Méthode de rejet

Résumé

La méthode du rejet est une méthode utilisée dans le domaine des probabilités. But La méthode de rejet est utilisée pour engendrer indirectement une variable aléatoire X, de densité de probabilité f, lorsqu'on ne sait pas simuler directement la loi de densité de probabilité f (c'est le cas par exemple si f n'est pas une densité classique, mais aussi pour la loi de Gauss). Soit (Y,U) un couple de variables aléatoires indépendantes tirées selon une loi uniforme, i.e. (Y,U) est un point tiré uniformément dans le carré unité. On peut alors montrer que la distribution de X est la loi conditionnelle de Y sachant l'événement M={U \le f_{X}(Y)}. Autrement dit, {f_{X}(x)=f_{Y}(x|M)}. Pour simuler une suite de variables aléatoires réelles \left(X_n\right)_{n\ge 1} de distribution identique à celle de X, il suffit donc, dans

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Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.

Oxford University Press2016

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Adaptive rejection of unstable disturbances

Sergio Valentinotti

EPFL2001

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Some impulse radio UWB (IR-UWB) networks may allow concurrent transmissions without power control (for example MAC protocols that do not use power control, or co-exisiting, non-coordinated piconets). In such cases, it has been proposed to mitigate multi-user interference (MUI) at the physical layer, but existing proposals for interference mitigation do not account for the multipath nature of UWB channels. We address this problem and propose a receiver that employs a combination of statistical interference modeling and thresholding to mitigate MUI. We find that in a multipath environment the proposed receiver significantly outperforms existing receiver designs that either completely neglect the effect of MUI or only use a simple threshold to reject samples from interfering users. Further, in contrast to successive interference cancellation schemes, our receiver does not require active decoding of each interferer. Thus there is no need to synchronize the receiver with all the interfering users, which would be impractical in an IR-UWB system that is likely to be run in ad hoc mode. To model MUI we consider a hidden Markov model (HMM) and a Gaussian mixture model (GMM). We find that the HMM models interference better than the GMM. However, the resulting performance difference is not huge and comes at the cost of increased receiver complexity.

2006

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