108 research outputs found
On the length of one-dimensional reactive paths
Motivated by some numerical observations on molecular dynamics simulations,
we analyze metastable trajectories in a very simplecsetting, namely paths
generated by a one-dimensional overdamped Langevin equation for a double well
potential. More precisely, we are interested in so-called reactive paths,
namely trajectories which leave definitely one well and reach the other one.
The aim of this paper is to precisely analyze the distribution of the lengths
of reactive paths in the limit of small temperature, and to compare the
theoretical results to numerical results obtained by a Monte Carlo method,
namely the multi-level splitting approach
Efficient large deviation estimation based on importance sampling
We present a complete framework for determining the asymptotic (or
logarithmic) efficiency of estimators of large deviation probabilities and rate
functions based on importance sampling. The framework relies on the idea that
importance sampling in that context is fully characterized by the joint large
deviations of two random variables: the observable defining the large deviation
probability of interest and the likelihood factor (or Radon-Nikodym derivative)
connecting the original process and the modified process used in importance
sampling. We recover with this framework known results about the asymptotic
efficiency of the exponential tilting and obtain new necessary and sufficient
conditions for a general change of process to be asymptotically efficient. This
allows us to construct new examples of efficient estimators for sample means of
random variables that do not have the exponential tilting form. Other examples
involving Markov chains and diffusions are presented to illustrate our results.Comment: v1: 34 pages, 8 figures; v2: Typos corrected; v3: More mathematical
version containing technical modifications in Assumption 2, Assumption 3, and
Eq. (53) needed in some of the proof
Nearest neighbor classification in infinite dimension
Let be a random element in a metric space (\calF,d), and let be a random variable with value or . is called the class, or the label, of . Assume i.i.d. copies (X_i,Y_i)_1\leqi\leqn. The problem of classification is to predict the label of a new random element . The -nearest neighbor classifier consists in the simple following rule : look at the nearest neighbors of and choose or for its label according to the majority vote. If (\calF,d)=(R^d,||.||), Stone has proved in 1977 the universal consistency of this classifier : its probability of error converges to the Bayes error, whatever the distribution of . We show in this paper that this result is no more valid in general metric spaces. However, if (\calF,d) is separable and if a regularity condition is assumed, then the -nearest neighbor classifier is weakly consistent
A multiple replica approach to simulate reactive trajectories
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
On the Rate of Convergence of the Functional -NN Estimates
Let be a general separable metric space and denote by \mathcal D_n=\{(\bX_1,Y_1), \hdots, (\bX_n,Y_n)\} independent and identically distributed -valued random variables with the same distribution as a generic pair (\bX, Y). In the regression function estimation problem, the goal is to estimate, for fixed \bx \in \mathcal F, the regression function r(\bx)=\mathbb E[Y|\bX=\bx] using the data . Motivated by a broad range of potential applications, we propose, in the present contribution, to investigate the properties of the so-called -nearest neighbor regression estimate. We present explicit general finite sample upper bounds, and particularize our results to important function spaces, such as reproducing kernel Hilbert spaces, Sobolev spaces or Besov spaces
Sur la vitesse de convergence de l'estimateur du plus proche voisin baggé
International audienceOn s'intéresse dans cette communication à l'estimation de la fonction de
New insights into Approximate Bayesian Computation
International audienceApproximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with ABC, which is an interesting hybrid between a k-nearest neighbor and a kernel method
On the Hill relation and the mean reaction time for metastable processes
We illustrate how the Hill relation and the notion of quasi-stationary
distribution can be used to analyse the error introduced by many algorithms
that have been proposed in the literature, in particular in molecular dynamics,
to compute mean reaction times between metastable states for Markov processes.
The theoretical findings are illustrated on various examples demonstrating the
sharpness of the error analysis as well as the applicability of our study to
elliptic diffusions
Mathematical statistics.
Approximate Bayesian Computation (abc for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with abc, which is an interesting hybrid between a k-nearest neighbor and a kernel method
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