## 3 Dynamic Expected Threat (DxT) Model Theory
### Methodology for Calculating Probability of Shooting/Moving Decision
#### 3.2.2 Proposed Process for Building the Probability Function to Shoot  
The construction of the probability function f, which represents the likelihood that a player opts to shoot rather than pass the ball, is fundamentally based on the Expected Goals (xG) value for the given situation, denoted as \(xG_{KOS_{i,j}}\). This probability reflects the player's decision-making process, which is contingent upon the likelihood of scoring from that specific context. Classical xT models calculate the frequency of shots and moves performed within each grid cell of the pitch to determine these probabilities. In contrast, our method evaluates shooting or passing probabilities by segmenting the range [0,1] of xG values and analyzing the frequency of shots and passes within each xG sub-segment. This methodology follows a five-step process, as outlined below:  
**Step 1: Partitioning the [0,1] Interval into Multiple Sub-Intervals**  
Given that xG values fall within the range of 0 to 1, we partition the interval [0,1] with a step size \(\Delta\). Each sub-interval is denoted as I\({}_{k}\) for k \(\in\) [1,N].  
For this study, we use \(\Delta=0.05\) and \(N=20\), resulting in the following intervals: \(I_{1}=[0,0.05]\), \(I_{2}=[\ 0.05,0.1]\),..., \(I_{20}=[0.95,1]\).  
**Step 2: Determining the Sub-Interval to which xGValues of Historical Actions Belong**We analyze n historical actions (both shots and moves) and compute their xG values using the KOS xG model. Each xG value is then assigned to the corresponding sub-interval \(I_{k}\) as shown in Figure 1.  
**Step 3: Measuring the Frequency of Shots in Each Sub-Interval**  
For \(k\in\)\([1,20]\), we calculate the frequency of shots within each sub-interval \(I_{k}\). This is achieved by determining the ratio of the number of historical shots with xG values in \(I_{k}\) to the total number of historical actions (shots and moves) within \(I_{k}\). We denote this ratio as \(freq_{I_{k}}\), computed as  
\[For\;k\in\;[1,20];\;freq_{I_{k}}=\frac{N_{s_{I_{k}}}}{N_{s_{I_{k}}}+N_{m_{I_{ k}}}} \tag{7}\]  
where \(N_{s_{I_{k}}}\) represents the number of shots with xG values in \(I_{k}\) and \(N_{m_{I_{k}}}\) represents the number of moves with xG values in \(I_{k}\).  
**Step 4: Calculating \(xG_{KOS_{i,j}}\)and Determining Its Sub-Interval**  
Using the KOS xG model, we calculate the value of \(xG_{KOS_{i,j}}\) (for the action we aim to compute the DxT for). We then identify the sub-interval to which \(xG_{KOS_{i,j}}\) belongs and denote this interval as \(I_{XG_{KOS_{i,j}}}\).  
**Step 5: Deducing a Formulation for the Probability Function to Shoot f**  
For similar scoring probabilities, as modeled by the sub-intervals \(I_{k}\) in this study, the likelihood that players decide to take a shot is relatively consistent within these sub-intervals. This holds true, especially when \(\Delta\) is very small. Furthermore, as the size of the training dataset used in Step 2 increases, the probability of players deciding to shoot can be approximated by the frequency of shots falling within the sub-interval \(I_{xG_{KOS_{i,j}}}\). Therefore:  
\[f\left(xG_{KOS_{i,j}}\right)=\frac{N_{s_{I_{xG_{KOS_{i,j}}}}}}{N_{s_{I_{xG_{ KOS_{i,j}}}}}+N_{m_{I_{xG_{KOS_{i,j}}}}}} \tag{8}\]  
where \(N_{s_{I_{xG_{KOS_{i,j}}}}}\) represents the number of historical shots having xG values included in the interval \(I_{xG_{KOS_{i,j}}}\) and \(N_{m_{I_{xG_{KOS_{i,j}}}}}\) the number of historical moves having xG values included in the interval \(I_{xG_{KOS_{i,j}}}\).