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Concept# Computational statistics

Résumé

Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computing) specific to the mathematical science of statistics. This area is also developing rapidly, leading to calls that a broader concept of computing should be taught as part of general statistical education.
As in traditional statistics the goal is to transform raw data into knowledge, but the focus lies on computer intensive statistical methods, such as cases with very large sample size and non-homogeneous data sets.
The terms 'computational statistics' and 'statistical computing' are often used interchangeably, although Carlo Lauro (a former president of the International Association for Statistical Computing) proposed making a distinction, defining 'statistical computing' as "the application of computer science to statistics",
and 'computational statistics' as "aiming at the design of algorithm for implementing
statistical methods on computers, including the ones unthinkable before the computer
age (e.g. bootstrap, simulation), as well as to cope with analytically intractable problems" [sic].
The term 'Computational statistics' may also be used to refer to computationally intensive statistical methods including resampling methods, Markov chain Monte Carlo methods, local regression, kernel density estimation, artificial neural networks and generalized additive models.
Though computational statistics is widely used today, it actually has a relatively short history of acceptance in the statistics community. For the most part, the founders of the field of statistics relied on mathematics and asymptotic approximations in the development of computational statistical methodology.
In statistical field, the first use of the term “computer” comes in an article in the Journal of the American Statistical Association archives by Robert P. Porter in 1891.

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Cours associés (23)

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Computational statistics

Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computing) specific to the mathematical science of statistics. This area is also developing rapidly, leading to calls that a broader concept of computing should be taught as part of general statistical education.

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Méthode de Monte-Carlo par chaînes de Markov

Les méthodes de Monte-Carlo par chaînes de Markov, ou méthodes MCMC pour Markov chain Monte Carlo en anglais, sont une classe de méthodes d'échantillonnage à partir de distributions de probabilité. Ces méthodes de Monte-Carlo se basent sur le parcours de chaînes de Markov qui ont pour lois stationnaires les distributions à échantillonner. Certaines méthodes utilisent des marches aléatoires sur les chaînes de Markov (algorithme de Metropolis-Hastings, échantillonnage de Gibbs), alors que d'autres algorithmes, plus complexes, introduisent des contraintes sur les parcours pour essayer d'accélérer la convergence (Monte Carlo Hybride, Surrelaxation successive).

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Isogeometric Analysis (IGA) is a computational methodology for the numerical approximation of Partial Differential Equations (PDEs). IGA is based on the isogeometric concept, for which the same basis functions, usually Non-Uniform Rational B-Splines (NURBS), are used both to represent the geometry and to approximate the unknown solutions of PDEs. Compared to the standard Finite Element method, NURBS-based IGA offers several advantages: ideally a direct interface with CAD tools, exact geometrical representation, simple refinement procedures, and smooth basis functions allowing to easily solve higher order problems, including structural shell problems. In these contexts, repeatedly solving a problem for a large set of geometric parameters might lead to high and eventually prohibitive computational cost. To cope with this problem, we consider in this work the Reduced Basis (RB) method for the solution of parameter dependent PDEs, specifically for which the NURBS representation of the computational domain is parameter dependent. RB refers to a technique that enables a rapid and reliable approximation of parametrized PDEs by constructing low dimensional approximation spaces. In this work, for the construction of the reduced spaces we adopt two different strategies, namely the Proper Orthogonal Decomposition and the greedy algorithm. In this thesis we combine RB and IGA for the efficient solution of parametrized problems for all the possible cases of NURBS geometrical parametrizations, which specifically include the NURBS control points, the weights, and both the control points and weights. In particular, we first focus on the solution of second order PDEs on parametrized lower dimensional manifolds, specifically surfaces in the three dimensional space. We consider geometrical parametrizations that entail a nonaffine dependence of the variational forms on the spatial coordinates and the geometric parameters. Thus, depending on the parametrization at hand and in order to ensure a suitable Offline/Online decomposition between the reduced order model construction and solution, we resort to the Empirical Interpolation Method (EIM) or the Matrix Discrete Empirical Interpolation Method (MDEIM), by comparing their performances. As application, we solve a class of benchmark structural problems modeled by Kirchoff-Love shells for which we consider NURBS geometric parametrizations and we apply the RB method to the solution of this class of fourth order PDEs. We highlight by means of numerical tests, the performance of the RB method applied to standard IGA approximation of parametrized shell geometries.

2015Pascal Frossard, Nikolaos Thomos, Nicolae Cleju

Network coding has been proposed recently as an efficient method to increase network throughput by allowing network nodes to combine packets instead of simply forwarding them. However, packet combinations in the network may increase delay, complexity and even generate overly redundant information when they are not designed properly. Typically, the best performance is not achieved when all the nodes perform network coding. In this paper, we address the problem of efficiently placing network coding nodes in overlay networks, so that the rate of innovating packets is kept high, and the delay for packet delivery is kept small. We first estimate the expected number of duplicated packets in each network node. These estimations permit to select the nodes that should implement network coding, so that the innovating rate increases. Two algorithms are then proposed for the cases where a central node is aware of the full network statistics and where each node knows the local statistics from its neighbor, respectively. The simulation results show that in the centralized scenario the maximum profit from network coding comes by adding only a few network coding nodes. A similar result is obtained with the algorithm based on local statistics, which moreover performs very close to the centralized solution. These results show that the proper selection of the network coding nodes is crucial for minimizing the transmission delay in streaming overlays.

2010Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.