## The Annals of Statistics

### Adaptive multiscale detection of filamentary structures in a background of uniform random points

#### Abstract

We are given a set of n points that might be uniformly distributed in the unit square [0,1]2. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with Cα-norm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T(α,β)>0, if the number of points sampled from the curve is smaller than T(α,β)n1/(1+α), reliable detection is not possible for large n. We describe a multiscale significant-runs algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T*(α,β)n1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T*/T.

At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will have area 2/n and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices all are strips of the same anisotropy and whose edges link such strips to their “good continuations.” The point-cloud data are reduced to counts that measure membership in strips. Each anisotropy graph is reduced to a subgraph that consist of strips with significant counts. The algorithm rejects H0 whenever some such subgraph contains a path that connects many consecutive significant counts.

#### Article information

Source
Ann. Statist., Volume 34, Number 1 (2006), 326-349.

Dates
First available in Project Euclid: 2 May 2006

https://projecteuclid.org/euclid.aos/1146576265

Digital Object Identifier
doi:10.1214/009053605000000787

Mathematical Reviews number (MathSciNet)
MR2275244

Zentralblatt MATH identifier
1091.62095

Subjects
Primary: 62M30: Spatial processes
Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties

#### Citation

Arias-Castro, Ery; Donoho, David L.; Huo, Xiaoming. Adaptive multiscale detection of filamentary structures in a background of uniform random points. Ann. Statist. 34 (2006), no. 1, 326--349. doi:10.1214/009053605000000787. https://projecteuclid.org/euclid.aos/1146576265

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