Adaptive multiscale detection of filamentary structures in a background of uniform random points
| dc.creator | Arias-Castro, Ery | |
| dc.creator | Donoho, David L. | |
| dc.creator | Huo, Xiaoming | |
| dc.date | 2006-05-18 | |
| dc.date.accessioned | 2026-07-07T08:07:50Z | |
| dc.date.available | 2026-07-07T08:07:50Z | |
| dc.description | 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 $\mathbf{H}_0$ whenever some such subgraph contains a path that connects many consecutive significant counts. | |
| dc.description | Published at http://dx.doi.org/10.1214/009053605000000787 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0605513 | |
| dc.identifier | http://arxiv.org/abs/math/0605513 | |
| dc.identifier | Annals of Statistics 2006, Vol. 34, No. 1, 326-349 | |
| dc.identifier | doi:10.1214/009053605000000787 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/131063 | |
| dc.subject | Statistics Theory | |
| dc.subject | 62M30 (Primary) 62G10, 62G20 (Secondary) | |
| dc.title | Adaptive multiscale detection of filamentary structures in a background of uniform random points | |
| dc.type | text |