## 2 Materials and Methods
### Algorithm for Dynamic Weighting Between Estimates of Kalman Filter and GERT Model
The denominator is supplemented with \(\ \left(1-P_{track}^{GERT}(t)\right)\), which reflects the probability of loss of support according to the results of the GERT analysis of phase transitions described in the previous subsection. Thus, with high reliability of local estimates and a stable filter, \(\alpha(t)\) approaches 1; with an increase in the risk of loss of support, it decreases. The structure of the formula is the result of the authors' construction and performs the function of a flexible merger weight regulator, which takes into account the real state of the system at time t.  
The stability of the Kalman filter \(S_{KF}(t)\) can be defined as the inverse of the standard deviation of the innovation:  
\[S_{KF}(t)=\frac{1}{\sigma_{r}(t)+\varepsilon}, \tag{3}\]  
where \(\sigma_{r}(t)\) is the standard deviation of the residual (innovation) at the current cycle,  
\(\varepsilon\) is a small number that prevents division by zero. The \(D_{meas}(t)\) measurement confidence is defined as the ratio of the number of reliable measurements to the total number of measurements in a given time window:  
\[D_{meas}(t)=1-\frac{N_{lost}(t)}{N_{total}(t)}, \tag{4}\]  
where \(N_{lost}(t)\) is the number of missed or invalid measurements and  
\(N_{total}(t)\) is the total number of measurements in the time window. The tracking probability provided by the GERT model, \(P_{track}^{GERT}(t)\), is formed as the product of the probabilities of passing through the following critical states in the network: detection, confirmation, and tracking retention. It can be represented as:  
\[P_{track}^{GERT}(t)=P_{\text{detection}}\cdot P_{\text{confirmation}}\cdot(1-P_ {\text{\emph{lost}}}), \tag{5}\]  
where each of the probabilities is determined based on the analysis of the paths in the GERT network at the current cycle. Thus, the calculated value \(\alpha(t)\) enables the intelligent module to adaptively choose the degree of confidence in each of the estimates depending on the real state of tracking, ensuring a balance between the accuracy of the Kalman filter and the stability of the GERT forecast. This enables the system to remain effective even in cases of loss of measurements, increased interference, or deterioration of the target dynamics. Figure 3 presents graphs of the dependence of the coefficient \(\alpha(t)\) on the stability indicators of the Kalman filter and confidence in measurements for different scenarios. The graph shown in Figure 3 illustrates a typical simulation result for the evolution of the coefficient \(\alpha(t)\) under different interference scenarios. The values of \(\alpha(t)\) are not pre-set but are dynamically calculated during simulation runs using the proposed formulae (2)-(5). These simulations involve variable measurement reliability, innovation fluctuation, and tracking loss probability. The resulting \(\alpha(t)\) curves reflect the system's adaptive behavior in real time and demonstrate how the intelligent module shifts the confidence weight between Kalman and GERT estimates according to current signal quality and uncertainty. In Figure 3, scenario 1 (red line) reflects a situation of gradual deterioration of the tracking quality. Over time, the stability of the Kalman filter, the confidence in the measurements, and the probability of tracking according to the GERT model decrease. As a result, the value of \(\alpha(t)\) gradually decreases from \(\approx\)0.9 to \(\approx\)0.15, which demonstrates the transition of the system from the priority of the Kalman filter to reliance on the probabilistic estimation of GERT. Figure 3: Illustration of the behavior of the coefficient \(\alpha(t)\) in different interference scenarios. Scenario 2 (blue dashed line) simulates a sudden failure in measurements, in which confidence in the sensor data is completely lost from the 50 th step. In this case, the stability of the Kalman filter and GERT remain high.