A machine learning model the prediction of athlete engagement based on cohesion …

1. where ({y}_i) is the truth value of the ith sample and (mathop ylimits_i left( {c_{m} ,g_{n} } right)) is a function of C and g. We terminate the optimization process when the predictive model yields the model we need.Houyuan, Z., Qi, Y., Shengren, C. & Bonan, Z. Research on prediction model of amateur runners’ sports performance based on FMS. C. Sports Biomech. Branch. 449–451 (2023).
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3. JM Keller MR Gray JA Givens 1985 A fuzzy k-nearest neighbor algorithm IEEE Trans. Syst. Man Cybern. 4 580 585This study aims to predict the level of athlete engagement by examining both team and individual dimensions. At the team level, cohesion is one of the most frequently cited concepts in sports science research. It refers to the solidarity and sense of belonging that individuals feel in a team, which is a key factor influencing team performance. Strong team cohesion predicts the level of athlete engagement. In sporting events, cohesion provides crucial social support for athletes²³. Teams with high levels of cohesion usually result in higher levels of member engagement and increased support and trust in each other. Research suggests that cohesion is associated with satisfying psychological needs, promoting intrinsic motivation, and elevating athlete engagement levels. Increased team cohesion is an effective way to increase athlete engagement among athletes. It can be concluded that cohesion not only affects the emotional experience of athletes, but also plays an significant role in their behavioral performance. At the individual level⁹, passion and mental toughness play crucial roles in influencing athlete engagement. Hegel once asserted, “Nothing great in the world has ever been accomplished without passion.” Passion is not only a potential indicator of teacher engagement among secondary school²⁴ but also significantly influences students’ learning engagement²⁵. In sports, passion, internalized into personal identity²⁶, fosters strong engagement, enabling athletes to persist in their activities. Research suggests that we should focus on promoting harmonious passion tendencies within achievement situations in order that there is an increased probability of adaptive behavioral outcomes ensuing and being able to promote healthy sport engagement²⁷. Also, the two types of passion (harmonious and obsessive) should lead to engagement that, in turn, should lead to improved sport performance. Thus with passion, people are able to focus completely on the task at hand, give their full commitment to the sport, and experience positive results from their participation. Mental toughness is indispensable in both protecting and enhancing athlete engagement, representing a crucial psychological asset for achieving exceptional sporting performance. Research suggests that mental toughness is essential for athletes to cope with difficulties successfully, avoid the effects of stress and maintain their performance development²⁸. It not only has a significantly positive predictive influence on athlete engagement²⁹ but also correlates positively with an individual’s sports performance. Athletes with a high level of mental toughness can participate in the process of training and competing with a positive state of mind and body, and have a high level of athlete engagement (e.g., a high level of mental toughness). Athletes with high mental toughness have a strong degree of confidence in their own abilities and respond positively when they encounter difficulties, so they tend to approach training and competition with a more positive physical and mental condition³⁰ and have a high level of athlete engagement. Therefore, mental toughness plays a central role in the foundation of sport success and is a proximal factor in the athlete engagement.
4. DOI: https://doi.org/10.1038/s41598-025-87794-y
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5. SVR is a regression algorithm based on the Support Vector Machine (SVM) algorithm. SVR maps the input data to a high-dimensional feature space and constructs a hyperplane that maximizes the boundary between the predicted and actual values. In the given sample set (T={left(x_{1},{y}_{1}right),cdots ,left(x_{N},{y}_{N}right)}), there are N input variables (x_{t}) and corresponding output variables ({y}_{t},t=1,cdots ,N). The form of SVR can be expressed as follows Eq. (17).Different machine learning algorithms predict the outcome of the MAE.
6. Xin Zhang & Song Guwhere (w) is an element in the high-dimensional space, (varphi left(xright)) is a nonlinear mapping function, and b is a threshold. Then, the above regression problem seeks to minimize the risk of f by minimizing the Euclidean norm of w to reduce the complexity as follows Eq. (18).
7. Different machine learning algorithms predict the outcome of R².Supervised learning models can be classified into regression and classification techniques when used to predict continuous and discrete target variables. LR is a typical regression model that predicts targets through the following equation⁴⁵ . Its formula is expressed in Eq. (7).
8. Previously, researchers have delved into the factors affecting athlete performance from a number of perspectives, encompassing not only the traditional technical, tactical and physical aspects, but also cutting-edge areas such as exercise fatigue, recovery mechanisms and psychological factors. Therefore, analyzing and predicting athlete performance is a complex and dynamic endeavor that involves multiple considerations. And machine learning technology plays an important role in analyzing data on athlete performance. The application of this technology is changing the way athletes are monitored in terms of health and performance, thus impacting their training, competition and career management. However, in previous studies, many researchers and coaches have focused on reducing exercise fatigue and burnout, so that they overlooking the fact that athletes who maintain a high level of engagement with training and competition are the ones who respond positively to the stress of competition and perform well in training or competition. Engagement represents a sustained, shared affective-cognitive state of concentration on any object, event, person or behavior. When athletes have a sense of control over their ego state, it helps them achieve their goals. Athlete engagement studied in this paper serves as an important indicator of positive athletic performance and can reflect positive and healthy mental states of athletes. Athletes’ mental health is a major resource throughout their athletic career and beyond. However, the diversity and complexity of sports performance and the nonlinear relationships between psychological traits complicate the task of analysis and prediction using traditional methods. Therefore, the use of effective machine learning models may help to achieve highly accurate predictions of athletic performance.K Goodger T Gorely D Lavallee C Harwood 2007 Burnout in sport: A systematic review J. Sport Psychol. 21 127 151
9. Through Table 3, the pso-SVR model was identified as the best hyperparameter combination. As shown in Fig. 6, the best hyperparameters for this machine learning model with respect to the features of cohesion, passion and mental toughness were C = 6.6636 and g = 0.0436. With this hyperparameter combination, the model achieved a prediction accuracy of 92.62%, demonstrating its effectiveness in predicting athlete engagement.Here ({(j,s)}^*) represents the selected best feature (j) and the cut point (text{s}), while ({R}_{1} (j,s)) and ({R}_{2} (j,s)) denote the two subregions obtained after the cut respectively. ({y}_i) is the target value of the ith sample, ({c}_{1}) and ({c}_{2}) are the average of the target values in the two subregions, which are calculated using the formula in Eq. (14).
10. C Lonsdale K Hodge S Jackson 2007 Athlete engagement: II. Development and initial validation of the athlete engagement questionnaire Int. J. Sport Psychol. 38 4 471 492This study was conducted according to the guidelines of the Declaration of Helsinki and approved by the Ethics Committee of Zhejiang Normal University (ZSRT2022079). All methods were carried out in accordance with relevant guidelines and regulations. Informed consent has been obtained from the participants. We express our great gratitude to all the authors of the scales and all the participants in the present study.
11. RFE is a feature selection technique that relies on a specific machine learning model to achieve a predefined number of features by iteratively eliminating features with lower weights. It sorts features based on the model’s coefficient weights or importance scores, recursively removing a certain proportion of redundant or irrelevant features at each iteration, ultimately retaining the subset of features that contributes most to the model’s predictive capability. However, a limitation of the traditional RFE method is the need to manually set a threshold for retaining the number of features, which could lead to the loss of important features or the retention of unnecessary ones. To overcome this drawback, this study adopted RFECV. This method evaluates the performance of each feature subset and selects the feature combination with the highest score, thereby ensuring that the chosen feature subset maximizes its contribution to model performance and effectively avoids the accuracy loss that may result from manually setting thresholds⁴³. Using the RFECV method, we screened out features with higher weights and determined that the three features of cohesion, passion and mental toughness were critical for model performance by comparing different combinations of features, as detailed in Table 2.AC St-Louis M Rapaport PL Chénard RJ Vallerand S Dandeneau 2021 On emotion regulation strategies and well-being: The role of passion J. Happiness Stud. 22 1791 1818
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14. Feature selection of machine learning model. The study decided to select the features for the machine learning model from the athletes’ cohesion, passion and mental toughness. Cohesion consists of four scores: ATG-T, ATG-S, GI-T and GI-S; Passion consists of three scores: passion criteria, harmonious passion and obsessive passion; Mental toughness consists of three scores: confidence, constancy and control.
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17. To validate the performance of the model, the available database was divided into two distinct groups: a training set for training predictive models and a testing set for evaluating its performance. After comparing the experimental results of different data partition ratios, we found that the 8:2 partition ratio has the lowest RMSE, as detailed in Table 1. During the model training phase, we improve the parameters continuously and use the test set to evaluate the performance of the model for further refinement. The test set not only evaluates the data, but also confirms the reliability and generalization ability of the model. In addition, employing five-fold cross-validation³⁵, we guarantee the ability of model to generalize and successfully mitigate the danger of overfitting.Article 
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20. M Sheard J Golby A Wersch van 2009 Progress toward construct validation of the sports mental toughness questionnaire (SMTQ) Eur. J. Psychol. Assess. 25 3 186 193 https://doi.org/10.1027/1015-5759.25.3.186Figures 9 and 10 displayed the MSE and MAE of seven machine learning algorithms. Both MSE and MAE were crucial indicators for assessing model quality, directly reflecting the accuracy of the model’s predictions. By comparing these error metrics, the overall effectiveness of the different prediction models could be determined. Before assessing model performance, lower MSE and MAE values indicated that the model’s predictions were closer to the actual data. Based on the results in Fig. 9, we can see that the PSO-SVR model has the best performance with an MSE of 0.046. This is followed by the GA-SVR model with an MSE of 0.0153. The LR model has an MSE of 0.0177. According to the results in Fig. 10, the LR model has an MAE of 0.0618 which is the best performance. This is followed by the PSO-SVR model with an MAE of 0.0656. The MAE of the RFR model was 0.0708, while the MSE and MAE values of the other four algorithms were higher, indicating larger prediction errors. Therefore, based on MSE and MAE values, the PSO-SVR model demonstrated the best performance in predicting athlete engagement, with the lowest error metrics, meeting the research requirements. This suggested that the PSO-SVR model was more effective in predicting athlete engagement while maintaining lower prediction errors.
21. Data preprocessing and normalizing techniques are employed to enhance the dataset’s quality. These methods aim to optimize the performance of machine learning algorithms by minimizing data noise and decreasing data dimensions⁴². The input dataset comprises the cumulative scores for cohesion, passion and mental toughness, whereas the output dataset comprises the overall score for athlete engagement. Input and output data normalized in the range 0 to 1. Equation (1) was used for normalization of data.
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23. S Kent K Kingston KF Paradis 2018 The relationship between passion, basic psychological needs satisfaction and athlete burnout: Examining direct and indirect effects J. Clin. Sport Psychol. 12 75 96Optimal hyperparameters for machine learning feature models.
24. In sports, the application of machine learning and AI technologies is continually broadening the boundaries of sports talent development, athlete performance enhancement and event decision-making. Through these advanced technologies, coaches and sports teams are able to achieve accurate predictions of athlete performance, driving a shift in the decision-making process from intuitive experience to quantitative data-based analysis¹⁰. Researchers have used smart technologies to play an significant role in several aspects of selection of basketball players, analysis of game activity, prediction of results and so on, improving team competitiveness significantly^11,12. The soccer club case further demonstrates the practical application of sports data analytics in predicting injury prevention, assessing optimal performance and optimizing training¹³. It is clear that predictions utilizing machine learning that enable timely assessment of athletes’ reaction and performance to different scenarios are critical for improving skills, promoting team synergy and guiding sports teams towards success. This highlights the significant value of sports prediction modeling in contemporary sports.Table 1 presented a summary of the predictive accuracy, RMSE, MSE and MAE of the prediction models constructed using seven different machine learning algorithms. Figure 11 displayed a radar chart generated from Table 3, which was used to assess the overall effectiveness of the prediction models established by these algorithms. To visually compare the performance of the different machine learning algorithms’ prediction models in terms of statistical accuracy, the radar chart was used to show the models’ combined performance across four dimensions: RMSE, MSE, MAE and predictive accuracy. In this radar chart, PSO-SVR and LR perform well on all assessment metrics, especially their R² values are close to 1, showing high predictive accuracy. Although GBR performs well on R², its RMSE and MSE are not as good.
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    where ({|left|wright||}^{2}) is the Euclidean norm of the weight vector, which is used to control the complexity of the model. ({R}_{reg}) is the empirical risk, (C) is the penalty parameter used to balance the weights of the descriptive function and the empirical risk, and ({left|{y}_i-fleft(xright)right|}_{varepsilon }) is the insensitive loss function used to measure the difference between the predicted value (fleft(x_iright)) and the true value ({y}_i). By introducing the non-negative slack variable ξ and the Lagrangian function, the function (fleft(xright)) is described as follows using the Karush–Kuhn–Tucker (KKT) condition Eq. (19).
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27. The study was funded by Zhejiang Provincial Philosophy and Social Sciences Work Office, 23NDJC118YBThis study comprehensively considered key characteristics such as cohesion, passion and mental toughness, and developed a machine learning model aimed at predicting athlete engagement. Through an in-depth comparison and analysis of the six base algorithms and four optimization algorithms, we observe that the PSO-SVR model outperforms the other models in terms of prediction accuracy significantly. In particular, it has demonstrated a remarkable ability to deal with non-linearities, adapt and optimize the feature space, and provide more accurate and reliable predictions of athlete engagement. The PSO-SVR model performed impressively in predicting athlete engagement with a prediction accuracy of 0.9262. In contrast, other models such as GA-SVR, RGS-SVR, SVR, LR and SWO-GBR are less predictive slightly. The results of this research not only highlight the great potential of the PSO-SVR model in predicting athlete engagement, but also provide a solid scientific basis for the field of athletic training and management. In summary, the wide application of the model is expected to provide scientific decision support in the field of sports training and management, and thus improve the individual performance of athletes and the overall competitiveness of sports teams.
28. LR struggles to capture the complex nonlinear relationships of psychological features and has limited predictive power. KNN has reduced accuracy on small datasets and is sensitive to feature scaling. GBRT and DTR are susceptible to overfitting by noise and outliers. RFR performs poorly when dealing with highly correlated features. SVR has stringent kernel function selection and preprocessing requirements. RGS is computationally expensive in high dimensional spaces. GA requires fine tuning of parameters and multiple iterations to converge^31,32. SWO as a new algorithm, lacks empirical guidance for parameter tuning and may not be as effective as mature algorithms³³. PSO is easy to fall into local optimum in high dimensional space, and is very sensitive to the initial setting and parameter adjustment³⁴.ZA Sejuti MS Islam 2023 A hybrid CNN–KNN approach for identification of COVID-19 with 5-fold cross validation J. Sens. Int. 4 100229
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30. C Tedi 2023 Coach leadership style, coach-athlete relationships and engagement in team sports: The mediating role of athlete burnout Revista iberoamericana de psicología del ejercicio y el deporte. 18 5 687 692The SMTQ was compiled by Sheard, Golby and Wersch (2009)³⁸. The Chinese version of the questionnaire demonstrated good reliability and validity in its application within China. There are 14 items in total, including three dimensions of self-confidence, constancy and control. A higher score indicates greater mental toughness in the athlete. The Cronbach’s ɑ coefficient for the questionnaire in this study was 0.849, signifying its strong reliability.
31. As an algorithmic discipline, machine learning analyzes and assimilates large volumes of pre-existing or synthesized datasets to make predictions and assessments. This provides sports scientists, coaches and athletes with an objective analysis tool for evaluating their performance, thus assisting them in making optimal decisions. Through personalized predictions from “learning algorithms”, it is possible to conduct in-depth analysis of each athlete’s performance and develop customized training plans to help them achieve optimal athletic performance. This allows athletes to gain a more comprehensive understanding of their physical and mental health, enabling timely adjustments and improvements in their performance during training and competitions. Coaches, for their part, can use predictive modeling to monitor athletes engagement levels regularly. Based on the prediction results, real-time information on the athlete’s true athletic performance can be obtained, and measures can be taken timely, such as adjusting the training load or adding psychological counseling. Changes in athlete engagement before and after training can also be analyzed to assess the effectiveness of training interventions and optimize future training strategies continuously. Such data-driven decision-making is often more reliable than subjective judgment. Sports managers can leverage insights from machine learning research to better support coaches and athletes, optimize resource allocation, and enhance the overall competitiveness of sports teams or programs. For example, predictive modeling of athletic engagement is utilized to develop scientific selection criteria to ensure that athlete reserves are recruited that match the team’s culture; Counseling and support resources can be allocated according to the psychological needs of athletes to ensure that athletes receive the support they need during training and competition.
32. In the field of sports, the utilization of machine learning technology has become a key strategy for data exploration and extraction, leveraging its interdisciplinary nature to integrate the advantages of sports science, computer science and statistics. This field not only demonstrates multi-layered and multi-domain characteristics but also highlights the importance of data analysis in specific training and competition scenarios, reflecting the overall trend toward precise sports training and management⁵². Big data analysis in the field of physical activity using AI techniques is justified as one of the important and effective tools to improve the quality of health and sports performance of professional athletes. As a fundamental application of artificial intelligence, machine learning, with its integration of various interdisciplinary domains and excellent performance, has emerged as a prominent topic in sports training method research.
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    The GEQ compiled by Carron (2010) and translated by Ma Hongyu³⁶ was adopted. This questionnaire represents a specialized tool for assessing cohesion in sports, which has good reliability and validity in Chinese use. The questionnaire comprises 15 items, encompassing the dimensions of Attraction to Group Task (ATG-T), Attraction to Group Social (ATG-S), Group Interaction Consistency (GI-T), and Group Social Interaction Consistency (GI-S). The Cronbach ɑ coefficient of test–retest reliability is 0.869, which meets the needs of this study.
34. Faculty of Information Science and Technology, Zhongkai University of Agriculture and Engineering, Guangzhou, 510225, ChinaDownload citation
35. In order to determine the best technology, we use the Root Mean Square Error (RMSE), Mean Square Error (MSE), Mean Absolute Error (MAE) and coefficient of determination (({R}^{2})) as evaluation indicators⁴¹. In the following formula, ({X}_i) represents the predicted value of the (i) th element, while ({Y}_i) denotes the actual values of the (i) th element. The regression method forecasts the ({Y}_i) value corresponding to the ({X}_i) value in a real dataset. The average of the true values⁴⁴. Its formula is expressed in Eq. (2).Zhikang Lin
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37. Given that athletes might be affected by various circumstances, including physical exhaustion and meteorological conditions. These factors may lead to noise or fluctuations in the statistics. By calculating an average of every 8 data points, the impact of short-term variations can be reduced effectively and unwanted noise is minimized. This facilitates the observation of long-term trends and alterations. In some cases, athlete data may show higher densities, complicating analysis. High frequency data can be reduced to lower frequencies to make the analysis process more efficient⁴¹. Data preprocessing is an essential step in tailoring the dataset to meet the unique requirements of various machine learning algorithms. It plays a pivotal role in assessing the predictive capabilities of different models.Our goal is to determine a function, (Fleft(xright)), which minimizes a specific loss function (L(y,Fleft(xright))). GBRT approaches the optimal solution (widehat{F}(x)) step by step by weighted combination of these simple regression trees (h(x_{t})). In GBRT, each basic regression tree (h(x_{t})) is constructed based on the input variable X and the negative gradient of the loss function. Together, these trees allow the model to capture the complex nonlinear relationships in the data more accurately. The model is formulated as follows⁴⁹ Eqs. (9) and (10).
38. Machine learning-based predictive models do show great potential in providing personalized strategies for athlete training and enhancing athlete engagement. However, this predictive model faces a number of challenges. For example, the PSO algorithm may encounter the problem of local optimal solutions when applied, the choice of algorithm parameters has a significant impact on the final results, and also its search efficiency may decrease when dealing with high-dimensional feature spaces. At the same time, the PSO-SVR model’s dependence on self-reported data may lead to biased predictions, as these data are often influenced by the subjective feelings of individual athletes. In addition, data privacy and security issues are critical when deploying models. We need to take measures such as data desensitization and encryption to protect the privacy of research subjects and to improve model transparency and interpretability.Reprints and permissions
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40. As for the combination of PSO and SVR, the following is an explanation of the optimal configuration of the hyperparameters: the regularization parameter C was set to 6.66, a choice that highlights the fact that we struck a good balance between model complexity and training error. This aims to prevent model overfitting while ensuring the model’s ability to fit the training data efficiently; the coefficient of the kernel function is 0.0436, and this low gamma value implies that the model prefers to use smoothed decision boundaries, which helps to capture the overall structural characteristics of the data; and the kernel type is chosen to be “rbf”, the radial basis function kernel, which specializes in revealing nonlinear relationships in the data, which is essential for enhancing the predictive power of the model. By carefully optimizing these key hyperparameters, the prediction accuracy of the SVR model is significantly improved. This makes it stand out in the comparison of many optimization algorithms and show excellent performance.Regarding the predictive accuracy of various models, the PSO-SVR model achieved the highest accuracy at 92.62%. The variables chosen to represent this study are cohesion, passion and mental toughness, which are significant potential factors influencing athlete engagement, although other factors may also affect their engagement. The PSO-SVR model excels in handling cohesion, passion and mental toughness due to its adaptability to data features and patterns, strong ability to handle high-dimensional features, and effectiveness in managing small samples and category imbalances.
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    The authors declare no competing interests.
42. Different machine learning algorithms predict the outcome of the RMSE.The AEQ was compiled by Lonsdale et al.³⁹(2007) and revised by Ye Lv’s translation²⁹. The questionnaire comprises 16 items, encompassing the four dimensions of self-confidence, dedication, vigor and enthusiasm. A higher score indicates a greater level of engagement from the athlete. The Cronbach’s alpha coefficient for this scale is 0.959, which meets the research requirements.
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45. Overall results indicated that the PSO-SVR algorithm performed the best in terms of accuracy, and RMSE and MSE scores are relatively low (see Table 3). It can be concluded that this model not only has high predictive accuracy but also exhibits low prediction errors. Compared to PSO-SVR, the LR algorithm has lower accuracy. However, LR outperforms PSO-SVR in terms of MAE, possibly because the errors in LR are symmetrically distributed, whereas the increased complexity of PSO-SVR may result in asymmetric error distribution. GA explores the solution space in depth through crossover and mutation so that better Mae metrics can be found. The stochastic nature of RSG can be an advantage in complex parameter spaces, especially when optimizing MSE.Different machine learning algorithms predict the outcome of the MSE.
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    Machine learning, a powerful interdisciplinary technology that incorporates multiple disciplines, has the remarkable capability to learn autonomously from experience.It constructs predictive models using “learning algorithms” to make accurate, personalized predictions for new scenarios. As a core component of artificial intelligence, machine learning plays a vital role by extracting valuable insights from diverse and complex data sets. Its applications extend to numerous fields, including medicine¹, bioinformatics² and automation technology³.
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51. College of Physical Education and Health Sciences, Zhejiang Normal University, Jinhua, 321004, ChinaIn this study, RFECV was employed to identify the optimal number of features for the model. The input features considered included mental toughness, cohesion, passion, sports satisfaction and psychological collectivism.
52. To assess the predictive performance of each model, four types of metrics were used: RMSE MSE, MAE and ({R}^{2}).

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