# Analyzing Momentum Shifts in Tennis: A Machine-Learning Approach to Predicting Match Outcomes
## 3 Results
### Visualizing the Process of Momentum Establishment  
In this section, data from the first semifinal match of the 2023 Wimbledon Championships (men's singles) was analyzed. Based on this match and the trend of change in the players' momentum, an attempt was made to solve the linear optimization model. Figure 1 provides the fitting results of this momentum model. The regression fitting results showed that out of the 334 test data points selected, 320 fell near the fitting results and could be considered valid results. The effective rate of the results reached 95.8%, which can be considered a valid fit. The linear regression index is listed in Table 5.  
By reviewing the data presented in the table, it is evident that the actual mean value is 92.17, while the mean value predicted by the regression model is 91.25. The close proximity of these values indicates a high level of accuracy in the model's fit. The variance of the data is 21.9, suggesting a moderate degree of dispersion. Furthermore, the F-statistic is 208, which strongly supports the adequacy of the model and indicates an excellent fit to the observed data. Overall, these results suggest that the linear regression model effectively captures the underlying trends within the dataset and demonstrates strong predictive performance. Apart from that, the \(p\)-value also has very good properties. With the \(p\leq 0.001\), it means that our regression model is of high feasibility and accuracy. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Rank Mean** & **Median** & **Kendall’s W** & \(\mathbf{x^{2}}\) & \(\mathbf{p}\) \\ \hline momentum & 1.593 & 26.742 & 0.035 & 254.335 & 0.000 *** \\ difference & 1.407 & 1 & 0.035 & 254.335 & 0.000 *** \\ \(p_{v}\) & 1.407 & 1 & 0.035 & 254.335 & 0.000 *** \\ \hline \hline \end{tabular}  
* represents 1% significance level.  
\end{table} 
Table 4: The Outcome of Kendall’s W.  
Figure 1: Linear Regression Figure of Momentum Model. The optimal value of \(k_{i}\) is listed in Table 6 (to two decimal places). The Optimal Value of K.  
Leave one decimal place for the \(K_{i}\) values, we can obtain Table 7.  
It should be noted that the parameter \(k\) here is derived from the training set. When the match venue changes, the value of \(k\) may vary. Due to the small difference in momentum between the players in the 0.1 dimension, it is not convenient for visual comparison. Therefore, we multiply the weight values of the variables affecting momentum by a factor of 10 so that the unit value of the weight becomes 1. In this way, we obtain the quantitative definition of momentum, which can be seen in the equation. \[ \begin{array}{llll}M_{i}=&2*p_{iw}+4*p_{ig}+6*p_{is}+p_{ia}+p_{inetpuron}\\ &+p_{ib}+p_{ibw}+s-d-3*p_{id}-s_{n}-p_{iu}-p_{ibm},i=1,2.\end{array}  \tag{9}\]  
Figure 2 illustrates the momentum changes in the two players, [PERSON] (blue line) and [PERSON] (orange line). Despite fluctuations, the overall momentum for both players increased over time, reflecting the escalating intensity of the match. Generally, momentum surged when a player capitalized on the opponent's mistakes or achieved a winning streak, while unexpected errors or referee warnings caused a drop in momentum. By the end of the match, [PERSON]'s momentum was significantly higher than [PERSON]'s, aligning with his eventual victory. At the start of the match, [PERSON] gained an early momentum advantage, breaking [PERSON]'s first service game to take a 3-0 lead. However, around the 50-point mark, [PERSON] regained momentum due to a penalty issued against [PERSON] for a grunt during a stroke, followed by a warning for serving delays, disrupting [PERSON]'s rhythm. Momentum fluctuated between the players thereafter, but in the final