11,703 research outputs found
Nonparametric ridge estimation
We study the problem of estimating the ridges of a density function. Ridge
estimation is an extension of mode finding and is useful for understanding the
structure of a density. It can also be used to find hidden structure in point
cloud data. We show that, under mild regularity conditions, the ridges of the
kernel density estimator consistently estimate the ridges of the true density.
When the data are noisy measurements of a manifold, we show that the ridges are
close and topologically similar to the hidden manifold. To find the estimated
ridges in practice, we adapt the modified mean-shift algorithm proposed by
Ozertem and Erdogmus [J. Mach. Learn. Res. 12 (2011) 1249-1286]. Some numerical
experiments verify that the algorithm is accurate.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1218 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Nonparametric Ridge Estimation
We study the problem of estimating the ridges of a density function. Ridge estimation is an extension of mode finding and is useful for understanding the structure of a density. It can also be used to find hidden structure in point cloud data. We show that, under mild regularity conditions, the ridges of the kernel density estimator consistently estimate the ridges of the true density. When the data are noisy measurements of a manifold, we show that the ridges are close and topologically similar to the hidden manifold. To find the estimated ridges in practice, we adapt the modified mean-shift algorithm proposed by Ozertem and Erdogmus [J. Mach. Learn. Res. 12 (2011) 1249–1286]. Some numerical experiments verify that the algorithm is accurate
Theoretical Analysis of Nonparametric Filament Estimation
This paper provides a rigorous study of the nonparametric estimation of
filaments or ridge lines of a probability density . Points on the filament
are considered as local extrema of the density when traversing the support of
along the integral curve driven by the vector field of second eigenvectors
of the Hessian of . We `parametrize' points on the filaments by such
integral curves, and thus both the estimation of integral curves and of
filaments will be considered via a plug-in method using kernel density
estimation. We establish rates of convergence and asymptotic distribution
results for the estimation of both the integral curves and the filaments. The
main theoretical result establishes the asymptotic distribution of the uniform
deviation of the estimated filament from its theoretical counterpart. This
result utilizes the extreme value behavior of non-stationary Gaussian processes
indexed by manifolds as .Comment: 55 pages, 1 figur
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