# 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
### Evaluation of Player Match Performance
#### 4.1.1 Weight Allocation for Key Indicators  
The entropy weight method is an objective weight calculation approach based on the principle of information entropy. This method is predominantly employed to evaluate the likelihood or significance of a specific state materialising among a multitude of independent potential states. Through data observation and analysis, seven key indicators are selected and categorised into positive and negative indicators. These two types of indicator data are then normalised, as shown in Table 5, where \(X_{min}=\text{min}\big{(}X_{1j},X_{2j},\ldots,X_{nj}\big{)}-0.0001\), \(X_{max}=\text{max}\big{(}X_{1j},X_{2j},\ldots,X_{nj}\big{)}+0.0001\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Index Type**} & \multirow{2}{*}{**Index Name**} & \multicolumn{2}{c}{**Normalisation**} & \multirow{2}{*}{**Description**} \\  & & & & **Formula** & \\ \hline
1 & Positive & Momentum & & Momentum score \\
2 & Positive & Ace & \(Z_{ij}=\frac{X_{ij}-X_{min}}{X_{max}-X_{min}}\) & Winning serve \\
3 & Positive & Winner & & Winning shot \\ \hline
4 & Negative & Double-fault & & Double miss \\
5 & Negative & Unf-err & & Unfored error \\
6 & Negative & Rally-count & \(Z_{ij}=\frac{X_{max}-X_{ij}}{X_{max}-X_{min}}\) & The number of strokes back and forth \\
7 & Negative & Dtop & & The difference in the distance run between the two players during a point \\ \hline \hline \end{tabular}
\end{table}
Table 5: Indicator meaning table.  
Pursuant to the results of the normalisation process set out in Table 5, the entropy value \(e_{j}\) and the information entropy redundancy \(d_{j}\) for the \(j\)-th indicator are calculated, as demonstrated in Equations (6) and (7), respectively.  
\[e_{j}=-k\sum_{i=1}^{m}P_{ij}\ln\bigl{(}P_{ij}\bigr{)},j=1,\ldots,m, \tag{6}\]  
\[d_{j}=1-e_{j},j=1,\ldots,m, \tag{7}\]  
Ultimately, the weight proportion of each indicator is determined using the indicator weight formula, thereby establishing the importance ranking of each influencing factor. The indicator weight formula is presented in Equation (8).  
\[w_{j}=\frac{d_{j}}{\sum_{j=1}^{m}d_{j}},j=1,\ldots,n, \tag{8}\]  
The entropy weight method results for Players P1 and P2 are presented in Table 6.  
As demonstrated in Table 6, it can be concluded that the weights attributed to Ace and Winner were relatively elevated, signifying that these two metrics more accurately reflected players' performance in the match. This is attributable to the elevated degree of data dispersion observed for Ace and Winner, in conjunction with their pronounced capacity to differentiate between players' skill levels. Conversely, the Momentum and Unf_err, which exhibit intermediate weight values, were found to be significant in evaluating player performance from the perspectives of match dynamics and technical stability. Conversely, Double_fault, Rally_count, and Dtop, which possess lower weights, demonstrated greater randomness and thus served as supplementary factors in the comprehensive evaluation of player performance.