## 3 Results  
In this section, the optimization process will be described, and it will be proved that the proposal of two separate dynamic parameters, the short- and long-term momentum, is \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & _Phist_ & _qhist_ & _mP_ & _mP_ & _mP_ & _mP_ & _mP_ & _mP_ & _mP_ & _mP_ & _mP_ & _mP_ \\  & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** \\ \hline
1 & 41.55 & 28.55 & 2.95 & 11.73 & 13.07 & 15.17 & 24.64 & 43.84 & 134.0 & 91 & 98 \\
2 & 39.34 & 29.38 & 6.23 & 11.45 & 9.40 & 9.61 & 23.40 & 28.48 & 139.5 & 86 & 91 \\
3 & 38.63 & 30.93 & 3.02 & 7.50 & 7.09 & 8.07 & 21.03 & 31.57 & 162.0 & 80 & 86 \\
4 & 37.27 & 32.20 & 4.74 & 15.47 & 23.58 & 9.23 & 19.56 & 39.01 & 174.0 & 74 & 80 \\
5 & 36.34 & 32.58 & 4.44 & 12.05 & 27.68 & 11.59 & 17.89 & 28.71 & 181.0 & 70 & 74 \\
6 & 36.67 & 33.14 & 5.49 & 9.43 & 24.13 & 8.26 & 17.57 & 31.13 & 180.0 & 63 & 70 \\
7 & 34.91 & 33.83 & 4.58 & 19.86 & 33.46 & 6.40 & 12.42 & 33.34 & 182.0 & 56 & 63 \\
8 & 35.41 & 34.35 & 1.44 & 16.37 & 32.74 & 5.33 & 16.30 & 30.20 & 180.0 & 50 & 56 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Group profiles. supported not only by domain knowledge and logical reasoning, but also by the objective results of the analysis. For optimization and result demonstration, the analysis of the two arguably most popular betting types, handicap and total points played in the match, will be conducted. Handicap and total points are the main features in building any pre-match model as they represent the strength ratio between two teams (handicap) as well as the length of the match (total points). In the following sections, the _real distribution_ of the total number of points played in a group of matches and the handicap with the _simulated distributions_ obtained by conducting the following four different simulations will be compared:  
1. Simulation based on the iid assumption;
2. Simulation that incorporates short-term momentum;
3. Simulation that incorporates long-term momentum;
4. Simulation that incorporates both short- and long-term momentums (_the evolving probability method_). To present the results in a visually interpretable manner, overlapping total points/ handicap histograms for each group of matches are plotted. One histogram shows the real distribution of the total number of points played in a group of matches/handicap obtained from the real data, while the other represents the simulated total points/handicap distribution. A large overlapping area naturally represents closer adherence between simulated and real distributions. A two-level Euclidean distance measure (L2-norm) was used to numerically support the graphs. The overlap was leveraged in the training process to optimize the parameters used by the _evolving probability method_. As stated earlier in this paper, the only parameters that need to be optimized are the parameters of the linear function, \(k\) and \(l\), necessary for the calculation of the long-term momentum (all other parameters shown in Table 2 are calculated directly from the historical data). For the purpose of optimization, the traditional empirical algorithmic approach, grid search, was chosen. The optimal combination of parameters \(k\) and \(l\) is the combination that minimizes the simulation error of all groups, in both total points and handicap overlapping histograms. Due to the nature of the optimization problem a two-level L2-norm (Euclidean distance measure) was chosen to solve the problem. The norm representation is shown below in Equations (9)-(11). In order to find the optimal values for \(k\) and \(l\), 70% of the entire dataset was used, leaving the rest for evaluation.