Central Trajectories

TL;DR

This paper introduces the concept of a central trajectory for summarizing sets of similar trajectories, providing algorithms to compute such trajectories efficiently in one and higher dimensions.

Contribution

It defines a new central trajectory measure for trajectory clustering and develops algorithms with complexity bounds for computing it in various dimensions.

Findings

Optimal central trajectory in 1D has complexity Θ(τ n^2)

Algorithm computes central trajectory in O(τ n^2 log n) time in 1D

Complexity in higher dimensions is at most O(τ n^{5/2}) with computation time O(τ n^3)

Abstract

An important task in trajectory analysis is clustering. The results of a clustering are often summarized by a single representative trajectory and an associated size of each cluster. We study the problem of computing a suitable representative of a set of similar trajectories. To this end we define a central trajectory $\mathcal{C}$, which consists of pieces of the input trajectories, switches from one entity to another only if they are within a small distance of each other, and such that at any time $t$, the point $\mathcal{C}(t)$ is as central as possible. We measure centrality in terms of the radius of the smallest disk centered at $\mathcal{C}(t)$ enclosing all entities at time $t$, and discuss how the techniques can be adapted to other measures of centrality. We first study the problem in $\mathbb{R}^1$, where we show that an optimal central trajectory $\mathcal{C}$ representing $n$ trajectories, each consisting of $\tau$ edges, has complexity $\Theta(\tau n^2)$ and can be computed in $O(\tau n^2 \log n)$ time. We then consider trajectories in $\mathbb{R}^d$ with $d\geq 2$, and show that the complexity of $\mathcal{C}$ is at most $O(\tau n^{5/2})$ and can be computed in $O(\tau n^3)$ time.