10,220 research outputs found
Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality
The problem of univariate mean change point detection and localization based
on a sequence of 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 on the noise variance, the minimal spacing
between two consecutive change points and the minimal magnitude of the
changes, are allowed to vary with . We first show that consistent
localization of the change points, when the signal-to-noise ratio , is impossible. In contrast, when
diverges with at the rate of at least
, we demonstrate that two computationally-efficient change
point estimators, one based on the solution to an -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
. We further show that such rate is minimax
optimal, up to a term
Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction
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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