## 3 Materials and Methods
### Data Processing
Considering the positioning in DGNSS and SPP for each observation time, it is possible to create ten databases consisting of the sub-datasets: E, N, U, SDE, SDN, SDU, \(\Delta\)E, \(\Delta\)N, \(\Delta\)U, as follows:  
\[\text{Xiaomi MI }10_{\text{Method,Time}}=\{\text{E,N,U,SDE,SDN,SDU, \(\Delta\)E, \(\Delta\)N, \(\Delta\)U}\} \tag{13}\]  
For each sub-dataset \(\Delta\)E, \(\Delta\)N, \(\Delta\)U, a normality test of Shapiro-Wilk can be conducted, using the following formula [24]:  
\[\text{W }=\frac{\left(\sum_{i=1}^{n}\text{a}_{i}\text{x}_{(i)}\right)^{2}}{ \sum_{i=1}^{n}\left(\text{x}_{i}-\overline{\text{x}}\right)^{2}} \tag{14}\]  
where:  
* \(\text{x}_{(i)}\) i-th is the smallest value (\(\Delta\)E\({}_{(i)}\), \(\Delta\)N\({}_{(i)}\), \(\Delta\)U\({}_{(i)}\)) of the sub-dataset (\(\Delta\)E, \(\Delta\)N, \(\Delta\)U);
* \(\overline{\text{x}}\) is the arithmetic mean of the values (\(\Delta\)E\({}_{i}\), \(\Delta\)N\({}_{i}\), \(\Delta\)U\({}_{i}\)) of the sub-dataset (\(\Delta\)E, \(\Delta\)N, \(\Delta\)U);
* \(\text{a}_{i}\) is equal to \[\text{a}_{i}\ =\left(\text{a}_{1}\dots\text{a}_{n}\right)\ =\frac{\text{m}^{ \prime}\text{V}^{-1}}{\left(\text{m}^{\prime}\text{V}^{-1}\text{V}^{-1}\text{ m}\right)^{\frac{1}{2}}}\] (15)
* \(\text{m}^{\prime}=\text{m}_{1}\),..., \(\text{m}_{n}\)
* \(\text{m}_{1}\),..., \(\text{m}_{n}\) are expected values of the ranks of a standardised random number;
* V is the matrix of the covariance of ranks. Through RStudio software, the Shapiro-Wilk test can be easily performed, analysing the returns of the W and \(p\) values. The test aims to verify the following hypotheses:  
* H0: Distribution of the dataset normal;
* H1: Distribution of the dataset not normal. In establishing a threshold value equal to 0.05, the following occur:
* If the \(p\)-value > 0.05, the null hypothesis H0 is accepted;
* If the \(p\)-value < 0.05, the null hypothesis H0 is rejected. According to the Shapiro-Wilks tests, the sub-datasets of \(\Delta\)E, \(\Delta\)N, \(\Delta\)U for which normality is verified are represented by the normal distribution of Gauss:  
\[\text{f}(\text{x})=\frac{1}{\text{SD}\sqrt{2\pi}}\text{e}^{\frac{(\text{x}- \overline{\text{x}})^{2}}{2\text{SD}^{2}}} \tag{16}\]  
Then, the arithmetic means for the sub-datasets \(\Delta\)E, \(\Delta\)N, \(\Delta\)U were calculated, taking into account all the x\({}_{i}\) values (\(\Delta\)E\({}_{i}\), \(\Delta\)N\({}_{i}\), \(\Delta\)U\({}_{i}\)) included in them, using the following formula:  
\[\overline{\text{x}}=\frac{1}{\text{n}}\sum_{\text{n}=i}^{n}\text{x}_{i} \tag{17}\]  
To identify possible outliers in the sub-datasets, two methods were chosen. In the first method, the xi outliers of the sub-datasets \(\Delta\)E, \(\Delta\)N, \(\Delta\)U were found and eliminated by applying the Tukey method [25; 26; 27]. Following the method, two fences, one lower and one upper, were identified:  
\[\text{Lower Fence}=\text{Q}_{1}-(1.5\ *\ \text{IQR}) \tag{18}\]  
\[\text{Upper Fence}=\text{Q}_{3}+(1.5\ *\ \text{IQR}) \tag{19}\]  
where:  
* Q\({}_{1}\) is the first quartile;Q3 is the third quartile;  
In this way, it is possible to eliminate the xi values lower and higher than the two fences and, consequently, three new sub-datasets \(\Delta\)E\({}_{\text{Mm}}\), \(\Delta\)N\({}_{\text{Mm}}\), \(\Delta\)U\({}_{\text{Mm}}\) are generated. Normality tests were performed using Equations (14) and (17). In the second method, the possible outliers xi are identified by applying an unsupervised classification algorithm called K-means, imposing some exclusion conditions based on distance [28]. The concept is to consider as outliers the values farthest from the cluster centres. The application of the K-means algorithm consists of the following steps [28; 29]:  
1. Choice of the number of classes K (K = 2);
2. Setting of a mean cluster centre Ci (i = 1, 2) for elements xi of classes K;
3. Assignment of the elements xi to one of the two classes Ci respecting the following condition: \[\left|\text{x}_{\text{i}}-\text{C}_{\text{i}}\right|=\text{ min};\] (20)
4.