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Articles

• [hal-04427506] Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data

  25 avril, par ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
  This work examines the asymptotic characteristics of a conditional set-indexed empirical process composed of functional ergodic random variables with missing at random (MAR). This paper’s findings enlarge the previous advancements in functional data analysis through the use of empirical process methodologies. These results are shown under specific structural hypotheses regarding entropy and under appealing situations regarding the model. The regression operator’s asymptotic (1−α)-confidence interval is provided for 0<α<1 as an application. Additionally, we offer a classification example to demonstrate the practical importance of the methodology.

• [hal-04282819] Gaussian-Smoothed Sliced Probability Divergences

  22 avril, par ano.nymous@ccsd.cnrs.fr.invalid (Mokhtar Z. Alaya), Mokhtar Z. Alaya
  Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the double empirical distribution for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance and prove that it converges with a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation.

• [hal-04543367] A Semi-Markov Model with Geometric Renewal Processes

  12 avril, par ano.nymous@ccsd.cnrs.fr.invalid (Jingqi Zhang), Jingqi Zhang
  We consider a repairable system modeled by a semi-Markov process (SMP), where we include a geometric renewal process for system degradation upon repair, and replacement strategies for non-repairable failure or upon N repairs. First Pérez-Ocón and Torres-Castro studied this system (Pérez-Ocón and Torres-Castro in Appl Stoch Model Bus Ind 18(2):157–170, 2002) and proposed availability calculation using the Laplace Transform. In our work, we consider an extended state space for up and down times separately. This allows us to leverage the standard theory for SMP to obtain all reliability related measurements such as reliability, availability (point and steady-state), mean times and rate of occurrence of failures of the system with general initial law. We proceed with a convolution algebra, which allows us to obtain final closed form formulas for the above measurements. Finally, numerical examples are given to illustrate the methodology.

• [hal-04282819] Gaussian-Smoothed Sliced Probability Divergences

  3 avril, par ano.nymous@ccsd.cnrs.fr.invalid (Mokhtar Z. Alaya), Mokhtar Z. Alaya
  Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the double empirical distribution for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance and prove that it converges with a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation.

• [tel-04500378] Méthode eulérienne faiblement diffuse appliquée à un modèle totalement conservatif pour la simulation des écoulements multifluides

  12 mars, par ano.nymous@ccsd.cnrs.fr.invalid (Vincent Mahy), Vincent Mahy
  Dans ce manuscrit, nous développons une méthode numérique adaptée à la simulation d’écoulements de fluides compressibles non miscibles. Pour modéliser ces écoulements, nous analysons un système totalement conservatif original comptant six équations, ferme par une équation d’état stiffened-gas et une équation d’équilibre en pression. Nous introduisons également un schéma numérique d’ordre 2, en espace et en temps, spécialement conçu pour la capture des interfaces entre les fluides dans des configurations à plusieurs dimensions. Pour atteindre l’ordre 2, nous mettons au point une méthode de reconstruction de pente multidimensionnelle basée sur le critère de stabilité : local extremum diminishing (LED). Le schéma d’ordre 2 associé au modèle totalement conservatif entraine l’apparition d’oscillations dans les profils de pression. Pour éviter ces oscillations parasites, nous démontrons un ensemble de propriétés essentielles. Tout d’abord, nous trouvons des conditions de stabilité, de type CFL, imposées par les reconstructions de pente. Puis, nous démontrons un théorème garantissant la consistance entre l’équation d’énergie et le transport des fractions volumiques. Ensuite, nous proposons une reconstruction de la pression en deux temps pour assurer la positivité de l’énergie interne. Enfin, nous développons une méthode numérique à une seule étape adaptée à la simulation d’écoulements faisant intervenir plus de deux fluides. L’ensemble des résultats présentés dans ce document est illustré par des cas test, à une, deux ou trois dimensions d’espace.

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