A Refined Definition for Groups of Moving Entities and its Computation

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. [JoCG, 2015] 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 n moving entities in R^1, specified by linear interpolation in a sequence of tau time stamps, we show that all maximal groups can be computed in O(tau^2 n^4) time. A similar approach applies if the time stamps of entities are not the same, at the cost of a small extra factor of alpha(n) in the running time. In higher dimensions, we can compute all maximal groups in O(tau^2 n^5 log n) time (for any constant number of dimensions). We also show that one tau factor can be traded for a much higher dependence on n by giving a O(tau n^4 2^n) 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 n and linear dependence on tau

An Experimental Evaluation of Grouping Definitions for Moving Entities

One important pattern analysis task for trajectory data is to find a group: a set of entities that travel together over a period of time. In this paper, we compare four definitions of groups by conducting extensive experiments using various data sets. The grouping definitions are different by one or more of three different characteristics: whether they use the measured sample points or continuous movement, how distance is used to decide if entities are in the same group, and whether the duration of the group is measured cumulatively or as one contiguous time interval. We are interested in the differences between the definitions and comparisons to human-annotated data, if available. We concentrate on pedestrian data and on different crowd densities. Furthermore, we analyze the robustness of the definitions with respect to their dependence on different sampling rates. We use two types of trajectory data sets: synthetic trajectories and real-life trajectories extracted from video surveillance. We present the results of the quantitative evaluations. For experiments with real-life trajectories, we augment them with a qualitative evaluation using videos that show groups in the trajectories with a color coding

Processing aggregated data : the location of clusters in health data

Spatially aggregated data is frequently used in geographical applications. Often spatial data analysis on aggregated data is performed in the same way as on exact data, which ignores the fact that we do not know the actual locations of the data. We here propose models and methods to take aggregation into account. For this we focus on the problem of locating clusters in aggregated data. More specifically, we study the problem of locating clusters in spatially aggregated health data. The data is given as a subdivision into regions with two values per region, the number of cases and the size of the population at risk. We formulate the problem as finding a placement of a cluster window of a given shape such that a cluster function depending on the population at risk and the cases is maximized. We propose area-based models to calculate the cases (and the population at risk) within a cluster window. These models are based on the areas of intersection of the cluster window with the regions of the subdivision. We show how to compute a subdivision such that within each cell of the subdivision the areas of intersection are simple functions. We evaluate experimentally how taking aggregation into account influences the location of the clusters found.Peer ReviewedPostprint (published version

Computations and Measures of Collective Movement Patterns Based on Trajectory Data

Typically, movement data of a moving entity (e.g., people, animal, or other moving objects) is described as a trajectory: a path made by a moving entity as it travels through space over a period of time. Various advanced tracking technologies such as Global Positioning Systems (GPS) able to gather trajectory data as a series of time-stamped locations where the position of the moving entity was recorded. One of the important tasks in the analysis of trajectory data is to recognize various patterns that may emerge from the movement of entities. These movement patterns are essential since they can reflect the behavior of an individual entity or show the relationship among multiple moving entities. In this thesis, we mostly focus on one particular type of pattern: the collective movement pattern or grouping. This pattern occurs when multiple entities travel together for a sufficiently long period of time. We begin our study on the collective movement with the analysis of various measures for a group of trajectories. For a single trajectory, these measures give a single value for the whole trajectory, for example, the average speed and the global direction. We extend measures for a single trajectory to a group of trajectories and add other measures specific for groups that do not exist for an individual trajectory, such as the density of a group. We show that a few tasks related to trajectory analysis, like the visualization of a large collection of trajectories, may use measures to improve its results. Next, we describe a new definition to model a collective movement. We present examples that in dense environments, we argue that our proposed definition corresponds better to human intuition. We formalize the model and give efficient algorithms to compute the refined groups from a set of trajectories. Finally, we compare four definitions of a collective movement experimentally, including our refined definition of groups. We evaluate the differences between each definition and human-annotated data quantitatively. We also examine the dependency of the grouping definitions on the density of the entities and the sampling rate of the input trajectories. For the qualitative analysis, we developed a novel visualization system that shows the reported groups with a color-coding in video footage. In addition to the collective movement pattern, we also study the polyline simplification problem. In particular, we look into the polyline simplification problem where given an input polyline, compute the output polyline that resembles the input and use the least number of vertices of the input such that the distance between the input and output polyline is at most ε (ε>0). In this thesis, we use two distance measures: the Hausdorff and Fréchet distance. Using these two distance measures, we first compare the two well-known algorithms for polyline simplification problems, the Douglas-Peucker and the Imai-Iri, with the optimum simplification possible. Finally, we consider the computation of the optimal simplification using the Hausdorff and Fréchet distance

An Experimental Comparison of Two Definitions for Groups of Moving Entities (Short Paper)

Two of the grouping definitions for trajectories that have been developed in recent years allow a continuous motion model and allow varying shape groups. One of these definitions was suggested as a refinement of the other. In this paper we perform an experimental comparison to highlight the differences in these two definitions on various data sets

Computations and Measures of Collective Movement Patterns Based on Trajectory Data

Typically, movement data of a moving entity (e.g., people, animal, or other moving objects) is described as a trajectory: a path made by a moving entity as it travels through space over a period of time. Various advanced tracking technologies such as Global Positioning Systems (GPS) able to gather trajectory data as a series of time-stamped locations where the position of the moving entity was recorded. One of the important tasks in the analysis of trajectory data is to recognize various patterns that may emerge from the movement of entities. These movement patterns are essential since they can reflect the behavior of an individual entity or show the relationship among multiple moving entities. In this thesis, we mostly focus on one particular type of pattern: the collective movement pattern or grouping. This pattern occurs when multiple entities travel together for a sufficiently long period of time. We begin our study on the collective movement with the analysis of various measures for a group of trajectories. For a single trajectory, these measures give a single value for the whole trajectory, for example, the average speed and the global direction. We extend measures for a single trajectory to a group of trajectories and add other measures specific for groups that do not exist for an individual trajectory, such as the density of a group. We show that a few tasks related to trajectory analysis, like the visualization of a large collection of trajectories, may use measures to improve its results. Next, we describe a new definition to model a collective movement. We present examples that in dense environments, we argue that our proposed definition corresponds better to human intuition. We formalize the model and give efficient algorithms to compute the refined groups from a set of trajectories. Finally, we compare four definitions of a collective movement experimentally, including our refined definition of groups. We evaluate the differences between each definition and human-annotated data quantitatively. We also examine the dependency of the grouping definitions on the density of the entities and the sampling rate of the input trajectories. For the qualitative analysis, we developed a novel visualization system that shows the reported groups with a color-coding in video footage. In addition to the collective movement pattern, we also study the polyline simplification problem. In particular, we look into the polyline simplification problem where given an input polyline, compute the output polyline that resembles the input and use the least number of vertices of the input such that the distance between the input and output polyline is at most ε (ε>0). In this thesis, we use two distance measures: the Hausdorff and Fréchet distance. Using these two distance measures, we first compare the two well-known algorithms for polyline simplification problems, the Douglas-Peucker and the Imai-Iri, with the optimum simplification possible. Finally, we consider the computation of the optimal simplification using the Hausdorff and Fréchet distance

On optimal polyline simplification using the Hausdorff and Fréchet distance

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given ε > 0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most ε.We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than we require for the optimum solution. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the (continuous) Fréchet distance, we give an $O(kn^5)$ time algorithm that requires $O(kn^2)$ space, where $k$ is the output complexity of the simplification

A Refined Definition for Groups of Moving Entities and its Computation

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 n moving entities in R1, specified by linear interpolation in a sequence of τ time stamps, we show that all maximal groups can be computed in O(τ2n4) time. A similar approach applies if the time stamps of entities are not the same, at the cost of a small extra factor of α(n) in the running time, where α denotes the inverse Ackermann function. In higher dimensions, we can compute all maximal groups in O(τ2n5logn) 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 τ factor can be traded for a much higher dependence on n by giving a O(τn42n) 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 nand linear dependence on τ