Difference Between Trajectory Computing and Time Series Analytics

A trajectory is commonly a time series of locations in some metric space. This leads to a lot of purely geometric approaches to trajectories and their interrelations. On the opposite, the definition of time series and the associated algorithms (time-frequency analysis, time series data mining, etc.) do not provide this geometric interpretations as the events in a time series need not come from a metric space and geometric approaches might fail.

Trajectory vs. Time Series

This figure illustrates the difference for a specific segment. For trajectory computing algorithms, the dashed line to the left defines, for example, the distance between this line segment and the other trajectory. From a time series perspective, only the dashed lines on the right of the figure can be accessed.

At first, it seems to be very pedantic to make a distinction between time series and trajectories. But, time series represent a much wider concept as compared to trajectories. While time series do not assume anything about the values the series takes between samples, trajectories represent the assumption of continuity. Therefore, a trajectory computing algorithm will not only argue about the sampled points (though it might calculate only on them), instead it will argue about some interpolation of the samples.

One predominant interpolation in this context is given by the piecewise linear interpolation. This interpolation connects all samples locations in a trajectory by linear line segments. It is widely used, because more complex interpolations such as splines don’t allow of efficient algorithms.

The borders between trajectory computing and time series becomes blurred by the fact that many applications will work using algorithms from the other domain. There are many examples, where some time series analytics on trajectories are successful. Especially, when the number of samples in the trajectory is high enough such that forgetting about the geometry is not too harmful. Vice versa, analyzing time series from non-continuous domains by assuming linear interpolation can be very successful.

Hence, there is no practicable distinction between what perspective on a given dataset should be used. It is even the case, that many (most?) trajectory computing algorithms exist in a modified form which works on the point set alone, therefore, in the time series domain.

This is often done in order to reduce computational complexity. Those variants are often called like the original with the additional word discrete.

For example, the Discrete Hausdorff Distance of trajectories is not defined to be the Hausdorff-distance of the set of points in the linear interpolation of the points in the trajectory. Instead, it approximates this measure by using only the sets of points used to model the trajectory. Similarly, the Fréchet distance can be approximated by the Discrete Fréchet Distance calculated on the point set alone.

These discrete approximations to useful distance measures for spatial trajectories have been introduced to reduce computational complexity. On the one hand side, these two algorithms are much simpler as the relative geometry of points is used instead of the relative geometry of line segments. However, the relative geometry of points is sufficiently specified by their distance. This reduces computational overhead a lot. Additionally, in large datasets, spatial indexing for points can be used. Points have the welcome property that they are local. Therefore, sets of points are easy to separate into disjoint subsets. Most spatial indices are based on this and allow for sublinear access to nearest neighbors and all points in a given spatial range.

In summary, trajectory computing is concerned with full geometric objects given by interpolating the samples while time series analytics restricts algorithms to the set of points, alone. In practice, many trajectory computing algorithms are approximated by time series algorithms in order to handle large amounts of data and the distinction becomes blurry.