phénomènes aléatoires par chaînes de Markov pose le problème du contrôle de convergence des algorithmes de simulation. Les méthodes de simulation par chaînes de Markov ergodiques s'appuient sur la loi des grands nombres, qui stipule que pour toute distribution initiale et toute fonction f, la moyenne empirique converge vers la moyenne de f, calculée avec l'unique probabilité invariante de la chaîne. Il convient alors, de déterminer un nombre suffisant de pas de simulation pour approximer, de façon relativement précise, la moyenne d'une certaine fonction par sa moyenne empirique. Plusieurs travaux ont étudié la vitesse de convergence de la chaîne vers son régime stationnaire. Cependant même en régime stationnaire, le problème du contrôle demeure, puisqu'il s'agit de calculer un intervalle de confiance pour un niveau fixé. Deux démarches existent pour déterminer le nombre de pas suffisants. Soit majorer directement la probabilité de déviation entre la moyenne empirique et la moyenne de la fonction sous la probabilité invariante, soit utiliser le théorème central limite. La première majoration est appelée borne de Chernoff et la seconde méthode invoque une borne de Berry-Esséen. Le point de départ de cette thèse fut l'inégalité de Gillman et plus particulièrement la méthode utilisée ; à savoir les outils présentés dans le livre de Kato sur la théorie des perturbations des opérateurs linéaires. L'exploitation plus poussée de cette méthode nous a permis d'obtenir les résultats suivant :•amélioration des bornes de Gillman et Dinwoodie : mise en évidence des comportements gaussien pour des petites déviations et poissonien pour les grandes déviations ;•extension au temps continu ;•extension aux espaces d'états quelconques, sous l'hypothèse d'un trou spectral pour le noyau de la chaîne et le générateur infinitésimal du processus ;•obtention d'une borne inférieure à la probabilité de déviation lorsque le processus (la chaîne) est réversible et d'espace d'états fini ;•amélioration et extension de la borne de Berry-Esséen obtenue par B. Mann

Multilevel branching splitting algorithm for estimating rare event probabilities

International audienceWe analyze the splitting algorithm performance in the estimation of rare event probabilities in a discrete multidimensional framework. For this we assume that each threshold is partitioned into disjoint subsets and the probability for a particle to reach the next threshold will depend on the starting subset. A straightforward estimator of the rare event probability is given by the proportion of simulated particles for which the rare event occurs. The variance of this estimator is the sum of two parts: with one part resuming the variability due to each threshold, and the second part resuming the variability due to the number of thresholds. This decomposition is analogous to that of the continuous case. The optimal algorithm is then derived by cancelling the first term leading to optimal thresholds. Then we compare this variance with that of the algorithm in which one of the threshold has been deleted. Finally, we investigate the sensitivity of the variance of the estimator with respect to a shape deformation of an optimal threshold. As an example, we consider a two-dimensional Ornstein–Uhlenbeck process with conformal maps for shape deformation

Effects of aircraft trajectories geometrical features upon air traffic controllers' conflict judgments

This work is a twofold contribution to the analysis of conflict detection process in Air Traffic Controllers (ATCos). The first one addresses methodological aspects and proposes a way to get responses as close as possible to controllers' actual expertise without using artifacts such as rating scales or inferring judgments from verbal material. The second objective is to compare the influence of three geometrical features of aircraft encounters and their capacity to alter an accurate perception of conflicts. The proposed methodology appeared to be useful for collecting expertise as controllers quickly appropriated it, and led to get coherent data. Its use can be envisaged when a reliable representation of mental picture of ATCos is essential. Concerning the geometrical features of aircraft trajectories, aircraft attitudes i.e., the fact they are stable, climbing of descending, entailed significant differences on detection accuracy. To a lesser extent, catch-ups and segmented trajectories showed a capacity to make an accurate perception of conflicts more difficult. These results must be interpreted as tendencies more than precise or quantified results. As the objective of this experiment was to be a pre-experiment in preparation for future collecting in the framework of the European project SESAR, a few different choices concerning the trajectories to be used in the traffic scenarios will help to precise these results