# Towards Moving Objects Behavior Analysis: Region Speed Limit Rate Measure
## 2 Definitions and Methods  
**Temporal segmentation:** Time is divided into consecutive temporal segments (mutually exclusive) of size (duration) \(dt\), see Figure 1. For example, suppose that \(dt=15\) min, then [14 June 2022 7:00, 14 June 2022 7:15) and [14 June 2022 7:15, 14 June 2022 7:30) are two (consecutive) temporal segments. **Spatial segmentation:** The region of interest (i.e., the geographical space) is segmented into cells (square tessellation, square tiling) of side \(l\), \(l\in\mathbb{I}^{+}\) (positive integers). The region of interest is represented by a bidimensional array \(R\) of order \(p\times q\), \(p\), \(q\in\mathbb{I}^{+}\). The element \(R[v,w]\), \(0<v\leq p\), \(0<w\leq q\), \(v\), \(w\in\mathbb{I}^{+}\); corresponds to a cell. We assume that the neighborhood relationships between the cells in the real world correspond to those established in the array \(R\) (according to rows and columns). Thus, e.g., the adjacent cells of the cell \(R[1,1]\) are \(R[1,2]\) (on the right), \(R[2,1]\) (on the bottom), and \(R[2,2]\) (diagonally). Figure 2 shows a region (array of order \(3\times 4\)). **Spatiotemporal scenario:** A _spatiotemporal scenario_\(STS\) is a 2-tuple (\(R\), _TempSeg_) where \(R\) is a segmented region and _TempSeg_ is a temporal segment. **Coordinated Point (CP):** A coordinated point is a 3-tuple (\(x\), \(y\), \(t\)), \(x\), \(y\in\mathbb{R}^{+}\), \(t\) is a timestamp, where \(x\) represents the latitude, \(y\) the longitude, and \(t\) the time when the CP was generated. For example, (10, 20, '15 June 2022 13:30:00') is a CP. **Trajectory:** A trajectory \(T\) is a sequence of \(n\) CPs, \(n\in\mathbb{I}^{+}\), \(T=\) <(\(x_{1}\), \(y_{1}\), \(t_{1}\)), (\(x_{2}\), \(y_{2}\), \(t_{2}\)),..., (\(x_{n}\), \(y_{n}\), \(t_{n}\))> where \(\forall i\), \(j\in\mathbb{I}^{+}\): \(0<i<n\), \(1<j\leq n\), \(i<j\), \(t_{i}<t_{j}\), i.e., the CPs are ordered temporally (by the ascending time of the CPs). Figure 1: Temporal segmentation. Segment of a trajectory in a cell of a spatiotemporal scenario: let us consider:  
* A spatiotemporal scenario \(STS(R,\)_TempSeg_) where \(R\) is a region of order \(p\times q\). * A trajectory \(T=\) < (\(x_{1}\), \(y_{1}\), \(t_{1}\)), (\(x_{2}\), \(y_{2}\), \(t_{2}\)),..., (\(x_{n}\), \(y_{n}\), \(t_{n}\))\(>\). * A subtrajectory \(Subt\) of \(T,\)\(Subt=\) < (\(x_{j}\), \(y_{j}\), \(t_{j}\)), (\(x_{i+1}\), \(y_{j+1}\), \(t_{j+1}\)),..., (\(x_{j+m}\), \(y_{j+m}\), \(t_{j+m}\))\(>\), \(0<j<n\), \(m\in\mathbb{1}^{\ast}\), (\(j+m\)) \(\leq n\), i.e., \(\mid\)\(Subt\mid>1\) (i.e., \(Subt\) has at least two CPs). That is, a subtrajectory is a subsequence of at least two CPs of a trajectory. We say that \(Subt\) is a segment of \(T\) in \(R[v,\)\(v]\) (\(R[v,\)\(w]\) is a cell of \(STS\)), \(0<v\leq p\), \(0<w\leq q\), if it meets the following conditions: 1. \(\forall\) CP (\(x_{k}\), \(y_{k}\), \(t_{k}\)) \(\in\)\(Subt\), \(k\in\mathbb{1}^{\ast}\), \(j\leq k\leq(j+m)\), (\(x_{k}\), \(y_{k}\)) is inside \(R[v,\)\(w]\), i.e., the cell \(R[v,\)\(w]\) contains the CP (\(x_{k}\), \(y_{k}\), \(t_{k}\)) (spatial inclusion [21]). 2. \(\forall\) CP (\(x_{k}\), \(y_{k}\), \(t_{k}\)) \(\in\)\(Subt\), \(k\in\mathbb{1}^{\ast}\), \(j\leq k\leq(j+m)\), \(t_{k}\in\)_TempSeg_, i.e., the segment _TempSeg_ includes the CP (\(x_{k}\), \(y_{k}\), \(t_{k}\)) (temporal inclusion [22]). Informally, condition (a) guarantees the spatial inclusion of all the CPs of the segment \(Subt\) in the cell \(R[v,\)\(w]\) and condition (b) in turn, guarantees the temporal inclusion of all the CPs of the segment \(Subt\) in the temporal segment _TempSeg_. Figure 3 shows a segment of \(T\) in a cell of a spatiotemporal scenario. We say that \(Subt\) is a _maximal segment_ in \(R[v,w]\) if, in addition to conditions (a) and (b), it meets the following:  
1. The first CP of \(Subt\), i.e., (\(x_{j}\), \(y_{j}\), \(t_{j}\)), is the first CP of \(T\) (i.e., \(j=1\)), or if \(j>1\) then, (\(x_{j-1}\), \(y_{j-1}\)) is not inside \(R[v,\)\(w]\) or \(t_{j-1}\
otin\)_TempSeg_.