# A Study on the Optimisation of Tennis Players' Match Strategies from the Perspective of Momentum
## 2 Construction of the Tennis Players' Momentum Quantification System
### Construction of the Dynamic Momentum Quantification System  
In light of the divergent psychological attributes and personal capabilities exhibited by players, the transfer probability matrix of the Markov chain is employed to dynamically quantify the momentum of tennis players. In the Markov chain model, the probability of the future state is dependent solely on the current state, and is independent of past states. The Formula (1) is as follows:  
\[P\Big{(}X_{n+1=j|X_{n-1}=i_{n-1},\ldots,X_{0}=i_{0}}\Big{)}=P(X_{n+1}=j|X_{n}=i)= P_{ij}, \tag{1}\]  
where \(X_{n+1}\) represents the future state, i.e., the win or loss of the next point in a tennis match, \(X_{n}\) represents the current state, i.e., the win or loss of the current point, and \(P_{ij}\) is the transition probability, which satisfies \(\sum_{j=1}^{m}P_{ij}=1\) for all i. In the context of match No. 1701, a detailed analysis was conducted to ascertain the probability of player 1 winning this point after losing the previous point (\(P_{01}\)). Through the utilisation of appropriate calculation methods, it was determined that the probability of player 1 winning this point after losing the previous point is 0.5060. Similarly, the probability of player 1 winning this point after winning the previous point (\(P_{11}\)) was calculated to be 0.5030. Furthermore, the probability of player 2 winning this point after losing the previous point (\(P_{01}\)) was determined to be 0.4970, and the probability of player 2 winning this point after winning the previous point (\(P_{11}\)) was found to be 0.4940. The calculation formulas for \"momentum\" are thus summarised in Formulas (2)-(4) by combining the calculation results from the above Markov chain with data on the serving side, break points, and winning streaks. \[MS_{P1}(t)=\left\{ \begin{array}{ll}(MS_{P1}(t-1)+P_{server\_win}+T)\times S,& \quad i=1\\ (\Delta C_{P12}+T)\times S,&\quad i=0\end{array} \right., \tag{2}\]  
\[T=\left\{ \begin{array}{ll}P_{break\_win},&\quad j=1\\ 0,&\quad j=0\end{array} \right., \tag{3}\]  
\[S=\left\{ \begin{array}{ll}P_{01},&\quad q=1\\ P_{11},&\quad q=0\end{array} \right., \tag{4}\]  
where the meaning of each parameter is presented in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Classification** & **Number of Indicators** & **Indicator Description** \\ \hline
1 & Basic Match Information & 12 & match\_id, player1/2, set\_no, game\_no, point\_no, etc. \\ \hline
2 & Player Performance & 16 & p1/p2\_acc, p1/p2\_winner, p1/p2\_double\_fault, \\  & Statistics & & p1/p2\_unfl\_err, p1/p2\_break\_pt, etc. \\ \hline
3 & Match Event Information & 18 & p1/p2\_sets, p1/p2\_games, p1/p2\_score, server, \\  & & & p1/p2\_distance, etc. \\ \hline \end{tabular}
\end{table} 
Table 1: Data classification table of the dataset. The application of the aforementioned formulae to the substituted data from match No. 1701 yields the \"momentum score\", a pivotal metric in the assessment of player performance. The results are illustrated in Figure 2.  
As demonstrated in Figure 2, the momentum scores of both players exhibit fluctuations throughout the tennis match, thereby reflecting the intensity of the competition and the dynamic interplay between the two athletes during each point. The fluctuations in momentum scores thus serve to highlight significant moments and critical junctures within the match. Furthermore, when a player's momentum score consistently exceeds that of their opponent, it can be deduced that said player may exert a greater degree of influence over the match's rhythm during that particular period. The analysis of the results indicates a close alignment between the fluctuating characteristics of the momentum scores and the actual match events. For instance, the momentum score advantage of Player 1 at the commencement of the fourth game in the fifth set directly corresponds to their actual performance of winning four consecutive points, thereby validating the effectiveness of the momentum score in quantifying the ability to control the match rhythm.