Showing papers by "Arnaud Guyader published in 2010"

On the Rate of Convergence of the Bagged Nearest Neighbor Estimate

Abstract: Bagging is a simple way to combine estimates in order to improve their performance. This method, suggested by Breiman in 1996, proceeds by resampling from the original data set, constructing a predictor from each subsample, and decide by combining. By bagging an n-sample, the crude nearest neighbor regression estimate is turned into a consistent weighted nearest neighbor regression estimate, which is amenable to statistical analysis. Letting the resampling size kn grows appropriately with n, it is shown that this estimate may achieve optimal rate of convergence, independently from the fact that resampling is done with or without replacement. Since the estimate with the optimal rate of convergence depends on the unknown distribution of the observations, adaptation results by data-splitting are presented.

Rates of Convergence of the Functional $k$ -Nearest Neighbor Estimate

Abstract: Let F be a separable Banach space, and let (X, Y) be a random pair taking values in F × R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate rn (x) of the regression function r(x) = E[Y|X = x], based on n independent copies of the pair (X, Y). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[rn(X) - r(X)]2, and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces, and reproducing kernel Hilbert spaces.

A multiple replica approach to simulate reactive trajectories

Abstract: A method to generate reactive trajectories, namely equilibrium trajectories leaving a metastable state and ending in another one is proposed. The algorithm is based on simulating in parallel many copies of the system, and selecting the replicas which have reached the highest values along a chosen one-dimensional reaction coordinate. This reaction coordinate does not need to precisely describe all the metastabilities of the system for the method to give reliable results. An extension of the algorithm to compute transition times from one metastable state to another one is also presented. We demonstrate the interest of the method on two simple cases: a one-dimensional two-well potential and a two-dimensional potential exhibiting two channels to pass from one metastable state to another one.