# The Interaction of Fitness and Fatigue on Performance in Football
## 3 Data Analysis  
### Definition of Fitness, Fatigue Indices and Performance Score  
The fitness (FI) and fatigue (FA) indices were calculated using a machine learning (ML) approach, as detailed in previous studies [10, 11]. The FI was determined by comparingthe real HR (expressed as percentage of individual HRmax) recorded during training drills (specifically, small-sided games) to the HR predicted by an ML model (Random Forest Regression model), developed following the methodology described by [PERSON] et al. (2010). To evaluate the HR responses, game simulations (games performed with two regular goals and goalkeepers) and possession games (possession drills performed without the goals) were considered. A positive FI (FI > 0) indicated that the actual HR was lower than the predicted HR, suggesting that the player demonstrated a good fitness status by minimizing HR responses relative to the external load. Conversely, a negative FI indicated a poor fitness status, where the player exhibited a higher HR response than predicted. The concurrent validity of the index was previously evaluated, demonstrating a large correlation (r = 0.7) with a submaximal fitness run test ([PERSON] et al., 2010). Both the actual and predicted HR values were expressed as a percentage of individual HRmax. The FA index was computed as the difference between the predicted PL values, generated by the ML model (Random Forest Regression model), and the actual PL values recorded throughout the training session. Following the procedures detailed by [PERSON] et al. (2010), we posited that a higher predicted PL compared to the real PL indicated the player's efficiency in locomotor activity, minimizing body load in response to external demands. In contrast, a higher actual PL suggested decreased locomotor efficiency, indicative of a potential state of fatigue ([PERSON] et al., 2010). The FA index demonstrated sensitivity to the period of the season, day of the week, and cumulative load from the prior week, showing a significant decrease following increased weekly load ([PERSON] et al., 2010). In addition, the FA index exhibited different behaviors according to different weekly periodization strategies ([PERSON] et al., 2016). To account for individual variability, both FI and FA indices were expressed in relation to the individual's absolute average and normal variation throughout the entire seasons using a z-score transformation (z-FI, z-FA). The z-FI was assessed weekly on match-day minus 3 (MD-3), typically used for metabolic training involving small-sided games, making it ideal for evaluating specific player fitness. The z-FA was assessed on MD-1, after a recovery day (MD-2), to reflect the players' readiness before the match. In alignment with the fitness-fatigue model ([PERSON] et al., 2016), players' performance is determined by the summation of positive (fitness) and negative (fatigue) responses to the training stimulus ([PERSON] et al., 2010). Using the summation, the signs of the numbers involved are preserved. This means that no changes are made to the signs of the values during the operation. Each number contributes to the total sum based on its original sign--positive or negative. Thus, an individual performance score (\(PS_{individual}\)) was calculated using the following formula:  
\[PS_{individual}=z\text{-}FI+z\text{-}FA\]  
Based on the weekly \(PS_{individual}\), players were categorized into three performance conditions:  
* Poor performance condition: \(PS\leq-1\). * Intermediate performance condition: \(PS\) > \(-1\) and \(PS\) < 1. * Optimal performance condition: \(PS\geq 1\). The \(PS_{individual}\) was calculated individually each week and at the team level to quantify the overall performance of players participating in the match (\(PS_{\text{team}}\)). The \(PS_{\text{team}}\) was computed as the average of the different \(PS_{individual}\) scores. To ensure a robust team-level analysis, only matches that included complete FI and FA training datasets from at least five players were considered.