## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 10, Number 2 (2016), 3693-3722.

### Convergence and rates for fixed-interval multiple-track smoothing using $k$-means type optimization

Matthew Thorpe and Adam M. Johansen

#### 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 $n^{-1/2}$.

#### Article information

**Source**

Electron. J. Statist., Volume 10, Number 2 (2016), 3693-3722.

**Dates**

Received: October 2015

First available in Project Euclid: 3 December 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1480734075

**Digital Object Identifier**

doi:10.1214/16-EJS1209

**Mathematical Reviews number (MathSciNet)**

MR3579199

**Zentralblatt MATH identifier**

1357.62203

**Subjects**

Primary: 62G20: Asymptotic properties

**Keywords**

Asymptotics $k$-means non-parametric regression rates of convergence variational methods

#### Citation

Thorpe, Matthew; Johansen, Adam M. Convergence and rates for fixed-interval multiple-track smoothing using $k$-means type optimization. Electron. J. Statist. 10 (2016), no. 2, 3693--3722. doi:10.1214/16-EJS1209. https://projecteuclid.org/euclid.ejs/1480734075