# Discovering Moving Flock Patterns in Movement Data: A Reeb Graph-Based Approach
## 2 Improved Definition of the Moving Flock  
Several definitions of the term \"flock\" have been proposed. Among these, the definition by [PERSON] et al. (2013) is adopted in this paper, as it is widely used in existing research. The definition can be stated as follows. **Definition 1** (Flock). : _Given a set of \(n\) trajectories, an (\(r\), \(m\), \(k\))-flock \(F\) during a time interval \(I=\left[t_{i},\ t_{j}\right]\) where \(j-i\geq k\) consists of at least \(m\) objects such that for each discrete timestamp \(t_{l}\in I\ (i\leq l\leq j)\) there exists a disk of radius \(r\) containing all \(m\) objects._  
In this definition, note that \(r\) denotes proximity (or radius) within which members are considered part of the flock, \(m\) represents the number of members in the flock, and \(k\) specifies the duration over which the flock exists. However, this definition cannot distinguish between stationary and moving flocks. To address this, [PERSON] (2017) and [PERSON] et al. (2018) proposed an enhanced definition of moving flock based on Definition 1. To the best of our knowledge, this was the first formal definition of the moving flock. The specific definitions are described in detail below. **Definition 2** (Spatial extent of a flock). : _Given an (\(r\), \(m\), \(k\))-flock \(F\) during a time interval \(I\), its spatial extent, \(ext(F,I)\) is defined as \(ext(F,I)=max\{l,\ w\}\), where \(l\) and \(w\) are the length and width, respectively, of the minimum bounding rectangle (MBR) of the set of \(sub\) trajectories belonging to the flock._  
**Definition 3** (Moving flock). : _Given a set of \(n\) trajectories, an (\(r\), \(m\), \(k\))-moving flock \(F_{M}\) during a time interval \(I=\left[t_{i},\ t_{j}\right]\) where \(j-i\geq k\) consists of at least \(m\) objects such that for each discrete timestamp \(t_{l}\in I\ (i\leq l\leq j)\) there exists a disk of radius \(r\) containing all \(m\) objects, and the spatial extent of \(F_{M}\) satisfies the following condition \(ext(F_{M},I)\geq r\)._  
Despite these definitions, ambiguity remains, as a stationary flock can still be misinterpreted as a moving flock. For example, suppose a group of \(m\) objects forms a flock during a time interval \(I=\left[t_{i},\ t_{j}\right]\) (\(j-i\geq k\)), but remains stationary during \(\left[t_{i},\ t_{j-1}\right]\) and only moves during \(\left[t_{j-1},\ t_{j}\right]\) (satisfying the condition \(ext(F_{M},I)\geq r\)), according to Definition 2 and Definition 3, this group would be classified as a moving flock during \(\left[t_{i},\ t_{j}\right]\). However, strictly speaking, it was only a moving flock during \(\left[t_{j-1},\ t_{j}\right]\), not during the entire interval \(\left[t_{i},\ t_{j}\right]\). To address this limitation, an improved definition of the moving flock is proposed in this paper. It is described in detail as follows. **Definition 4** (Spatial extent of a flock between two consecutive timestamps). : _Given an (\(r\), \(m\), \(k\))-flock \(F\) during a time interval \(I=\left[t_{i},\ t_{j}\right]\) where \(j-i\geq k\), assume \(t_{u}\) and \(t_{u+1}\) (\(i\leq u\leq j-1\)) are two consecutive timestamps, the spatial extent of \(F\) between \(t_{u}\) and \(t_{u+1}\) is defined as \(ext(F|t_{u},F|t_{u+1})=d\left\{p_{u},\ p_{u+1}\right\}\) where \(p_{u}\) and \(p_{u+1}\) are the geometric centres of the MBRs of \(F\) at \(t_{u}\) and \(t_{u+1}\), respectively, and \(d\left\{p_{u},\ p_{u+1}\right\}\) is the Euclidean distance between \(p_{u}\) and \(p_{u+1}\)._  
Note that \(d\) denotes displacement. Figure 1 illustrates this concept. Here, \(ext(F|t_{u},F|t_{u+1})\) denotes the spatial extent of a flock \(F\) (comprising three objects: \(O_{1}\), \(O_{2}\), and \(O_{3}\)) over two consecutive timestamps \(t_{u}\) and \(t_{u+1}\). **Definition 5** (Improved moving flock).