# Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques
## 2 Materials and Methods  
### Point Process Analysis  
Considering a 2D space S where \(S\subset R^{d}\) with d the dimension of the data, a region of interest could be introduced. This region could be representing rectangular or non-rectangular areas of investigation based on the particular analysis (Figure 1). Data could be presented inside the 2D region in a coordinate format denoted as (x,y). Every coordinate consists of a corresponding position s; (s) is denoted as the value for each z in position s. The z(s) is defined as a random variable at each position and the spatial model corresponding to the random variable is denoted as \(\{\mathbf{z}(\mathbf{s}):\mathbf{s}\in S\}\). The data consists of n positions s\({}_{1}\)\(\dots\) s\({}_{n}\), each based on an (x,y) coordinate in 2D space. Pattern analysis, based on the spatial associations between positions, is represented by the following types: 1. independent, 2. regular and 3. clustered (Figure 2) [7, 8, 9]. The **Poisson random process** with distribution is given by Equation (1) [10]:  
\[p[N(A)]=\frac{\lambda a(A)}{k!}e^{-\lambda a(A)},k=0,1,2,\dots \tag{1}\]  
where \(\lambda\) is the value of the intensity and \(a(. )\) is the unit area for a specific region and \(A\)\(\in\)\(S\)\(\subset\)\(R^{2}\). The random variable \(N(A)\) is the number of events in the set \(A\)\(\subset\)\(X\) defined as a random process (Figure 3a). A point process **Z(s)** is called a **determinantal** random point process if for a random subset \(A\) drawn according to Z(s), the process holds for every \(A\)\(\subset\)\(X\) (Equation (2)):  
\[p[A\subset X]=\det(K) \tag{2}\]  
where \(K=\left[K_{ij}\right]_{j\in A}\) is the matrix with rows and columns indexed by the elements of the set \(A\). In our case, the point process is based on the form \(p[N(A)\subset X]=\det(K)\), where \(p[N(a)]\) is given by the Poisson random point process. Alternatively, a Poisson point process  
Figure 1: (**a**) Non-rectangular and (**b**) Rectangular regions of interest. Figure 2: Types of spatial patterns. where distance between two positions is taking place could be considered by defining it as an **inhibition process**[10] with distance \(\delta\), where \(\delta\) is the threshold distance by deleting all pairs of events that are closer than \(\delta\) (Figure 3b). Finally a point process with diversion called **determinantal point process**[11, 12, 13] is illustrated in Figure 3c. A determinantal point process is often used to induce diversity or repulsiveness among the points of a sample. Figure 3a illustrates simulated coordinates (x,y) of 100 positions of a **Poisson point process** with estimated \(\widehat{\lambda}=2.018\) simulated on the unit square. Figure 3b illustrates simulated coordinates (x,y) of 100 points of an **inhibition point process** with estimated \(\widehat{\lambda}=0.781\) and \(\delta=0.1\) simulated on the unit square. Figure 3c illustrates simulated coordinates (x,y) of 100 points of a **determinantal point process** with estimated \(\widehat{\lambda}=3.01\) simulated on the unit square. The main properties for the Poisson process are [11, 12, 13]:  
* The number of positions in a region A has a Poisson distribution with mean \(\lambda N(A)\)
* The positions of these points are i.i.d. and uniformly distributed inside \(A\)
* The contents of two disjoint regions A and B are independent  
Consider the locations of base stations, i.e., antennas, of the mobile network in Paris [14] considering all the base stations, of all operators and for all operating frequencies (Figure 4). Based on the previous point process types, it is clear that the real data are close to the Poisson point process leading to a similar simulating pattern. Figure 4: Real data locations of base stations, antennas of the mobile network in Paris [14]. Figure 3: Point process types: (**a**) Poisson; (**b**) inhibition Poisson; and (**c**) determinantal.