## 2 Materials and Methods
### Validation Dataset
#### 2.7.1 Group Profiling  
It is a well-known fact that the final results can only be as good as the quality of input data allows it, regardless of the predictive model used. In the analysis of volleyball matches, the service proved to be one of the most important features. When models are implemented in betting systems, analyses at the team, player, or tournament level have to be periodically re-run to ensure model precision. Simple ranking systems and averaged statistics are often not enough, and their results tend to be misinterpreted even by some of the leading betting houses. Home-away statistics and reactions fluctuate from player to player, team to team, and coach to coach. In an ideal scenario, enough historical data to construct a profile of team performance over a longer time for each team would be available. Then, these data can be used to calculate the likelihood of winning the point on own serve, the parameters of the short-term momentum for each team, as well as the rough estimate of the expected number of total points used to calculate the long-term momentum. These parameters can then be used as input parameters of the proposed _evolving probability method_ to estimate the handicap and the total number of points that will be played between two teams. In the case of this paper, however, the heterogeneity of the available dataset did not allow leveraging this simple approach. The collected dataset did not contain enough information on the historical matches of most teams. Since acquiring a significantly larger dataset was not feasible, it was decided to devise with a method that would most efficiently compensate for the relatively low number of observations (when taking into account the needs for team profiling). The approach proposed in this paper was to exchange the notion of individual team profiling with building group profiles, but in a way that does not group teams--individual matches in which opposing teams have a **similar strength ratio** are grouped. For example, one of the groups contains matches with a strong favorite, while the other contains matches in which teams of similar strengths are opposed. It is important to notice that in this way it is possible to leverage the dataset much more efficiently and that now, individual teams are naturally represented in more than one group since they have different strength ratios when pitted against different teams. In this way of profiling, the strength of the opposing team is taken into account, and it is not necessary to take additional care of it. This means that the same team will have a different profile depending on the opposing team. The first step in the dataset division was performing the Shin transformation ([59; 60; 61]) of pre-match winning odds to calculate the pre-match probabilities of winning the match for both teams. These probabilities were then used to determine the \"home\" team and the \"away\" team in each match, as was explained before. The Shin transformation was required because some betting houses adapt their betting odds much more to market trends than the real state in the game. Consequently, there is no way to calculate the real probability of winning the match from the odds. It has been empirically demonstrated that Shin's normalization is optimal to transform the odds into probabilities when it comes to the outcome of volleyball matches [60]. After the real probability of the match outcome was approximated from the odds, the dataset was sorted by the calculated likelihood of the favorite team winning the match. Then, groups of matches were defined with constraints that every group needs to have a similar number of matches, and should not be too wide. The number of groups to choose is a trade-off between the size of the dataset and the similarity of the matches within a particular group. The number eight was arbitrarily chosen, but it is expected that similar results will be obtained with a different number of groups, as long as the number of groups is not too small nor too large.