## 2 Materials and Methods
### Long-Term Momentum  
Short-term momentum will not adequately represent situations when events on the field strongly deviate from historical statistics. This can happen for various reasons, which may or may not be reflected in data about the match--for example, a strong team could be missing key players due to injuries, resulting in a poorer performance compared to what historical statistics show, and the model which uses only historical data may continue to favor them more strongly than it should. To account for situations like these, it is important to combine historical match data and available data about the match whose outcome is being predicted. Several papers have already proposed approaches for updating the pre-match data with new information in tennis [40; 41; 42]. In this paper, the notion of \"long-term momentum\" is proposed. This momentum dynamically adjusts the historical serve-winning probability using the updating rule which leverages empirical Bayes estimator and currently simulated match results. The idea is to simulate all potential score sequences of the match whose outcome is being predicted. This effectively means that the probability of winning a point is slightly adjusted after each point in the match simulation, so if the team is on a losing streak, the probability of winning the next point may gradually become significantly lower than the average point-winning probability gained from historical statistics, better reflecting what is actually going on in a match simulation. To integrate the long-term momentum in the proposed model, a new tuning parameter \(\lambda\) is introduced, which will represent the size of the long-term momentum effect on the point-winning probability. The long-term momentum has to behave in such a way that historical probability always has the largest influence at the beginning of the match, which then diminishes as the match simulation progresses--especially if the currently simulated match events strongly deviate from what historical statistics state. To model this without introducing a lot of unnecessary complexity, a simple linear interpolation is used, with the interpolation parameter being the current number of points in the match simulation (\(totalPts_{current}\)) divided by the parameter \(\lambda\). The new probabilities are called \(p_{lttm}\) and \(q_{lttm}\) (\(lttm\) denoting \"long-term momentum\")--see Equations (3) and (4). Quantities \(p_{current}\) and \(q_{current}\) are also introduced, which are the probabilities of winning one's serve using only information from the currently simulated match--in effect, a ratio between the points won on the team's own serve and the total number of serves, for each team, respectively:  
\[p_{lttm}=\frac{totalPts_{current}}{\lambda}*p_{current}+(1-\frac{totalPts_{ current}}{\lambda})*p_{ltt} \tag{3}\]  
\[q_{lttm}=\frac{totalPts_{current}}{\lambda}*q_{current}+(1-\frac{totalPts_{ current}}{\lambda})*q_{ltt} \tag{4}\]  
The choice of \(\lambda\) parameter allows the ability to choose how strongly the long-term momentum will influence the point-winning probabilities. Choosing larger values of \(\lambda\) makes historical statistics have a larger impact, while lower values put more emphasis on what is currently simulated. The logical question that arises is how to choose the value of this parameter, or, in other words, how to estimate which values of \(\lambda\) would work best based on the _a priori_ information about the match. Due to the nature of volleyball, the difference in team strengths directly influences the number of points. A simple yet effective approach is to choose the value of parameter \(\lambda\) concerning the rough estimate of the expected number of total points. There is logical reasoning behind this if the nature and rules of volleyball are considered. If two teams of similar strength and ability are playing, a much longer match is expected, compared to a match that pits severely imbalanced teams with an obvious favorite against each other.