# A Study on the Optimisation of Tennis Players' Match Strategies from the Perspective of Momentum
## 4 Construction of a Momentum-Based Player Match Strategy Optimisation Model
### Prediction of Match Trends
#### 4.2.5 Nonlinear Fitting for Predicting Match Trends  
The calculations pertaining to the multiple linear regression equation previously discussed have yielded an adjusted coefficient of determination \(AdjR^{2}\) of 0.797. This finding suggests that the model possesses the capacity to make predictions. However, in order to enhance the model's accuracy and generalisability, in this study, we established a \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Test NO.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Absolute ME (\(\times\)**10\({}^{-4}\)**) & 9.27 & 8.32 & 8.87 & 9.12 & 9.02 & 8.46 & 9.41 & 8.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Normalised Regression Coefficient Results. BP neural network prediction model using the 13 selected indicators for prediction. The hyperbolic tangent sigmoid function (tansig) is employed as the activation function for the hidden layer neurons, while the linear function (purelin) is used as the activation function for the output layer. The mathematical expressions of the neural network are shown in Equations (16) and (17). \[f(x)=W^{(o)}tansig\Big{(}W^{(h)}x+b^{(h)}\Big{)}+b^{(o)}, \tag{16}\]  
\[tansig:y=\frac{2}{1+e^{-2x}}-1, \tag{17}\]  
where W represents the weight matrix, and x and b are vectors. The superscript o denotes the input layer, and h denotes the hidden layer. This study involved the training of multiple models and employed the Levenberg-Marquardt (LM) algorithm, a non-linear least-squares optimisation method, to train the parameters of the neural network, calculate and control errors, and ultimately select the optimal model. The expressions for calculating the mean-squared error (MSE) and root-mean-squared error (RMSE) are shown in Equation (18). \[MSE=\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y_{i}})^{2},\ RMSE=\sqrt{\frac{1}{m} \sum_{i=1}^{m}\Big{(}y_{i}-\hat{y_{i}}\Big{)}^{2}}, \tag{18}\]  
In order to facilitate a comparison of the predictive performance of the BP neural network model and the multiple linear regression model, data from five competitions (ID: 1304, 1406, 1502, 1601, and 1701) were utilised for computational analysis. The multiple linear regression model yielded a mean-squared error (MSE) of 0.3268 and a root-mean-squared error (RMSE) of 0.5716. In comparison, the BP neural network prediction model achieved an MSE of 0.2006 and an RMSE of 0.4479. The findings suggest that the predicted values from the BP neural network model exhibit a stronger alignment with the actual values, thereby substantiating its superior predictive accuracy. The establishment of a quantitative match trend evaluation system and optimisation strategies was predicated on the calculation of the cumulative momentum score difference by the model. A thorough analysis of the match data revealed that, when the absolute value of the cumulative momentum score difference between players fell below 1.0, the match entered a state of heightened contestation, characterised by balanced scoring opportunities. In this phase, players are advised to prioritise shot consistency and tactical diversity, utilising precise placement control and rhythm variation to overcome deadlocks. It is recommended that coaches provide guidance to athletes in the adoption of steady defence and counterattack strategies, with a focus on maximising first-serve success rates and leveraging frequent placement changes to exhaust opponents' physical stamina. In instances where the absolute cumulative momentum score differential exceeds 1.0, the player who is in an advantageous position is likely to sustain their dominance over the subsequent 3-5 scoring rounds, thereby consolidating their advantageous position. At this stage, the leading player should sustain offensive pressure by enhancing serve-and-volley tactics and proactive line-changing strategies to extend gains.