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Convergence and rates for fixed-interval multiple-track smoothing using k-means type optimization
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Thorpe, Matthew and Johansen, Adam M. (2016) Convergence and rates for fixed-interval multiple-track smoothing using k-means type optimization. Electronic Journal of Statistics, 10 (2). pp. 3693-3722. doi:10.1214/16-EJS1209
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Official URL: http://doi.org/10.1214/16-EJS1209
Abstract
We address the task of estimating multiple trajectories from unlabeled data. This problem arises in many settings, one could think of the construction of maps of transport networks from passive observation of travellers, or the reconstruction of the behaviour of uncooperative vehicles from external observations, for example. There are two coupled problems. The first is a data association problem: how to map data points onto individual trajectories. The second is, given a solution to the data association problem, to estimate those trajectories. We construct estimators as a solution to a regularized variational problem (to which approximate solutions can be obtained via the simple, efficient and widespread k-means method) and show that, as the number of data points, n, increases, these estimators exhibit stable behaviour. More precisely, we show that they converge in an appropriate Sobolev space in probability and with rate n −1/2.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science > Statistics | ||||||
| Library of Congress Subject Headings (LCSH): | Asymptotic efficiencies (Statistics), Sobolev spaces | ||||||
| Journal or Publication Title: | Electronic Journal of Statistics | ||||||
| Publisher: | Institute of Mathematical Statistics | ||||||
| ISSN: | 1935-7524 | ||||||
| Official Date: | 3 December 2016 | ||||||
| Dates: |
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| Volume: | 10 | ||||||
| Number: | 2 | ||||||
| Page Range: | pp. 3693-3722 | ||||||
| DOI: | 10.1214/16-EJS1209 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Funder: | Engineering and Physical Sciences Research Council (EPSRC), Selex ES Ltd |
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