Robust visualization of trajectory data

2.1 Acquisition, processing, and analysis tasks

The raw data that is collected usually either is provided as a series of GPS fixes, i. e. composed of longitude and latitude values, and sometimes altitude values, or as x, y, z coordinates, e. g. from an inertial measurement unit (IMU) or video analysis in a local or global coordinate system. Data sets can be collected for small time periods, such as several minutes to analyze behavior in a controlled setting, or over large periods of time, up to several years, e. g. to analyze development or migration and to compare seasons and impact of environmental changes. Depending on the analysis goal and the available technology, the temporal resolution can range from a few data points per day to 1 Hz or even higher resolutions. The resulting data sets thus might range from only a few hundred data points to several tens of gigabyte in size with tens of millions of data points. The latter case in practice still requires reduction before further analysis or trajectory visualization can be performed, as scalability to such volume is currently not supported throughout the full analysis pipeline. Still, quality issues can occur on all levels of scale. Software on sensor tags might already preprocess the data, e. g. for error correction, and in particular IMUs need careful calibration and corresponding data might suffer from drift over longer periods of time [33]. Further preprocessing is required to ensure sufficient data quality for different analysis purposes, ranging from simple summary calculations to statistical methods and models, and visualization is recommended for all preprocessing steps, e. g. to ensure plausibility [32]. In addition, due the parameterization of the methods involved, an interactive visual analytics loop can greatly support the analysis and the understanding of raw data and the impact of preprocessing [34]. Visual analytics combines automated analysis techniques with interactive visualizations for effective understanding, reasoning and decision making of complex data sets. [35]. For larger projects, the analysis (see Figure 1) is often performed in a circular fashion, going from analysis of an initial data set either back to adjust cleaning, restructuring or pre-analysis, or to the collection and analysis of further data for extension or comparison reasons, e. g. new seasons or events. The location series then can be interpreted as the movement trajectory of the entity. In order to derive information from these trajectories, e. g. on human mobility or animal behaviour [37], [36], [38], a variety of analyses can be performed using the trajectory data, such as home range estimation, step selection, path selection, path construction, clustering, segmentation, as well as corridor construction and estimation.

Those analyses are then used to help solve movement-related tasks [38] and to answer movement-related research questions [39]:

1. Analysis of space use, e. g. home ranges of animals or activity spaces of humans.

2. Behaviour identification via quantifying moving patterns, such as resting, travelling etc. [14], and often supported by a segmentation of the trajectory to identify points of changing behavior.

3. Interaction and collective behavior in the case of multiple trajectories and entities.

4. Investigation of the processes that underlie movement patterns, such as identifying influence factors of variation in movement rates by the environment, or quantification of how sex and reproductive status influence the duration of, and transition among, different behavioural modes.

5. Prediction and modeling of movement.

2.2 Robustness requirements

The data quality issues can have a detrimental effect on the analysis for each of these tasks and questions: The extent to which space use is correctly identified, events and patterns can be detected, and further characterizations such as behaviour categories can be derived, is strongly affected by the coverage of movement features in the data that are characteristic for the targeted pattern or behavior. These could include turning points or spatio-temporal clusters, but also more complex features required in the categorization of movement or behavior. Detection methods and algorithms could be sensible to the absence or sparsity of such features, and thus might fail to detect events and patterns, or provide false results based on misinterpretation of the data. An analysis of the influence of location errors, as induced by different tracking technologies, on parameter estimation and subsequent biological pattern analysis for animal movement showed a significant decline in the ability to detect patterns [19]. In case of low resolution or missing data, interpolation methods can be used to fill the gap, but potentially even increase uncertainty or provide incorrect information. For example, dead reckoning can be used to calculate position information and fill up the gap by employing IMU data, but uncertainty in the resulting movement data can be further increased due to the imprecision and inaccuracy of measurement [19]. Sparsity of required movement features, and thus high uncertainty, might be caused simply by low temporal resolution in the acquisition, i. e. low sampling rate of sensors (or in a more general setting low resolution of the data) [18]. Models are developed with the aim to reduce the uncertainty of trajectories [17], such as Fuzzy C-Means (FCM), which is an clustering algorithm with noise, and further variants [16]. A second cause can be later removal or adjustment of data points in the preprocessing step, e. g. by probability sampling. In fact, some established methods might require resolutions lower than what is routinely collected these days, thus making data reduction steps mandatory [13]. In special cases, such as human mobility analysis, the restrictions of the built environments infrastructure can largely decrease the amount of data required for location estimation with high probability [12]. In contrast, in less restricted cases, such as animal social network analysis in the wild, different sampling rates can change the perceived structure of a network [20], and the simplified, binary method to construct a social network can lead to wrong interpretations about the social structure and wrong inferences about the position of individuals within the network [21].

Statistic frameworks and models are widely used in animal behavior analysis, such as maximum-likelihood estimation (MLE), a method to estimate distribution parameters based on an observed sample, or Bayesian inference which is based on Bayes’ theorem, where the probability for a hypothesis is updated when more evidence or information becomes available. The results of such methods based on the distributions and corresponding probability introduce uncertainty factors for the further analysis, yet those factors are usually not shown in the subsequent trajectory visualization. Adding these factors might help to increase analysis quality and trust. Section 4 gives examples on how visualizations can be enriched with uncertainty information to provide helpful input for the analysis.

Similar to the automated analysis, human interpretation of data might be mislead by uncertainty in the data or the visualization. The uncertainty stemming from the causes listed in Section 1 needs to be taken into account in the further processing, visualization, and interpretation in order avoid wrong conclusions. Both automated analysis methods and visualisation metaphors can be sensitive to changes in the input data characteristics. For example, methods for event or pattern detection in movement time-series might either miss events or assign wrong categories, e. g. for behaviour classes, to time windows when the temporal resolution of measured locations is not sufficient or the locations are off from the real location by a certain amount.

2.3 Example use case

We demonstrate robustness issues in animal behavior research, where animal movement tracks are analyzed to derive characteristics of animal decision making, social interaction, and influence of environmental factors. To this end, we show example trajectories that despite their small scale already exhibit features that allow to highlight visual analysis issues.

Figure 2

Goose migration trajectory. Left: using all data points from the data set. Right: using daily sampling rate. The zig-zag movement feature is removed (inside blue disc, see blue circle for a zoom-in).

Figure 3

Travel trajectory of a goose projected on a globe map using the Teamwise framework [10] (based on the cesium platform and using ESRI National geographic imagery, © ESRI). This visualization helps to take into account aspects of the surrounding environments which can help to interpret movement decisions, e. g. by showing topographic information.

Figure 2 shows two trajectory visualizations based on a goose migration data set (#12 from [30], [29]), one showing the full data set with a duration from end of January to August and a mean temporal resolution of around eighteen hours, and the other with a temporal resolution of one day. Even with this small change in resolution a major feature, a zig-zag movement pattern, is missing. Putting the movement into the context of the environment in which it happened is a common procedure for interpretation. Figure 3 shows the same trajectory projected on a globe visualization. It shows the zig-zag pattern happening at the Gulf of Riga, with a potentially important role for evidence of behavioral patterns. Such feature removals appear consistently for many different types of movement patterns when data reduction is performed, e. g. when following or circumflying natural resources such as rivers or mountains. Figure 4 shows three movement trajectories, where two are sampled from the original trajectory using the R adeHabitatLT package [6] with hourly and daily sampling resolutions. The red trajectory is generated using all data points in the file, the green with hourly sampling, and the blue trajectory using one data point each day. To generate the trajectory, the package requires a reference time data point, with the closest timepoint to the reference data point being picked daily or hourly sampling.

Figure 4

Sampling and data reduction effects: The red trajectory represents the full data set, the green one represents the a reduced data set with hourly rate, the blue one represents the daily rate trajectory.

Analyzing the movement using the blue line between data points is based on the hypothesis that a location is not only directly accessible from the previous one, but that also the direct way is taken in this case. However, there might be mountains between two locations that hinder direct access by the investigated animal, or rivers, roads, or lakes that it follows. Further causes of deviations could be bad weather or interaction with other animals. As a result, the animal will probably take a detour to reach the next location. In this case, the lowest sampling however still resembles closely the global shape of the movement, allowing the analyst to investigate the overall migration route. It however omits several curves that might contain important environmental features for animal decision making, most prominently the one leading from Voronezh to the region around Kharkiv (bottom left). Figure 5 shows a baboon movement with trajectory visualizations based on two temporal resolutions. The hourly resolution strongly changes the characteristics of important features such as river (dark green band from top left to bottom center) crossing points and orientation along roads (light orange band roughly following the river shape on the right hand side). The analysis based on time spend at locations, such as home range analysis or first passage time, without taking into account the environmental factors might draw an inaccurate conclusion based on the direct line trajectory [15].

Figure 5

Close-up of the baboon movement visualization from Figure 1, showing river and road structures that represent orientation or obstacles in animal movement. The red line shows the original temporal resolution, the green line hourly sampling.

3.1 Polyline simplification

Applying trajectory simplification, one needs to be careful that the simplification process does not erase important patterns and that the overall movement shape is sufficiently preserved. Therefore, the simplification is typically governed by a distance measure d ( T , T ′ ) which determines how similar the input trajectory T and the simplified trajectory T ′ are. The user then can specify an error threshold ε > 0, and the goal of the simplification process is to compute T ′ of smallest size, such that d ( T , T ′ ) ≤ ε. As the simplification is only concerned with the spatial aspects of the trajectories and not the temporal information, we will refer to trajectories as polylines in the remainder of the chapter. A polyline P is then simply a sequence of points p 1 , … , p n with p i ∈ R 2 and induced straight line segments between consecutive points. The size of a polyline equals the number of points n. Commonly used distance measures for polylines are the Hausdorff distance d H and the Fréchet distance d F . The latter is often deemed superior for movement analysis as it takes the order of the points along the polyline into account, while d H ignores this aspect. An optimal polyline simplification under d H can be computed in O ( n 2 ) [23] and under d F in O ( n 2 log n ) [24].

Figure 6

The blue line segment p i p j ‾ may only be part of the simplified polyline if it has a distance of at most ε to the subpolyline between p i and p j (marked black). Taking uncertainties into account (red disks), it is only valid if its distance towards any subpolyline realization within the given ordered disks is sufficiently small, including the one shown as dashed red line.

3.2 Uncertainty-aware simplification

In contrast to previous approaches which assume precise knowledge of the point locations, Buchin et al. [22] recently introduced the problem of uncertain polyline simplification, where the input is not a sequence of points but a sequence of regions. The regions are assumed to contain the true location. The goal of the simplification process is to compute a polyline P ′ such that d ( P , P ′ ) ≤ ε for all possible realizations of P given the input regions. We will focus here on the simple model in which all regions are disks. This complies with the fact that imprecision of GPS measurements can be expressed with an error radius r. As the error might differ depending on the absolute location, the received signal strength of the satellites and other aspects, the radius is not assumed to be homogeneous. Hence, the input is a sequence of disks D = D 1 , … , D n , where each disk D i = ( c i , r i ) is a tuple of a center point c i and an error radius r i ∈ R + . A polyline P = p 1 , … , p n is called a realization with respect to D if p i ∈ D i . See Figure 6 for an illustration of this concept.

Figure 7

Left: Input sequence of 50 disks with an exemplary polyline realization through their center points. Middle and right: Optimal simplifications for ε = 2 and ε = 3, respectively.

Figure 8

Migration routes of 18 geese.

Figure 9

Trajectory 3 from Figure 8 simplified with ε = 2. Circles indicate the uncertainty regions of the remaining points.

An example of a possible input disk sequence and the envisioned outcome is shown in Figure 7. The runtime to find the best simplified polyline in this setting is in O ( n 3 ) for both d H and d F [22]. But so far, no practical feasibility study was conducted.

We first conducted an experimental study on generated disk sequences. Using artificial data allows to control aspects as the number of disks n, the maximum radius r m a x , and the shape of the trajectory. Creating sequences with up to n = 1000 disks, and varying the value of ε, we observe that the running time in practice is quadratic in n and not cubic as predicted by the worst-case theoretical analysis. As a result, the running time of the algorithm is not much larger than that of the simplification algorithm that does not take uncertainty into account. In our experiments, the maximum increase in running time when considering disks was 25 %.

Based on the encouraging results of the feasibility study, we also conducted experiments on real-world data, particularly on the 18 trajectories from geese data set described in Section 2.3, which are depicted in Figure 8.

The data sets does not contain information about the uncertainty of the single locations. Hence we estimated error radii. We intentionally overestimated GPS uncertainty to make uncertainty-aware simplification – which needs to consider all possible realization – more difficult. Nevertheless, the simplification capability is quite pronounced even for small values of ε. For ε = 1, the average trajectory size was reduced to roughly 35 %, for ε = 2 to around 5 %, allowing for a cleaner visualization while still maintaining the overall shape. For larger values of ε, only very few data points remain and the trajectory becomes too oversimplified for visual analysis. But for sensible choices of ε, we get the desired granularity that allows for concise but meaningful visualization, see Figure 9 for an example.

Figure 10

Left image: Two animal trajectories (orange and green), that both visit a waterhole (blue) and then come close to each other (red disk). Middle: Simplification without context consideration. Right: Context-aware simplification.

Figure 11

Context-aware simplification of car trajectories. Colors indicate tree structures that were identified for faster consistent simplification.

3.3 Context-aware bundle simplification

When studying animal or human movement data, one is typically not only interested in a single trajectory but a collection of trajectories from different individuals, such as from a flock of birds or travels along a road network. If now each trajectory is simplified independently of the others, we might erase relevant interactions between the individuals, for example, groups of animals meeting at a location. Furthermore, as discussed for the example of baboon movement (see Figure 5) environmental elements as rivers might guide or influence their behaviour. Again, if all trajectory data points close to the river would be simplified away, the movement context might be lost for the viewer. Figure 10 (left and middle) shows an example where the analysis of the trajectory data might be heavily distorted after the simplification.

In [41], [42], we proposed the polyline bundle simplification problem. Here, given a collection of trajectories, we want to find the smallest simplification for the trajectories governed by a distance measure and an error threshold ε > 0 as before, but with the additional constraint that points that occur in multiple trajectories have either to be kept in all of them or deleted from all of them. This consistency criterion unfortunately makes the problem NP-hard. But we showed that an efficient bicriteria approximation algorithm exists and also devised an efficient heuristic based on partitioning the trajectories into bundles that exhibit a tree structure.

Note that in real-world data, trajectories rarely share exactly the same location measurements, even if the respective humans or animals are in the same place. To capture close-by points we can hence again use an uncertainty radius and then impose the consistency criterion whenever two points from different trajectories are within the respective circle. Furthermore, we can also make the criterion more strict by demanding that such points have to be kept in the simplified output. In that way, we could also easily safe points close to important landmarks (as rivers or waterholes) from being simplified away, see Figure 10, right. This stricter criterion does not only help to ensure that relevant parts of the trajectories are kept but also makes the problem easier again from a computational perspective. Experiments conducted on large data sets demonstrate that even large bundles can be processed within a few seconds. Figure 11 shows the produced solution on a set of several thousand car trajectories with over a million data points.