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A fully data-driven method for estimating the shape of a point cloud. (English) Zbl 1357.62228

Summary: Given a random sample of points from some unknown distribution, we propose a new data-driven method for estimating its probability support \(S\). Under the mild assumption that \(S\) is \(r\)-convex, the smallest \(r\)-convex set which contains the sample points is the natural estimator. The main problem for using this estimator in practice is that \(r\) is an unknown geometric characteristic of the set \(S\). A stochastic algorithm is proposed for selecting its optimal value from the data under the hypothesis that the sample is uniformly generated. The new data-driven reconstruction of \(S\) is able to achieve the same convergence rates as the convex hull for estimating convex sets, but under a much more flexible smoothness shape condition.

MSC:

62H35 Image analysis in multivariate analysis
62H15 Hypothesis testing in multivariate analysis
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference

Software:

alphahull
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Full Text: DOI arXiv

References:

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