## 4 Experimentation
### DxT Convergence Analysis  
To determine the optimal iteration number \(n\) at which the DxT values converge, we employed a convergence-based solution approach. Specifically, we introduced a convergence criterion to measure the relative evolution of the normalized DxT values between two consecutive iterations (\(k\) and \(k+1\)):  
\[DxT\text{ }Convergence\text{ }Criteria_{k\to k+1}\] \[=\sum_{i=1}^{L}\sum_{j=1}^{W}\frac{DxT_{i,j,k+1}}{\frac{max_{x_{ i,j^{\prime}}}\left(DxT_{i,j^{\prime},k+1}\right)}{max_{x_{i,j^{\prime}}} \left(DxT_{i,j,k}\right)}}{\frac{DxT_{i,j,k}}{max_{x_{i,j^{\prime}}}\left(DxT _{i,j^{\prime},k}\right)}} \tag{12}\]  
where:  
* \(DxT\text{ }Convergence\text{ }Criteria_{k\to k+1}\) is the chosen convergence criteria from the kth iteration to \(k+1\)th. * \(DxT_{i,j,k}\) is the Dynamic Expected Thread in the grid cell (\(i,j\)) after k iteration. * \(max_{x_{i,j^{\prime}}}\left(DxT_{i,j^{\prime},k}\right)\) is the maximum Dynamic Expected Thread on the pitch after k iteration. In addition to evaluating the convergence criterion as a function of the number of iterations, we also analyzed its standard deviation for different actions across iterations. Given the dynamic nature of DxT, we aimed to quantify the stability of threat values for each action. Specifically, we computed the standard deviation of DxT Convergence Criteria changes at each iteration to assess the degree of fluctuation before stabilization. The high standard deviation in early iterations observed in Figure 2 indicates significant updates in DxT values, whereas the gradual reduction in standard deviation suggests that the model is approaching convergence. This analysis led us to determine that 10 iterations were sufficient to achieve a stable and consistent DxT representation across all actions. Consequently, for the remainder of this study, we will adopt a fixed number of 10 iterations for the generation of DxT grids. Case Study: DxT to Evaluate the Sequence of Actions Leading to Albania's First Goal Against Croatia, Euro 2024  
To illustrate the inner workings of the DxT model--its dynamic nature and its ability to evaluate tactical situations more effectively than the xT framework--we applied it to Albania's first goal against Croatia during Euro 2024. Albania scored after a sequence of passes and carries, culminating in a shot that resulted in a goal. This sequence is depicted in Figure 3.  
Figure 3: Sequence Leading to Albania’s First Goal Against Croatia, Euro 2024. Figure 2: Evolution of the DxT Convergence Criterion and Its Standard Deviation as a Function of the Num-ber of Iterations. The first action in this sequence was a backward pass from [PERSON] to [PERSON], moving from grid (3,9) to grid (4,8), as shown in Figure 4. In this figure, opposing players are represented as red dots, while teammates appear as blue dots. The ball transitioned from a grid cell with a DxT starting value of 0.0906 to a cell with an end value of 0.0942. The DxT value of this first action is computed as the difference between the DxT values of these two grid cells:  
\[DxT_{Action\ 1}=0.0942-0.0906=0.0035\]  
[PERSON] then carried the ball laterally (Figure 5), generating a DxT value of:  
\[DxT_{Action\ 2}=0.0812-0.0824=0.0012\]  
The remaining actions and their corresponding DxT maps are shown in Figures A1-A4, while Table 2 summarizes the full sequence leading to the goal: