# Inter-Satellite Link Prediction with Supervised Learning: An Application in Polar Orbits
## 5 Results
### Fixed Distance Threshold  
The goal of this work is not to predict the satellite-to-satellite distance but to anticipate the encounters between satellite pairs and to compute the performance of this anticipation capability. To do so, the ground truth must be post-processed into a binary signal by defining a distance threshold (\(T\)) above which communication is not feasible. This threshold can vary depending on the user preferences and the antenna characteristics, but it must lie between 1000 km and 3000 km. For the following results, three scenarios coming from three different \(T\)s are presented, namely \(T\) = 1500 km, \(T\) = 2000 km, and \(T\) = 2500 km, which correspond to the normalized thresholds of 0.75, 0.5, and 0.25, respectively (see Equation (3)). Notice that these normalized thresholds are the ones used to transform the ground truth into a binary signal. \[T_{norm}=\frac{T-T_{max}}{T_{min}-T_{max}} \tag{3}\]  
where \(T_{max}\) and \(T_{min}\) are set to 3000 km and 1000 km, respectively. To quantify model performance, the harmonic mean of precision (\(P\)) and recall (\(R\)), called the F1 score (\(F1\)), is used. In our scenario, \(P\) is the probability of estimating a contact inside the real time slot, while \(R\) assesses the proportion of successful encounters relative to the total available communication time. In this sense, a precision lower than 1 means that some predicted encounters do not exist, which would be translated into wasted transmission energy. A recall lower than 1 means that some encounters are not detected and some communication opportunities are lost. Equations (4)-(6) present the mathematical formulations of the \(F1\), \(P\), and \(R\), respectively. \[F1=\frac{2\times P\times R}{P+R} \tag{4}\]  
Figure 5: Comparison of ground truth and model predictions (orange) for three distinct satellite pairs across both synthetic and Celestrak test sets. Figure 6: Obtained distribution of MAE depicting the model mean performance in predicting all samples in both synthetic and Celestrak test sets. \[P=\frac{TP}{TP+FP} \tag{5}\]  
\[R=\frac{TP}{TP+FN} \tag{6}\]  
where \(TP\), \(FP\), and \(FN\) stand for true positive, false positive, and false negative, respectively. It is important to notice that to compute \(TP\), \(FP\), and \(FN\), the model prediction must also be post-processed into a binary signal using a decision threshold. In this sense, all output values lower than the decision threshold are interpreted as 0 or no encounter instances, whereas higher values are set to 1 and interpreted as communication opportunities. Setting an appropriate threshold is important and hardly affects the performance of the model. Figure 7 illustrates the influence of the decision threshold on the \(P\), \(R\), and \(F1\) metrics for both synthetic and realistic datasets. To provide a more intuitive understanding, the decision threshold is also expressed in kilometers through the inverse of Equation (3). Furthermore, each figure presents three different scenarios coming from three different distance thresholds indicated with a dashed vertical red line, namely 1500 km, 2000 km, and 2500 km. As expected, as the decision threshold rises, the count of detected encounters decreases, resulting in a reduction in \(FP\)s and an increase in \(FN\)s. According to Equations (5) and (6), the decrease in \(FP\)s enhances the \(P\), elevating the likelihood of accurately estimating a contact within the real time slot. However, the increase in \(FN\)s negatively affects the \(R\), as it diminishes the proportion of successful encounters. It is worth noting that a similar  
Figure 7: Precision, recall, and F1-score curves obtained using both synthetic and Celestrak test sets across three different distance thresholds. (**a**) Synthetic data, \(T\) = 1500 km. (**b**) Synthetic data, \(T\) = 2000 km. (**c**) Synthetic data, \(T\) = 2500 km. (**d**) Celestrak data, \(T\) = 1500 km. (**e**) Celestrak data, \(T\) = 2000 km. (**f**) Celestrak data, \(T\) = 2500 km.