# Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques
## 3 Results  
Analysis of the spatial point pattern is to consider the neighboring structure of the region of interest based on distances between the positions. An appropriate measure to calibrate the spatial dependence is [PERSON]'s K-function given by the form [24; 25; 26; 27] (Equation (7)):  
\[K(d)=\lambda^{-1}E\text{ [number of extra points within distance of a randomly chosen arbitrary point] } \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline
**References** & **Brief Presentation** \\ \hline [PERSON] (2017) [14] & Focus on Poisson point process considering probabilistic modeling using data based on antennas of the mobile network in Paris \\ \hline [PERSON], [PERSON], [PERSON]. & Poisson random point analysis using determinant point process \\ [PERSON], [PERSON] (2014) [19] & Modeling the spatial distribution of cell phone antennas in the city of Buenos Aires (CABA) \\ \hline [PERSON] (2018) [21] & Probabilistic methods in Telecommunications \\ \hline [PERSON], [PERSON], [PERSON], and [PERSON] (2014) [22] & Modeling using Gauss–Poisson point process \\ \hline [PERSON], [PERSON], and [PERSON] (2015) [23] & Modeling using Ginibre point process \\ \hline \end{tabular}
\end{table} 
Table 1: Presentation of previous research into random point process involving antenna stations for telecommunications. where the expected number of event positions under the distance d is \(\lambda\pi d^{2}\), defining \(\lambda\) as value of the intensity under Poisson random point process; so \(K(d)=\pi d^{2}\). The value \(\lambda K(d)\) could be defined as the expected number of points within distance d of a randomly chosen arbitrary point; hence (Equation (8)):  
\[\lambda\widehat{K}(d)=\frac{1}{n}\sum_{i=1}^{n}(\text{number of events within distance d of the event i}) \tag{8}\]  
The number of event positions could be estimated by an indicator function \(I_{d}(d_{\bar{y}})\). Thus, estimation of the K-function is given by the form (Equation (9)):  
\[\widehat{K}(d)=\frac{1}{\lambda n}\sum_{i=1}^{n}\sum_{j=1}^{n}I_{d}(d_{ij}) \tag{9}\]  
The unbiased estimate of the K-function can be finally written by [19; 20; 21; 22] Equation (10):  
\[\widehat{K}(d)=\frac{1}{\lambda n}\sum_{i=1}^{n}\sum_{ \begin{array}{c}j=1\\ j\
eq i\end{array} }^{n}w_{ij}I_{d}(d_{ij}) \tag{10}\]  
where: \(d_{ij}\) is the distance between the \(i\)-th and \(j\)-th observed event position, which can be viewed as the radius of a circle centered at event position \(i\) and passing through \(j\); and \(w_{\bar{y}}\) is a weighting value equal to this circle's proportion of the entire area A. For specific distance value \(d\), \(I_{d}(d_{\bar{y}})\) is an indicator function which is 1 if \(d_{\bar{y}}\leq d\). It is clear that as \(d\) increases, \(w_{\bar{y}}\)\(\rightarrow\)\(\infty\). If \(K(d)>\pi d^{2}\), the theoretical K-function, then there is evidence of clustering. Investigation for a clustering pattern could be achieved by plotting \(\widehat{K}(d)\) and [PERSON]K(d)\), so spatial dependences could be illustrated. Figure 5b [PERSON] compares [PERSON]'s K-function and the theoretical one, and clearly shows evidence of regularity, as well as the Poisson random point process with 1000 positions (Figure 5a) [28; 29; 30]. This plot is given in Figure 5b, where we see possible evidence for clustering, because the observed K-function is above the curve corresponding to a random process (theoretical one); where blue is the observed K-function, black dashed is the theoretical K-function, and red dashed are the lower and upper limits for K-Function. Especially at small scales (\(d<5\)), the process does not show evidence of clustering. At other scales (\(d\)\(>\)5), we have evidence of clustering. The lower and upper limits for the K-functions are given by \(L(d)=\min\left\{\widehat{K}(d)\right\}\) and \(U(d)=\max\left\{\widehat{K}(d)\right\}\). Figure 5: (a) [PERSON] random point process; and (b) [PERSON]’s K-function.