8,274 research outputs found
Statistical Inference using the Morse-Smale Complex
The Morse-Smale complex of a function decomposes the sample space into
cells where is increasing or decreasing. When applied to nonparametric
density estimation and regression, it provides a way to represent, visualize,
and compare multivariate functions. In this paper, we present some statistical
results on estimating Morse-Smale complexes. This allows us to derive new
results for two existing methods: mode clustering and Morse-Smale regression.
We also develop two new methods based on the Morse-Smale complex: a
visualization technique for multivariate functions and a two-sample,
multivariate hypothesis test.Comment: 45 pages, 13 figures. Accepted to Electronic Journal of Statistic
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
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