## 2 Materials and Methods
### Short-Term Momentum  
The short-term momentum represents a brief above-average performance of a team after a certain motivating event happens in the match. Scientists have explained this through the psychological momentum phenomenon, better known as \"hot hand\" in the sports domain. In tennis, scientists usually consider this motivating event to be the win of a game or a set [29; 55; 56]. [PERSON] et al. [38] considered a more fine-grained approach in volleyball and examined the hot hand effects at the point level. They used two different measures for proving the existence of a hot hand in volleyball matches: conditional probability and runs test. The same approach has previously been seen in basketball [23]. The research described in this paper also uses conditional probability and available historical data to explore the team's reaction after winning one or more points in a row on their serve. Therefore, the short-term momentum of the first order is calculated as the conditional probability of winning the second point in a row after the team has won the first point. Similarly, the short-term momentum of the second order is calculated as the conditional probability of winning the third point in a row after the team has previously won two points in a row, etc. However, unlike [PERSON] et al. [38], the idea of this paper is not to test the existence of the hot hand, but to build a probabilistic model that implements the short-term boost phenomenon as an increase in point-winning likelihood on one's serve, and demonstrate that such a model performs better compared to one that does not recognize this phenomenon. Before modeling the short-term momentum itself, an extensive exploratory analysis on available discrete sports data was performed. Real-life volleyball match data were used to test the following two assumptions, which are required to rationalize the need for the short-term momentum:  
1. If a team wins a point, a short-term \"boost\" effect appears which can be represented as the probability of winning the next point becoming slightly larger;
2. The boost effect is cumulative, meaning that winning more points in a row will result in an even larger probability of winning the next point Before demonstrating the results, new terms will be defined. Motivating events that are primarily considered are one or more consecutive points won on a team's own serve which lead to winning the next point in sequence. Momentum order (\(i\)) indicates the number of such consecutive points. For example, \(i=2\) means that the serving team has won three points in a row, i.e., the motivating event of winning two consecutive points has occurred, and that the event led to winning the third point. The number of momentum occurrences (\(N_{i}\)) represents how many times the momentum of \(i\)-th order appeared (per team) during a match. Momentum value (\(m_{i}\)) is a number that indicates how much the probability of winning the next point on the team's own serve is increased, compared to the previously established serve probability, if the serving team has previously won \(i\) points in a row. When the team who is on serve wins \(i\) points on one's own serve, the probability of winning the next point on serve will be calculated using Equations (1) and (2). Equations (1) and (2) are based on previously calculated momentum values, \(m_{i}\) (the equations are explained in more detail below). Figure 2 shows the dependence of the momentum value (\(m_{i}\)) on the momentum order (\(i\)) of an average volleyball team. For example, the \(m_{2}\) value is a number that represents how much (in percentage) the probability of winning the third point will increase, in relation to the average probability of winning points on the team's own serve, if the team previously won two points in a row. Figure 2 confirms the hypothesis that a short-term boost effect exists and that it is cumulative. Winning more points in a row will result in a larger probability of winning the next point.