# A Spatio-Temporal Entropy-based Framework for the Detection of Trajectories Similarity

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

- spatio-temporal data mining and analysis approaches based on spatial, temporal and spatio-temporal functions and clustering analysis. Their objectives are oriented to the detection of movement patterns, trajectory similarities and outliers [10,11,12]. Such approaches have been particularly applied to derive behavioral patterns. Different criteria can be applied from distance-based measures [13,14] to the application of clustering algorithms [15,16,17];

## 3. Methodology

- Detection of the critical points according to some predefined parameters.
- Spatial and temporal entropy calculation.

#### 3.1. Critical Points Detection

**Curvature:**a curvature denotes a curve in a trajectory and a curvature point is defined as the central point of the curve.**Turning point**: a turning point is considered to be a changing direction point of a trajectory. For instance, a straight trajectory with no curve does not have any turning points while two turning points arise for each trajectory curvature.**Self-intersection point**: a self-intersecting point denotes a point where the trajectory passes through at least twice. A self-interacting point is considered to be a critical point as it encompasses some specific important properties: a self-intersecting point in a trajectory represents a node surely denoting the relative importance of that self-intersecting point. Among different parameters that can be considered for qualifying a self-intersecting point, the time and distance covered by the self-intersecting part of the trajectory between passing through twice can be mentioned.

**Rule**

**1**.

_{i}of a trajectory T if and only if:

_{i}) and V(t

_{i}) respectively represent the position and velocity of a trajectory point t

_{i}, µ

_{i}the velocity average for the trajectory points t

_{i,}t

_{i−1, …,}t

_{i−10}, β a parameter that denotes the expected magnitude of the velocity change (e.g., valued as 4/3 in the experiments developed).

_{i}) are detected and stored in a series of Abstract Trajectory Descriptor (ATD) representations that also consider the starting and ending points, as well as the critical points identified by the geometrical and semantic-based parameters. Figure 2A illustrates the principles behind the different trajectory critical points (i.e., self-intersections, direction changes, curvatures, stop points) and how such ATDs should be derived. Figure 2B gives a schematic representation of the trajectory according to these critical points. The main principle behind this approach is to favor a comparison of several trajectories according to the parameters identified, and considering the underlying spatial and temporal properties embedded in these critical points.

#### 3.2. Spatio-temporal Entropy Approach Principles

- Not only the usual spatial distribution of the intermediate trajectory points should be considered, but also their temporal distribution. Next, additional parameters such as velocity should be considered in order to reflect some intrinsic changes in the trajectory.
- Trajectories encompass some specific behavior reflected by critical points as identified by our approach (stop points, direction changes, curvature, self-intersections). Such critical points embed some spatial, temporal and semantic properties that represent some very specific behaviors that are crucial when cross-comparing several trajectories.
- The reasons mentioned above leads us to search for and apply a measure of entropy that will reflect the spatial, temporal and semantic distribution of all these parameters within the intimate representation of some trajectories. The objective is to provide a sort of quantitative evaluation of such qualitative properties that will help to evaluate trajectory similarities and differences, and thus according to different configurationally parameters (as another objective is to provide a flexible evaluation approach).

#### 3.3. Spatio-temporal Entropies

_{j}gives the average of the distances between the entities of this class j. Similarly, the external distance denoted by d

_{j}represents the average of the distances between the entities of this class j and the other entities that do not belong to this class j. These respective inner distance (2) and external distances (3) are given as follows:

_{j}represents the set of entities of class j, while n

_{j}represents the cardinality of c

_{j}. N is the total number of entities, d

_{i,j}represents the distance between entities i and j.

_{i}and t

_{j}of a given trajectory T. Hereafter, geometric and semantic descriptors introduced in the previous section are considered to be separate classes. Similarly, critical points are considered to be specific classes. This measure of spatial distance, as applied to the critical points of trajectory data, supports derivation of internal and external distances, and this for all classes of semantic and geometric descriptors. Let us therefore introduce the measure of spatial entropy as introduced in [62], and as based on the measures of internal and external distances:

_{s}is the measure of entropy, p

_{i}is the percentage of entities in the class i.

_{i}denotes the ratio of the critical points of the class i over all the classes.

_{1}, T

_{2},…, T

_{n}represent the id of trajectory 1, trajectory 2 and trajectory 2, respectively. Table 3 presents the general structure of this matrix and the different critical points considered for each trajectory.

## 4. Experimental validation

#### 4.1. Reference Data

#### 4.2. Implementation

#### 4.2.1. Critical Points Detection

#### Convex Hulls

#### Speed Change Points

#### Stop Points

#### 4.2.2. Spatial and Temporal Entropy

- According to the spatial and temporal entropies, as well as for the intrinsic geometrical parameters derived from shape and complexity, trajectories 66, 88 and 91 are relatively similar.
- A similar pattern appears for trajectories 77 and 85.
- Conversely, there is no evidence of similarity in the temporal dimension according to the values exhibited by the temporal entropies. This in fact denotes a valuable trend: exhibiting some spatial and geometrical similarities does not imply a similar trend when considering the temporal dimension.

## 5. Conclusions

- Similarities and differences are analyzed according to the intrinsic spatial and temporal properties of some selected trajectories, and according to a series of geometrical (direction changes, curvatures) and semantic parameters (stop points, speed changes). A peculiarity of the approach is that trajectory similarities are evaluated regardless of the trajectory beginning and end point both in space and time.
- A notion of ATD (Abstract Trajectory Descriptor) is introduced and can be considered to be a sort of geometrical and semantic signature for each considered trajectory. Not only does this provide a structural representation of a trajectory to derive different measures of entropy, but it also gives a simplified representation that is flexible as potentially extendable to take into account additional or different parameters.
- To facilitate the implementation and computation of approach, a prior filtering of the trajectory samples is processed and where critical points only are conserved according to the geometrical and semantic parameters identified and selected.
- The approach is flexible as not only the measures of entropy are derived in the spatial and temporal dimensions, but also as a series of intermediate measures of entropy are derived independently for each identified semantic and geometrical variable. This allows studying at the macro or micro level the diversity of these parameters at different levels.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Critical points and ATD representation of a given trajectory example. (

**A**) ATD of physical and geometric descriptors; (

**B**) General trajectory.

**Figure 5.**Two similar trajectories but with different CHs. Two different trajectories presented by Blue and purple colors.

**Figure 7.**

**Left**: similar geometries with different speed behaviors;

**Right**: different geometries with similar speed behaviors.

Current Approaches | Data Sources | |||
---|---|---|---|---|

GPS Data | Smart Card Data | Cell Phone Data | Other | |

Semantic-based | [4,8,9,20,21] | [22,23] | [1,24,25] | [26,27] |

Spatio-temporal data mining and analysis | [3,5,12,13,14,15,16,17,21,28,29,30,31,32,33,34,35] | [36,37,38,39] | [1,18,40,41,42] | [11,15,43,44] |

Graph-based | [45,46,47] | [39,48] | [49,50] | [51] |

Properties | Categories | Properties | Related Work |
---|---|---|---|

ST data mining and analysis | Spatial | Distance | [3,5,12,13,14,17,18,46,47,49] |

Direction | [1,4,16,17,18,26,28,32,41] | ||

Turning Angle | [5,11,12,32] | ||

Sinuosity | [1,11,12,29] | ||

Spatio-temporal | Velocity | [4,5,11,18,28,47] | |

Acceleration | [5,28,29] | ||

Temporal | Time of Occurrence | [8,9,15,17,22,36,38,42,44,49,50] | |

Semantic | Environment data | [1,11,15,20,21,28,36,37,38] | |

Stop Point | [14,16,17,18,26,28,37,45] | ||

POI | [1,3,14,21,22,30,31,32,33,36,41,42,43,44,45,46,47,50,51] | ||

People attributes | [4,20,21,36,37] |

T_{1} | T_{2} | T_{3} | … | T_{n} | |||
---|---|---|---|---|---|---|---|

Spatial | Spatial Entropy | V_{11} | V_{12} | . | … | V_{1n} | |

Semantic information measure | speed | . | . | ||||

stop | . | . | |||||

Geometric information measure | Curvature | . | . | ||||

Turning | . | . | |||||

Intersection | |||||||

Temporal | Temporal Entropy | ||||||

Semantic information measure | speed | ||||||

stop | |||||||

Geometric information measure | Curvature | . | . | ||||

Turning | . | . | |||||

Intersection | V_{61} | . | . | … | V_{6n} |

Entropy Type | Trajectory Id | Stop | Speed | Turning | Curvature | Aggregated Entropy |
---|---|---|---|---|---|---|

Spatial | 47 | 0.11 | 0.34 | 0. 38 | 0.53 | 0.376 |

56 | 0.14 | 0.51 | 0.46 | 0.18 | 0.287 | |

Temporal | 47 | 0.36 | 0.29 | 0.41 | 0.32 | 0.335 |

56 | 0.28 | 0.53 | 0.49 | 0.27 | 0.359 |

Category | Length of Trajectory (m) | N. of Trajectories | N. of Primary CH | N. Removed CH | Variance of Distance to CH Line | N. of Curvature Points | N. of Turning Points | N. of Intersection Points |
---|---|---|---|---|---|---|---|---|

1 | 0–1000 | 56 | 510 | 63 | 3.41 | 447 | 449 | 5 |

2 | 1000–4000 | 82 | 1245 | 235 | 7.33 | 1010 | 1012 | 26 |

3 | 4000–8000 | 127 | 3910 | 1376 | 14.57 | 2534 | 2536 | 58 |

4 | 8000–15,000 | 61 | 1833 | 639 | 17.20 | 1194 | 1196 | 31 |

For All Points | For Stop Points | |||||
---|---|---|---|---|---|---|

Minimum Value | Maximum Value | Average | Minimum Value | Maximum Value | Average | |

Belief | 0.09 | 0.945 | 0.864 | 0.723 | 0.945 | 0.834 |

Disbelief | 0.02 | 0.894 | 0.448 | 0.12 | 0.27 | 0.145 |

Uncertainty | 0.04 | 0.23 | 0.135 | 0.06 | 0.19 | 0.125 |

Spatial Distances (m) | |||||
---|---|---|---|---|---|

Semantic Parameters | Geometric Parameters | ||||

Stop | Speed | Turning | Curvature | Intersection | |

Minimum | 81.3 | 66.9 | 138.4 | 185.2 | 0 |

Maximum | 6590.1 | 1631.4 | 650.9 | 718.1 | 1903.8 |

Mean | 1073.6 | 589.3 | 315.7 | 377.6 | 661.5 |

Variance | 456.27 | 129.43 | 96.67 | 89.16 | 104.17 |

Temporal Distances (S) | |||||
---|---|---|---|---|---|

Semantic Parameters | Geometric Parameters | ||||

Stop | Speed | Turning | Curvature | Intersection | |

Minimum | 9 | 11 | 16 | 25 | 0 |

Maximum | 592 | 236 | 51 | 78 | 389 |

Mean | 68 | 125 | 34 | 47 | 235 |

Variance | 21 | 38 | 10 | 17 | 107 |

Semantic Parameters | Geometric Parameters | |||||
---|---|---|---|---|---|---|

Stop | Speed | Turning | Curvature | Intersection | ||

Spatial | Internal Distance | 1099.7 | 1425.2 | 1669.5 | 1351.8 | 744.5 |

External Distance | 1733.9 | 1886.5 | 1905.2 | 1922.8 | 2185.3 | |

Temporal | Internal Distance | 6720 | 8447 | 12971 | 9832 | 4063 |

External Distance | 21,849 | 45,216 | 66,213 | 54,470 | 83,416 |

Trajectory Id | Length (m) | Spatial Entropies | ||||
---|---|---|---|---|---|---|

Stop | Speed | Turning | Curvature | Aggregated Spatial Entropy | ||

66 | 5146 | 0.58 | 0.65 | 0.27 | 0.24 | 0.76 |

77 | 6990 | 0.75 | 0.24 | 0.16 | 0.11 | 0.46 |

85 | 8082 | 0.80 | 0.26 | 0.19 | 0.14 | 0.52 |

88 | 9079 | 0.52 | 0.69 | 0.30 | 0.27 | 0.81 |

91 | 11,806 | 0.44 | 0.61 | 0.26 | 0.23 | 0.78 |

Trajectory Id | Length (m) | Temporal Entropies | ||||
---|---|---|---|---|---|---|

Stop | Speed | Turning | Curvature | Aggregated Temporal Entropy | ||

66 | 5146 | 0.24 | 0.47 | 4.55 | 4.28 | 0.74 |

77 | 6990 | 0.11 | 0.63 | 3.12 | 2.77 | 0.77 |

85 | 8082 | 0.08 | 0.69 | 2.91 | 2.65 | 0.69 |

88 | 9079 | 0.45 | 0.63 | 5.76 | 3.91 | 0.91 |

91 | 11,806 | 0.32 | 0.48 | 7.24 | 5.39 | 0.82 |

**Table 12.**Comparison matrix of sample trajectories using shape (upper triangle) and complexity (lower triangle).

Trajectory No. | 66 | 77 | 85 | 88 | 91 |
---|---|---|---|---|---|

66 | 32.86% | 27.11% | 79.2% | 78.49% | |

77 | 38.29% | 83.26% | 42.18% | 39.95% | |

85 | 35.08% | 86.4% | 37.1% | 33.72% | |

88 | 79.51% | 38.48% | 41.23% | 80.68% | |

91 | 74.93% | 45.76% | 38.02% | 83.42% |

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## Share and Cite

**MDPI and ACS Style**

Hosseinpoor Milaghardan, A.; Ali Abbaspour, R.; Claramunt, C.
A Spatio-Temporal Entropy-based Framework for the Detection of Trajectories Similarity. *Entropy* **2018**, *20*, 490.
https://doi.org/10.3390/e20070490

**AMA Style**

Hosseinpoor Milaghardan A, Ali Abbaspour R, Claramunt C.
A Spatio-Temporal Entropy-based Framework for the Detection of Trajectories Similarity. *Entropy*. 2018; 20(7):490.
https://doi.org/10.3390/e20070490

**Chicago/Turabian Style**

Hosseinpoor Milaghardan, Amin, Rahim Ali Abbaspour, and Christophe Claramunt.
2018. "A Spatio-Temporal Entropy-based Framework for the Detection of Trajectories Similarity" *Entropy* 20, no. 7: 490.
https://doi.org/10.3390/e20070490