Sample and Data

The data from the study included detailed point-by-point data of 564 men's singles main draw matches during 2021-2023 U.S. Open and Wimbledon tournaments provided by Jeff Sackmann (2024) via Tennis Abstract (www.tennisabstract.com). In total, there were 135,110 points played by 211 individual players.

Procedures and Statistical Analysis

Initially, missing value and outlier were detected to ensure data quality. For instance, missing values in the serve speed (mph) were treated as null speeds when calculating each match's average serve speed, thereby maintaining the completeness and accuracy of dataset. Subsequently, cleaned point-by-point data were used to extract relevant indicators for analyzing player performance. The data processing workflow is illustrated in Figure 1.

Figure 1 Performance indicator screening flow chart

Based on the existing literature and information in the raw data, summary statistics of 12 indicators were extracted to represent player’s match performance. Additionally, a ranking gap was included as opposition effect factor on performance (see Table 1).

Research Framework

The study first developed a comprehensive research framework to capture, analyze, and predict momentum in tennis matches. Figure 2 provides an overview of the framework, detailing the key stages of the research process, from data preparation to model application. The process began with the extraction of key performance indicators, followed by the establishment of a mathematical model to quantify momentum. To validate the model, a consistency check was performed by comparing the player with the higher momentum at the end of each set to the actual match outcome. This validation step assessed the model’s alignment with actual results. Subsequently, a predictive model was developed using a Gradient Boosting Decision Tree (GBDT) regression approach. This model analyzed turning points within tennis matches, identifying areas where momentum shifts significantly impact player performance. The predictive model's accuracy and interpretability were further evaluated to generate actionable tactical insights. These insights can be applied across various contexts, including pre-match analysis, in-match tactical adjustments, and post-match reviews to optimize training and performance.

Figure 2 The Structure of the Research Framework for Tennis Match Momentum Analysis

Tennis Momentum Model

In the modeling stage, the research transformed the weights into additive changes according to the different indicators with different impact weights.

The study combines this with the fact that research will use the ranking as an initialization momentum. The initialization of momentum scores in equations (1) to (3) is grounded in the use of ATP rankings, which serve as a proxy for pre-match strength and reputation, an approach that has been utilized in prior research to set player baseline states (Pham & Bufi, 2023). First, we will use the variable Mi(t) (the momentum of player i (the i ^th player)at time t (the t ^th time point)), represents the change in momentum of player i with time, M ₁ (t) represents the change in momentum of player 1 with respect to time t, M ₂ (t) represents the change in momentum of player 2 with respect to time, t, p ₁ (p _i is the rank of player i)denotes the ranking of player 1, denotes the ranking of player 2. Thus, study can get the initialized momentum score, as shown in equations (1), (2), and (3):

Equation (4) defines momentum as a recursive function over time, drawing on basic principles of temporal state propagation found in time-series models such as AR and ARMA. The additive momentum increments in equation (5) are inspired by feature-based performance scoring systems common in sports analytics (Ma, 2024), where different actions (e.g., ace, break_point, point_won) contribute unequally to performance trends. As study traverse each time point t in the race, the momentum variable is updated according to equation (4):

Whenever a player wins a point, his performance status is expected to improve and therefore his momentum should be elevated. Such increase is represented by an increment, as shown in equation (5). For instance, if the player wins a point, his momentum increases by a fixed value d_j (j is the j ^th scoring item, and d_j is the scoring points for scoring item j); Similarly, when player breaks serve successfully or hits an ace, this brings a lot of momentum, so his momentum should have a different increase. Our study can represent this increase using other increments, which research will set to d ₂ , d ₃ respectively.

Furthermore, it was also assumed that the outcome of the previous game impacts the current game (Klaassen & Magnus, 2001). To reflect this, an initial momentum value is recalculated at the beginning of each game. This recalculation incorporates the player’s ranking and the weighted momentum value from the last point of the previous game, as shown in equation (6). Equation (6) introduces an inter-game adjustment mechanism based on weighted memory of previous performance, reflecting concepts from state transition models and psychological momentum theories (Ma, 2024), with the addition of a tunable weight β to model state carry-over. To balance the relationship between games and reflect players' state adjustments, experiments determined the optimal weighting coefficient β=0.2.. The equation for the initial momentum of each game is:

At this point, the study established a mathematical model that captures the flow of the situation as the game progresses. The model is represented by equation (7):

After getting the mathematical model of momentum, our study used python code to write the model and made the visualization of the data, taking d ₁ =1, d ₂ =1,d ₃ =1,s=1, Here, in order to show more intuitively the tendency of the players' momentum change when the match is in progress, this study introduced a concept of momentum difference, which is obtained by subtracting the momentum of player1 and player2, and at the same time.

To validate the model, a Kappa consistency test was conducted (Cohen, 1960). Momentum data from the 31 matches of the 2023 Wimbledon tournament were extracted and processed into a binary classification: values greater than 0 indicate that Player 1 had, while values less than 0 indicate that Player 2. Our research then calculated the match outcome for each point based on the momentum and filtered the results to obtain set-level outcomes. A Kappa coefficient model was established to compare these results with the actual set outcomes (game_winner), conducting a Kappa consistency test to assess alignment.

The Kappa coefficient was utilized to assess the agreement between predicted results derived from the model and actual outcomes from the raw data. As a widely recognized metric for measuring correlation between categorical data, the Kappa coefficient was appropriate for this binary classification task. This method provided a robust measure of the model’s consistency and alignment with real-world results. According to the widely accepted Kappa coefficient interpretation standard proposed by Cohen (1960), a Kappa value below 0 indicates no agreement, 0-0.20 indicates slight agreement, 0.21-0.40 indicates fair agreement, 0.41-0.60 indicates moderate agreement, 0.61-0.80 indicates substantial agreement, and values above 0.81 indicate almost perfect agreement. The Kappa value of 0.96 achieved in this study thus demonstrates an almost perfect alignment between the predicted and actual outcomes, validating the reliability of the proposed model.

Momentum Prediction Model

The study also aimed to predict critical factors influencing match dynamics. Building on the concept of momentum differential, a threshold of 0 was set to identify "swings" or turning points within matches. The GBDT regression model was trained on the data to predict these momentum swings, and feature importance was analyzed to determine the key predictors of momentum shifts. Several configurations of the GBDT model were tested to optimize performance. The final model parameters are presented in Table 2. Then, the feature importance is determined by calculating the contribution of each feature to the splits across all decision trees.

To evaluate the effectiveness and generalization ability of the proposed momentum prediction model, several alternative machine learning algorithms were tested alongside GBDT. These included LightGBM, XGBoost, and K-Nearest Neighbors (KNN). All models were trained on the same dataset using identical features and hyperparameter tuning strategies. Their performance was assessed using three key metrics: mean absolute error (MAE), root mean square error (RMSE), and the coefficient of determination (R²). The results, presented in Table 3, show that the GBDT model achieved the MAE (0.659) and RMSE (0.979), and the highest R² (0.828), indicating strong predictive accuracy and goodness of fit. LightGBM and XGBoost yielded comparable results, with only marginal differences from GBDT. These findings suggest that all three gradient-boosting frameworks are suitable for momentum prediction tasks, with GBDT performing slightly better overall. In contrast, the KNN model demonstrated weaker predictive performance across all metrics. This significant gap indicates that KNN, which lacks the ability to model complex feature interactions and sequential dependencies, is less suited for the nuanced task of modeling momentum dynamics in tennis.

Overall, these results reinforce the effectiveness of the proposed GBDT-based model while confirming the robustness of gradient-boosted ensemble methods for this specific prediction problem.

Factor Analysis

To assess the selected indicators, the study first evaluated their orientation. It was determined that indicators such as ranking gap and double faults (DF) were negatively oriented, where lower values corresponded to better performance. Conversely, all other indicators were positively oriented. To ensure consistency in the weight analysis, negatively oriented indicators were inverted, aligning all indicators to follow a consistent trend.

Before proceeding with factor analysis, an independent sample T-test was performed (see Table 3). The test for homogeneity of variances yielded a significant P-value of 0.024** for average serve speed, indicating a violation of the homogeneity assumption. Consequently, this indicator was excluded from subsequent analyses, leaving a total of 12 indicators for consideration.

To identify relevant features for the model, Pearson correlation coefficients were calculated between each indicator and the player score. The magnitude of these coefficients provided insight into which indicators were most strongly correlated with player performance. Based on this analysis, five key indicators were selected for momentum calculation model: initial ranking, right to serve, point scored, break point won, and ACE occurrence.

During the model development phase, the latest ATP rolling rankings (from one week prior to each tournament) were used as initial input values. The processed dataset included quantitative variables such as point_no, server (referring to the player currently serving), point_winner, game_winner, p1_ace, p2_ace, p1_winner, p2_winner, p1_double_fault, p2_double_fault, p1_unf_err, p2_unf_err, p1_break_pt_won, and the result_gap from the previous point.In the modeling framework, result_gap served as the dependent variable, while the remaining indicators were used as predictors. To better interpret the magnitude of the statistical differences between winning and losing players, effect sizes (Cohen’s d) were calculated alongside traditional significance tests.

Table 4 summarizes the statistical results for each performance factor, including F and P values, as well as corresponding effect sizes. Indicators such as Aces (d = 0.14), Distance Run (d = 1.50), and Winners Won Rate (d = 0.32) demonstrated varying degrees of impact. Particularly, Distance Run exhibited a large effect size, highlighting its relevance in distinguishing dominant performance patterns.

Kappa Coefficient Test

The Kappa coefficient of 0.96 demonstrates a nearly perfect alignment between the model's calculated momentum advantage and the actual set outcomes, validating its reliability. The test's significance levels (z = 33.081, p < 0.01) further confirm the statistical robustness of this alignment.

Model Testing Results and Evaluation

Figure 3 presents a partial prediction plot for the test dataset, comparing predicted values with actual values. In the plot, the blue line represents the true values, while the green line shows the predicted values. The model demonstrates a high degree of accuracy, as the predicted values closely follow the trend of the actual data, particularly around the 935th point of the season starting from the Round of 16. Some minor discrepancies are observed at smaller peaks and valleys, suggesting that while the model captures the general patterns effectively, there is room for refinement in specific intervals.

Figure 3 The Predicted Value Compared with The True Value

To systematically evaluate and address potential overfitting observed in Figure 3, the model was assessed using a 5-fold cross-validation approach. As shown in Table 5, the mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and R² scores of the cross-validation set closely mirrored those of both the training and test sets. Specifically, the training set achieved an MAE of 0.576 and an R² of 0.851, while the 5-fold cross-validation yielded an MAE of 0.609 and R² of 0.829. The test set recorded a comparable MAE of 0.659 and R² of 0.828. These minimal performance gaps indicate that the model generalizes well and does not exhibit signs of overfitting. Additionally, the RMSE difference between training (0.808) and test set (0.979) was small relative to the scale of the target variable, and the standardized effect size (Cohen’s d = 0.15) between the MAEs of training and test sets further confirmed negligible overfitting. Together, these results validate the stability and robustness of the proposed momentum prediction model.

For both the training and test sets, the corresponding values are illustrated in Figure 4.

Figure 4 Comparison of Error Metrics between Train and Test Sets

The close alignment between these metrics suggests that the model does not overfit the training data and maintains strong generalization capability. While this level of performance is satisfactory, further parameter optimization or exploration of alternative algorithms could enhance predictive accuracy on unseen data.

Feature Importance

The analysis of feature importance provided insights into the relative impact of various factors on momentum shifts. Figure 5 illustrates the key predictors identified by the model. The server variable ranked as the most significant factor influencing momentum swings, followed by point_no, which represents the progression of the match, and point_winner, which indicates the winner of a specific point. These features were determined to have the greatest impact on predicting momentum dynamics during a match.

Figure 5 Feature Importance Ratio

Model Application

Real-Time Momentum Visualization

To illustrate the model's effectiveness, the final match of the 2023 Wimbledon tournament between Carlos Alcaraz and Novak Djokovic was selected for visualization (Figure 6).

Figure 6 Momentum Visualization of the 2023 Wimbledon Men’s Final

The visualization shows that Djokovic initially held the momentum advantage, while Alcalaz gained control midway through the match, resulting in a significant momentum shift. The model accurately tracks this transition, with both players experiencing sharp momentum changes at critical points, indicating a close contest. Alcaraz's momentum eventually stabilized at a higher level, aligning with the actual match outcome, thereby demonstrating the model’s capability to depict shifts in competitive advantage over time. The momentum differential graph offers a clear view of each player’s relative advantage during the match.

The model was further applied to other matches, such as a round-of-16 match between Carlos Alcaraz and Nicolas Jarry (Figure 7), to illustrate different momentum patterns. By adjusting the match_id in the model code, momentum trajectories for various matches can be generated, showcasing the model’s adaptability to different match dynamics.

Figure 7 Momentum Visualization of a Round-of-16 Match

This visualization depicts momentum patterns from another match in the tournament. Unlike the final, this match exhibited more consistent momentum patterns, with fewer abrupt shifts, highlighting the variability in competitive dynamics across matches.

Pre-Match Analysis and Opponent Profiling

The findings of this study demonstrate the model’s suitability for pre-match analysis and opponent profiling. The model was used to calculate momentum turning points for Carlos Alcaraz and Novak Djokovic under various score scenarios during the U.S. Open and Wimbledon tournaments from 2022 to 2023, providing valuable insights into their strategic tendencies and performance shifts.

Figure 8 presents the momentum turning points for both Carlos Alcaraz across 22 matches and Novak Djokovic across 26 matches. Positive_Turning_Points indicate moments where momentum shifts from negative to positive, while Negative_Turning_Points represents shifts from positive to negative. For Carlos Alcaraz, at a score of 0:30, he encounters more negative turning points (75) than positive ones (62). Conversely, at 30:40, positive turning points (49) exceed negative ones (33). The lower sum values at scores such as 40:0, 40:15, and 40:30 reflect fewer momentum fluctuations, indicating greater stability at these moments. For Novak Djokovic, the total number of momentums turning points for Novak Djokovic is highest at a score of 15:0 (142), followed by 15:40 (132) and 15:15 (130). Negative turning points dominate at disadvantageous scores like 0:40 and 0:30, while critical moments such as 15:40 and 15:15 exhibit significant positive turning points.

Figure 8 Momentum Turning Points for Carlos Alcaraz and Novak Djokovic