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January 2017
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9 Reads
We provide an overview of the probability and statistical tools underlying the extreme value theory, which aims to predict occurrence of rare events. Firstly, we explain that the asymptotic distribution of extreme values belongs, in some sense, to the family of the generalised extreme value distributions which depend on a real parameter, called the extreme value index. Secondly, we discuss statistical tail estimation methods based on estimators of the extreme value index
February 2016
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106 Reads
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4 Citations
Simulation Modelling Practice and Theory
We analyse the splitting algorithm performance in the estimation of rare event probabilities and this 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 we get is the sum of two parts: one part resuming the variability due to each threshold and a second part resuming the variability due to the thresholds number. 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.
February 2016
We analyse the splitting algorithm performance in the estimation of rare event probabilities and this 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 we get is the sum of two parts: one part resuming the variability due to each threshold and a second part resuming the variability due to the thresholds number. 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.
January 2013
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6,259 Reads
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48 Citations
January 2011
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46 Reads
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2 Citations
Stochastic Processes and their Applications
A straightforward application of an interacting particle system to estimate a rare event for switching diffusions fails to produce reasonable estimates within a reasonable amount of simulation time. To overcome this, a conditional “sampling per mode†algorithm has been proposed by Krystul in [10]; instead of starting the algorithm with particles randomly distributed, we draw in each mode, a fixed number 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 as well as a central limit theorem for the estimate.
January 2010
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16 Reads
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5 Citations
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.
October 2009
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19 Reads
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.
March 2009
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190 Reads
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55 Citations
IntroductionPrinciples and implementationsAnalysis in a simplified setting: a coin-flipping modelAnalysis and central limit theorem in a more general settingA numerical illustrationReferences
January 2009
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167 Reads
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8 Citations
September 2008
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5 Reads
... This FPRAS is based on the Multilevel Splitting algorithm (MSA) used in rare event analysis [3]. The idea behind the Multilevel Splitting algorithm is to simulate a rare event in a large state space by dividing the state space into a sequence of nested sub-spaces each of which are not rare events themselves. ...
Reference:
Counting Candy Crush Configurations
February 2016
Simulation Modelling Practice and Theory
... Therefore, numerous distributions are examined by hydrologists in different parts of the world [40]. Although there are different theoretical distributions to fit the extreme data series, the generalized extreme value distribution (GEVD) is the most applied technique in rainfall frequency analysis [2,41]. The GEVD is the collective of three statistical distributions that are commonly applied for flood hazard analysis. ...
January 2013
... Nevertheless, it is difficult to assess their proper impact onto ATC operations. Indeed, measuring the risk perceived by ATCs involves cognitive science (Averty et al. 2006, Chaloulos et al. 2010 and is not in the scope of this paper. Recently, Vela et al. (2011) presented a mathematical approach to study the design and implementation of CD&R algorithms in relationship with ATCs' taskload which is defined as the number of conflict resolution maneuvers required to separate aircraft. ...
January 2006
... A number of techniques based on the above mentioned idea of state space decomposition and splitting of trajectories are available in the literature: multilevel splitting techniques (for a complete review and detailed list of references see (Glasserman et al., 1999;Lezaud et al., 2004)); empirical method RESTART Villén-Altamirano, 1991, 1994); and more recent Interacting Particle Systems (IPS) approaches (Cérou et al., 2002;Del Moral, 2004;Cérou et al., 2005). The IPS approach seems to be the most suitable for rare event estimation in stochastic dynamical systems. ...
September 2003
... Splitting [15], [16], [17], [18], [19] is a variance reduction technique for accurately estimating ζ in models where {κ B < κ A } is a rare event. In this method, the state space of {h(X(t))} is partitioned on the basis of a set of real values 0 = 0 < 1 < 2 < · · · < q = , as shown in Figure 5(a). ...
January 2009
... By last exit time, we mean that trajectories of interest are not allowed to re-enter I before hitting T . Such Markov trajectories have a wide range of applications, such as population genetics (Griffiths and Tavaré 1994;Stephens and Donnelly 2000;De Iorio and Griffiths 2004a, b;Birkner and Blath 2008;Hobolth et al. 2008;Griffiths et al. 2008;Birkner et al. 2011;Koskela et al. 2015), mathematical finance (Casella and Roberts 2008), neuroscience (Bibbona and Ditlevsen 2013), physics (Del Moral and Garnier 2005;Johansen et al. 2006), and engineering (Blom et al. 2007;Lezaud et al. 2010). This problem is non-standard in SMC (Fearnhead 2008) because the dimension of a particle can be random, and the intermediate target distribution of interest, namely that of a partially reconstructed chain conditioned on its eventual hitting of T before returning to I , is usually unavailable. ...
January 2010
... Feynman-Kac formulae and their particle interpretations are also commonly used in financial mathematics to model option prices, futures prices and sensitivity measures, and in insurance and risk models [4,5,33,42,44,43]. They are used in rare event analysis to model conditional distributions of stochastic processes evolving in a rare event regime [6,5,20]. ...
October 2006
... This Feynman-Kac setting subsequently supported the evaluation of the reach probability through sequential Monte Carlo simulation in the form of an Interacting Particle System (IPS), including proof of convergence [11]. Krystul et al. [28] have used the Feynman-Kac setting to prove convergence of IPS using sampling per mode for a switching diffusion. ...
January 2011
Stochastic Processes and their Applications
... The most recent advances and most rigorous proofs of convergence under the strong Markov assumption are given in [1] where the authors show that the splitting fits the framework of Feynman Kac distribution approximations. A few heuristics have been then worked out such as in [34], [35], [36] and [37]. A summary of these methods can be found in [38,39] and some comparisons in [40]. ...
October 2006
... 2.2). A logistic model (comparable to our ISA(n) characteristic) was used also by (Averty et al. 2008) for the analysis of air traffic controllers decision-making in conflict risk detection. ...
March 2008
Air Traffic Control Quarterly
