## 3 Methodology
### Path Planning Model
#### 3.3.5 Regular Activity Hotspot Selection  
In this paper, the sub-hotspot chain composed of the receiving point and the previous hotspot is taken as the passing path from the user to the receiving point. However, the preordered hotspots and path of the receiving point on the hotspot chain are not all regular hotspots and paths of the user, so in order to determine the starting point of the passing path, it is necessary to determine the regular activity hotspot of the user. In this paper, ee periodically update the regular activity hotspots by considering the time-space distance to other activity hotspots and receiving points and standardize this distance using the kernel density function. When users' travel patterns and hotspot locations change, updates are made through the KDE framework. Therefore, the regular activity hotspot selection model is constructed as in Equation (4) and finally selects the hotspot \(\varphi\).  
\[MAX(P)=\sum_{c^{{}^{\prime}}=1}^{C^{\prime}}\sum_{k^{{}^{\prime}}=1}^{K_{s}}H_ {c^{\prime}k^{\prime}}(p_{d}\big{(}c^{\prime}k^{\prime}\big{)}+p_{t}\big{(}c^{ \prime}k^{\prime}\big{)}+p_{z}\big{(}c^{\prime}k^{\prime}\big{)}) \tag{4}\]  
\[0<L\leq h \tag{5}\]  
\[0<DT\leq 86,400 \tag{6}\]  
\[0<c^{{}^{\prime}}<C \tag{7}\]  
\[0<K_{s}<K \tag{8}\]  
\[\sum_{c^{{}^{\prime}}=1}^{C^{\prime}}\sum_{k^{{}^{\prime}}=1}^{K_{s}}H_{c^{ \prime}k^{\prime}}=1 \tag{9}\]  
\[H_{c^{\prime}k^{\prime}}=\left\{\begin{array}{ll}0,&\text{Not a regular activity hotspot}\\ 1,&\text{regular activity hotspot}\end{array}\right.\quad\forall c^{{}^{\prime}}\in C^{{}^{ \prime}},k^{{}^{\prime}}\in K_{s} \tag{10}\]  
\[P=\sum_{c^{{}^{\prime}}=1}^{C^{\prime}}\sum_{k^{{}^{\prime}}=1}^{K_{s}}H_{c^{ \prime}k^{\prime}}(p_{d}\big{(}c^{\prime}k^{\prime}\big{)}+p_{t}\big{(}c^{ \prime}k^{\prime}\big{)}+p_{z}\big{(}c^{\prime}k^{\prime}\big{)}) \tag{11}\]  
where \(p_{d}(c^{\prime}k^{\prime})\) represents the spatial kernel density estimate between active hotspots (the calculation formula is shown in Equation (12)); \(p_{t}(c^{\prime}k^{\prime})\) represents the time kernel density estimate between active hotspots (the calculation formula is shown in Equation (15)); and \(p_{z}(c^{\prime}k^{\prime})\) represents the kernel density estimate of active hotspots and the user receiving point, and the calculation formula is shown in Equation (16).  
\[p_{d}\big{(}c^{\prime}k^{\prime}\big{)}=\frac{1}{K_{s}h^{2}}\sum_{i=1}^{C^{ \prime}}\sum_{j=1}^{K_{s}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\frac{L(X_{c^{ \prime}k^{\prime}}X_{ij})^{2}}{k^{2}}} \tag{12}\]where \(L\big{(}X_{c^{\prime}k^{\prime}},X_{ij}\big{)}\) represents the spatial distance between the hotspot \(X_{c^{\prime}k^{\prime}}\) and the hotspot \(X_{ij}\), and the calculation formula is  
\[L\big{(}X_{c^{\prime}k^{\prime}},X_{ij}\big{)}=\gamma*arccos(\varepsilon) \tag{13}\]  
\[\varepsilon=sin(\beta)*sin(\beta^{{}^{\prime}})*cos(\alpha-\alpha^{{}^{\prime }})+cos(\beta)*cos(\beta^{\prime}) \tag{14}\]  
where \(\gamma\) represents the radius of the Earth, \(\alpha\),\(\alpha^{\prime}\) indicates the angular value of the hotspot \(X_{c^{\prime}k^{\prime}},\ X_{ij}\) is the longitude transformation, \(\beta\), \(\beta^{\prime}\) indicates the angle value of \(X_{c^{\prime}k^{\prime}}\), and \(X_{ij}\) is the position latitude transformation.  
\[p_{t}\big{(}\big{(}c^{\prime}k^{\prime}\big{)}\big{)}=\frac{1}{K_{\delta}h^{2 }}\sum_{i=1}^{c^{\prime}}\sum_{j=1}^{K_{\delta}}\frac{1}{\sqrt{2\pi}}e^{-\frac {1}{2}\frac{L(X_{c^{\prime}k^{\prime}},X_{ij})}{8\theta,400}} \tag{15}\]  
where \(T_{d}\big{(}Y_{c^{\prime}k^{\prime}},Y_{ij}\big{)}\) represents the Euclidean distance between the hotspot \(Y_{c^{\prime}k^{\prime}}\) and the hotspot time \(Y_{ij}\).  
\[\text{P}_{x}\big{(}\big{(}c^{\prime}k^{\prime}\big{)}\big{)}=\frac{1}{2\text{ h}^{2}}\frac{1}{\sqrt{2\pi}}\text{e}^{-\frac{1}{2}\frac{L(X_{c^{\prime}k^{\prime}},X_{ \delta})^{2}}{\text{h}^{2}}} \tag{16}\]