A Refined Definition for Groups of Moving Entities and Its Computation

Abstract

One of the important tasks in the analysis of spatio-temporal data collected from moving entities is to find a group: a set of entities that travel together for a sufficiently long period of time. Buchin et al.2 introduce a formal definition of groups, analyze its mathematical structure, and present efficient algorithms for computing all maximal groups in a given set of trajectories. In this paper, we refine their definition and argue that our proposed definition corresponds better to human intuition in certain cases, particularly in dense environments. We present algorithms to compute all maximal groups from a set of moving entities according to the new definition. For a set of [Formula: see text] moving entities in [Formula: see text], specified by linear interpolation in a sequence of [Formula: see text] time stamps, we show that all maximal groups can be computed in [Formula: see text] time. A similar approach applies if the time stamps of entities are not the same, at the cost of a small extra factor of [Formula: see text] in the running time, where [Formula: see text] denotes the inverse Ackermann function. In higher dimensions, we can compute all maximal groups in [Formula: see text] time (for any constant number of dimensions), regardless of whether the time stamps of entities are the same or not. We also show that one [Formula: see text] factor can be traded for a much higher dependence on [Formula: see text] by giving a [Formula: see text] algorithm for the same problem. Consequently, we give a linear-time algorithm when the number of entities is constant and the input size relates to the number of time stamps of each entity. Finally, we provide a construction to show that it might be difficult to develop an algorithm with polynomial dependence on [Formula: see text] and linear dependence on [Formula: see text].

Multi-Granular Trend Detection for Time-Series Analysis

“…In this section we describe how the work on trajectory grouping by Buchin et al [5] and Van Kreveld et al [40] can be used to detect trends. We first revisit their definitions, and see how they relate to trends in time-series data in Section 3.1.…”

“…4. Therefore, van Kreveld et al [40] give a refined definition of group in which the entities in group G have to be ε-connected using only entities that are also in G. The number of maximal groups according to this definition is still O(τn 3 ). Computing all of them takes O(τ 2 n 4 ) time.…”

“…The algorithm by van Kreveld et al [40] can find all groups in a set of trajectories (or in an ensemble) in O(τ 2 n 4 ) time. We are interested in a more restricted setting where all groups start at the initial time t s .…”

“…We propose a geometric model for trend-detection in one-dimensional time-varying data, inspired by topological grouping structures for moving objects in two-or higher-dimensional space. Specifically, our model builds on the work by Buchin et al [5] and Van Kreveld et al [40]. Trends are detected using three natural parameters: granularity, support-size, and duration.…”

“…Kostitsyna et al [22] and Löffler et al [23] extend the work of Buchin et al to environments with obstacles. Van Kreveld et al [40] recently observed that in dense environments the original definition by Buchin et al may not capture the nature of a group well. They propose a slight variation on the definition for this setting.…”