## The Annals of Statistics

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

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

Ery Arias-Castro, David L. Donoho, and Xiaoming Huo

#### 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 *C*^{α}-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*)*n*^{1/(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*)*n*^{1/(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 **H**_{0} 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

**Permanent link to this document**

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

**Keywords**

Multiscale geometric analysis pattern recognition good continuation Erdös–Rényi laws runs test beamlets

#### 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