## 2 Materials and Methods
### Combining Short- and Long-Term Momentums  
The combined formula intends to update the serve probabilities of both teams in a volleyball match by combining the historical statistics and the statistics of the currently simulated match (i.e., long-term momentum), while at the same time updating the newly calculated probabilities depending on the occurrence of events which also cause a short-term change in the service statistics (i.e., short-term momentum).  
To implement the combined function for modeling the non-iid distribution, Formulas (1) and (5) are combined, treating \(p_{hist}\) from Formula (1) as \(p_{ltm}\) in Formula (5). The same formula can be written for the _away_ team, treating \(q_{hist}\) from Formula (2) as \(q_{ltm}\) in Formula (6):  
\[p=\left[\frac{total\)\(Pts_{current}}{k*total\)\(Pts_{exp}+l}*p_{current}+\left(1-\frac{total\)\(Pts_{current}}{k*total\)\(Pts_{exp}+l}\right)*p_{hist}\right]*\left(1+m_{i}^{p}\right) \tag{7}\]  
\[q=\left[\frac{total\)\(Pts_{current}}{k*total\)\(Pts_{exp}+l}*q_{current}+\left(1-\frac{total\)\(Pts_{current}}{k*total\)\(Pts_{exp}+l}\right)*q_{hist}\right]*\left(1+m_{i}^{q}\right) \tag{8}\]  
where the symbols denote the following:  
\(p_{hist}\) (\(q_{hist}\)) --the probability of winning a point on own serve for the home (guest) team calculated from historical data;  
\(p_{current}\) (\(q_{current}\))--the probability of winning a point on own serve in the current match for the home (guest) team;  
\(m_{i}^{p}\) (\(m_{i}^{p}\)) --the momentum parameter that determines how much the serve probability would further increase depending on the event of winning one, two or three points in row for the home (guest) team (if \(i=0\), \(m_{i}^{p}=0\) (\(m_{i}^{q}=0\)), if \(i>3\), \(m_{i}^{p}=m_{3}^{p}\), (\(m_{i}^{q}=m_{3}^{q}\)));\(total\)\(Pts_{exp}\) --the expected total number of points that will be played in the match between the teams of known strength ratio calculated from historical data; \(total\)\(Pts_{current}\) --the total number of points played in the current match; \(k\),1 --parameters of the linear function.  
Figure 4 shows a match tree that incorporates the proposed combined Formulas (7) and (8). The figure shows the changes of serve probabilities throughout the match for both teams. In order to demonstrate the example, a match with the following characteristics is selected: \(p_{hist}=35.03\%\), \(q_{hist}=34.47\%\), \(m_{1}^{p}=2.28\%\), \(m_{2}^{p}=21.77\%\), \(m_{3}^{p}=31.75\%\), \(m_{1}^{q}=5.51\%\), \(m_{2}^{q}=13.97\%\), \(m_{3}^{q}=34.58\%\) and \(total\)\(Pts_{exp}=179\). Values \(k=1\) and \(l=0\) are selected as the parameters of the linear function. In the approach proposed in this paper, after each rally, the non-constant input parameters of the proposed functions (7) and (8) are updated. Those parameters are the following: \(total\)\(Pts_{current}\), \(P_{current}\) and \(q_{current}\). After updating the values, Equations (7) and (8) are used to calculate the probabilities of winning a point on a team's own serve. Each state has exactly two exclusionary transition probabilities, so it is possible to describe a volleyball match as a binary tree where each leaf represents a non-unique match result and all nodes with the same number of total points are at the same tree depth.  
After the presentation and description of the formula that combines short- and long-term concepts, the next section will describe how to leverage this formula to produce match-winning predictions. For this purpose, a Monte Carlo simulation of a volleyball match is performed.