## Abstract

The analysis of large-scale trajectory data has tremendous benefits for applications ranging from transportation planning to traffic management. A fundamental building block for the analysis of such data is the computation of similarity between trajectories. Existing work for similarity computation focuses mainly on the spatial aspects of trajectories, but more rarely takes into account time in conjunction with space. A key challenge when considering time is how to handle trajectories that are sampled asynchronously or at variable rates, which can lead to uncertainty. To tackle this problem, we quantify trajectory similarity as an interval, rather than a single value, to capture the uncertainty that can result from different sampling rates and asynchronous sampling. Based on this perspective, we develop a new trajectory similarity measure, Trajectory Interval Distance Estimation, which models similarity computation as a convex optimisation problem. Using two real datasets, we demonstrate that our proposed measure is extremely effective for assessing similarity in comparison to existing state of the art measures.

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## Notes

- 1.
https://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm. Accessed 20 Jun 2018.

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## A Appendix

### A Appendix

### A.1 Accuracy using NDCG measure

In this section, we use Normalized Discounted Cumulative Gain (Järvelin and Kekäläinen 2002) to examine the accuracy of different measures that we mentioned in Sect. 5 for Cabspotting dataset.

For every measure, similar to Sect. 5, the ground truth is built by computing *k* nearest neighbours (*k*-NN) set for a given query trajectory from a given set of original reference trajectories *DB*. The trajectories in *k*-NN are ordered from most similar to less similar trajectory. Then we give the relevance value of *k* to the first trajectory (the most similar trajectory) and relevance value of 1 to the last trajectory. We compute *k*-NN’ for the same query trajectory from the dataset with lower sampling rate. We eliminate all the trajectories in set *k*-NN’ - *k*-NN (*k*-NN’ = *k*-NN’ \(\cap \)*k*-NN). Then, we assign a relevance to each of them based on their rank in *k*-NN’. For example, take \(T_1\) to \(T_{10}\) as reference trajectories and *Q* as a given query trajectory. Using one of measures 4-NN set for *Q* is \(<T_4, T_9, T_2, T_7>\) as the most similar trajectories in order (it means that \(T_4\) is the most similar trajectory to *Q* and \(T_9\) is the second most similar trajectory to *Q* and so on.). We give a relevance value based on their similarity rank (\(<4,3,2,1>\)). It means that relevance value for \(T_4\) is 4 which means it has highest relevance to *Q*. Also, we generate a lower sampled version of \(T_1\) to \(T_10\) by choosing \(50\%\) of their sampled points randomly and build trajectories \(T'_1\) to \(T'_10\). Then we extract 4-NN’ for the query trajectory *Q* from \(T'_1\) to \(T'_{10}\). The ideal situation is to extract the same set of similar trajectories with the same order so that 4-NN and 4-NN’ have the NDCG of 1. However, if a measure extracts \(<T_9, T_4, T_5, T_7> \)the same similar trajectories in order, we eliminate \(T_5\) as it is not in *k*-NN. Then, the relevance values for the set \(<T_4, T_9, T_2, T_7>\) using *k*-NN’ is \(<3,4,0,1>\) (\(T_4\) has the relevance of 3 in *k*-NN’ and \(T_9\) has the relevance of 4, \(T_2\) is not in *k*-NN’ and \(T_7\) has the relevance of 1. Indeed, the ideal relevance for the set \(<T_4, T_9, T_2, T_7>\) using that given measure is \(<4,3,2,1>\), however, for the lower sampled version of trajectories, it is \(<3,4,0,1>\). Then we compute DCG for \(<4,3,2,1>\) as the ideal DCG (IDCG) and for \(<3,4,0,1>\) as given DCG (GDCG). Then, we divide GDCG by IDCG.

Figure 13 shows the results for cabspotting dataset. Similar to the results of Spearman’s rank correlation, TIDE and TIDE* have higher accuracy in comparison to other measures. However, since NDCG does not penalize for “bad“ trajectories in the results, we see better results in comparison to Spearman’s rank correlation.

### A.2 The impact of estimated maximum speed

As discussed before, when we do not have information about speed limits of an object, we estimate the maximum speed of the object using sampled points of its trajectories (Sect. 3). In this experiment, we want to verify the impact of “estimating“ the maximum speed. In other words, we may underestimate the maximum speed and we want to see the impact of increasing the estimated maximum speed on the accuracy. The ground truth is the same as the previous, however, we increase the estimated maximum speed for the lower-sampled version of the Cabspotting dataset. The outcome is that, there is not a considerable impact on the accuracy. As an example, in Fig. 14, we show the results for different sampling rates experiment in Fig. 9c. TIDE-Speed1 and TIDE*-Speed1 show the results for the original speed estimation (Fig. 9c) and TIDE-Speed2 and TIDE*-Speed2 show the results for the increased speed (by 25 percent).

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Naderivesal, S., Kulik, L. & Bailey, J. An effective and versatile distance measure for spatiotemporal trajectories.
*Data Min Knowl Disc* **33, **577–606 (2019). https://doi.org/10.1007/s10618-019-00615-5

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### Keywords

- Spatiotemporal trajectory
- Similarity
- Distance
- Measure
- Uncertainty