10,220 research outputs found

    Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality

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    The problem of univariate mean change point detection and localization based on a sequence of nn independent observations with piecewise constant means has been intensively studied for more than half century, and serves as a blueprint for change point problems in more complex settings. We provide a complete characterization of this classical problem in a general framework in which the upper bound σ2\sigma^2 on the noise variance, the minimal spacing Δ\Delta between two consecutive change points and the minimal magnitude κ\kappa of the changes, are allowed to vary with nn. We first show that consistent localization of the change points, when the signal-to-noise ratio κΔσ<log(n)\frac{\kappa \sqrt{\Delta}}{\sigma} < \sqrt{\log(n)}, is impossible. In contrast, when κΔσ\frac{\kappa \sqrt{\Delta}}{\sigma} diverges with nn at the rate of at least log(n)\sqrt{\log(n)}, we demonstrate that two computationally-efficient change point estimators, one based on the solution to an 0\ell_0-penalized least squares problem and the other on the popular wild binary segmentation algorithm, are both consistent and achieve a localization rate of the order σ2κ2log(n)\frac{\sigma^2}{\kappa^2} \log(n). We further show that such rate is minimax optimal, up to a log(n)\log(n) term

    Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

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    We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.Comment: 28 pages, 12 figure
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