## 2 Materials and Methods
### Short-Term Momentum
Figure 3 shows the number of momentum occurrences (\(N_{i}\)) in an average match for an average team depending on the momentum order (\(i\)). For example, the \(N_{2}\) value represents the number that tells how many times the team has won three points in a row. By analyzing Figure 3, it is evident that momentums of the fourth order and beyond appear extremely rarely during the matches in the dataset used in this research. This means that the low number of observations pertaining to these momentums does not provide statistically relevant results. For this reason, the _evolving probability method_ proposed in this paper implements the short-term momentum only up to the third order, even though the method can be very easily extrapolated to work with momentums of higher orders if desired. Note: The values themselves on Figures 2 and 3 are not as significant in demonstrating the general trend of the relationship. These values are discussed in more detail in Section 2.7.1. Since the real-life data (see Figure 2) provide enough evidence to rationalize the introduction of the short-term momentum, an approach to mathematically model it using available historical data is devised. If there are enough historical data available, the historical probability of winning a point on serve for a particular team can be calculated. This will be referred as \(p_{\mathit{hist}}\) and \(q_{\mathit{hist}}\). If the existence of the short-term momentum is assumed, one can also calculate the  
Figure 2: The increase in the average service win probabilities up to the eighth order momentum. short-term momentum values as explained before (conditional probabilities). These values will be called \(m_{1}\), \(m_{2}\) and \(m_{3}\). Let us start with two variables, \(p_{hist}\) and \(q_{hist}\). In this paper, these values are calculated as probabilities of winning a point after a break. This was chosen simply because, by definition, the points after the break are the first points in a row and as such do not include the short-term momentum based on the way it is defined in this paper. Then, when the team who is on serve wins a point, the probability of winning the next point on serve is adjusted, exchanging \(p_{hist}\) with the new probability \(p_{stm}\), or \(q_{hist}\) with the new probability \(q_{stm}\), depending on which team was on serve (the letters \(stm\) denote \"short-term momentum\")--see Equations (1) and (2):  
\[p_{stm} =p_{hist}*(1+m_{i}^{p}) \tag{1}\] \[q_{stm} =q_{hist}*(1+m_{i}^{q}) \tag{2}\]  
The index \(i\) represents the momentum order and can acquire values 1, 2, or 3 depending on how many points in a row the team in question has won, and letters \(p\) and \(q\) refer to the home and away team, respectively, in accordance with the previously established notation (note: \(p\) and \(q\) in the expression do not signify the exponentiation, but the team the momentum value is related to). As stated previously, these momentum parameters are calculated from historical data before the match (if enough data are available). To keep the calculations simple, momentum values are kept constant during the match and do not  
Figure 3: The number of momentum occurrences up to the eighth order momentum. consider the potential importance of the specific point being won considering its role in the match, and how it may affect the momentum value itself. With enough historical data, implementing this approach should be rather straightforward. However, the specifics of each sport should be taken into consideration when handling the short-term momentums themselves. For example, in the case of volleyball, it is enough to analyze the short-term momentum after a team's serve. This stems from the fact that volleyball rules dictate rotating serves after each lost serve. Additionally, in this sport, the team line-up changes very frequently, so a short-term momentum happens when either a good server is on the turn to serve (so a point is immediately scored) or a well-trained lineup is doing a good job following that serve, turning it into a point.