# Statistical flaws of the fitness-fatigue sports performance prediction model

Alexandre Marchal, Othmène Benazieb, Yisakor Weldegebriel, Thibaut Méline, Frank Imbach

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Alexandre Marchal, Othmène Benazieb, Yisakor Weldegebriel, Thibaut Méline, Frank Imbach. Statistical flaws of the fitness-fatigue sports performance prediction model. Scientific Reports, 2025, 15 (1), pp.3706. 10.1038/s41598-025-88153-7. hal-04999009

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OPEN **Statistical flaws of the fitness-fatigue sports performance prediction model**

Alexandre Marchal<fnref n="1" />, Othmène Benazieb<fnref n="1" />, Yisakor Weldegebriel<fnref n="1" />, Thibaut Méline² & Frank Imbach<fnref n="1" />,³✉

Optimizing athletic training programs with the support of predictive models is an active research topic, fuelled by a consistent data collection. The Fitness-Fatigue Model (FFM) is a pioneer for modelling responses to training on performance based on training load exclusively. It has been subject to several extensions and its methodology has been questioned. In this article, we leveraged a Bayesian framework involving biologically meaningful priors to diagnose the fit and predictive ability of the FFM. We used cross-validation to draw a clear distinction between goodness-of-fit and predictive ability. The FFM showed major statistical flaws. On the one hand, the model was ill-conditioned, and we illustrated the poor identifiability of fitness and fatigue parameters using Markov chains in the Bayesian framework. On the other hand, the model exhibited an overfitting pattern, as adding the fatigue-related parameters did not significantly improve the model's predictive ability (p-value > 0.40). We confirmed these results with 2 independent datasets. Both results question the relevance of the fatigue part of the model formulation, hence the biological relevance of the fatigue component of the FFM. Modelling sport performance through biologically meaningful and interpretable models remains a statistical challenge.

**Keywords** Sport science, Bayesian statistics, Cross-Validation, Ill-conditioning, Overfitting, Fitness-Fatigue Model

Modelling the effects of training on performance has been extensively studied over the last five decades. A historical approach has been driving research with the statement of a dose response relationship between exercise and performance, through *fitness* and *fatigue* states variables<fnref n="1" />⁻⁴. The so-called Fitness-Fatigue Model (FFM) is deterministic and based on a schematisation of the biological processes underpinning physical exercise⁵. In its simple form, it assumes that a training dose induces two antagonistic responses, the fitness representing incremental, long-lasting positive adaptations of the body and its physiological processes; and the fatigue standing as intense, short-lasting inhibition of the performance potential. From a mathematical perspective, fitness and fatigue states variables are derived from first order differential equations in which the training dose is expressed as a discrete function and convolved with an exponential transfer function. Then, the predicted performance is given by the difference between states variables that serves as a proxy of global physiological adaptations to training doses, responsible for athletic performance⁵,⁶. Considering the explicit and algebraic formulation of the model, FFMs provide the explicit solution that would maximize the performance without performing simulations of training doses⁴.

Due to their interest for identifying an optimal training program in which training doses, scheduling and recovery lead the performance potential of each athlete, several extensions of the common FFM relate a collection of FFMs henceforth<fnref n="1" />,²,⁵⁻¹¹. Among these modifications, Busso⁸ added nonlinearity to the model by weighting the fatigue states from the accumulation of training stimuli. Two other alternatives relied on delayed effects of training, using delay and serial functions supported by physiological evidence⁶,¹¹.

Despite a meaningful field of application around sports performance, FFMs have been criticized due to their lack of accuracy and their unreliability for predicting athletic performances¹². A few authors questioned their foundation, highlighting methodological and mathematical flaws amongst ill-conditioning and failure to capture meaningful information regardless of the model complexity⁷,¹²⁻¹⁴. Estimating one or more states variables (e.g. fitness, fatigue) from a single input, FFMs suffer from explanation and predictive capability due to a lack of information. Since the model stems from prior knowledge (i.e. the physiological kinetics underpinning FFM

<footnote n="1">¹Seenovate, Paris 75009, France. ²Fédération Française des Sports de Glace, Paris, France. ³DMeM, INRAe, Univ Montpellier, Montpellier 34000, France. ✉email: frank.imbach@umontpellier.fr</footnote>
transfer functions), estimating model parameters in a Bayesian framework could reinforce the framework and benefit from a priori expertise.

Bayesian modelling is a statistical paradigm that allows for combining information from observed data (the likelihood) with information from background knowledge (the prior). It also enables solving models with Monte-Carlo Markov Chains (MCMC), an approach that unveils the whole distribution of the resulting, concordant posterior estimations. On one hand, the inclusion of a prior knowledge makes a special sense in the context of a deterministic model, from which the components have a biological meaning. On the other hand, unveiling the posterior distribution is a powerful tool for the diagnosis of statistical issues such as ill-conditioning or an overall lack of information¹⁵. Tackling FFMs from a Bayesian perspective has not been done until recently¹⁶. Peng et al. identified a gain in the plausibility of the parameter estimates as compared to the frequentist counterpart (i.e. non-Bayesian, based on pure likelihood). However, the predictive ability of the FFM fitted in a Bayesian framework was not investigated by these authors¹⁶.

In this paper, we investigated the relevance of a FFM applied to sport performance prediction. The task was decomposed using model of growing complexity (i.e. from one to two states variables). We investigated the estimability of the non-linear parameters (the time constants) of the model. A bayesian framework was employed to benefit from prior information on the time course of states variables. In the following, we will address the predictive ability of the models in the perspective to support the design of athletic training programs.

# Material & methods

## Data description

For the sake of robustness of analyses, we modelled the performance of world-class short-track speed skaters over two distinct datasets, collected in 2019 and 2016 respectively. For both datasets, participants were fully informed about data collection and written consent was obtained from them. Also, the study was performed in agreement with the standards set by the declaration of Helsinki (2013) involving human subjects. The protocol was reviewed and approved by the local research Ethics Committee (EuroMov, University of Montpellier, France). The present retrospective study relied on the collected data without causing any changes in the training programming of athletes.

### The 2019 dataset

Nine (9) national elite short-track speed skaters were recruited in a previous study¹². Two (2) athletes were injured during the training range and were discarded from the study. Details about the population of interest (i.e. 7 athletes) are provided by Imbach, Perrey, et al.¹².

The data collection spread over 10 weeks, from June to September. The period of study was interrupted by a two-week summer break, during which athletes fully rested. Skaters sustained a mean weekly training volume of 16.6±2.5 h during this period. A total of 375 training and 248 performance observations were collected, distributed as 53.6±1.6 training observations and 35.4±2.2 performances per athlete.

The performance data were collected each week and consisted in the time to perform a 166.68 m race after a standardised warm-up; hence a lower performance value was sought. Timing gate systems (Brower timing system, USA) were used to record valid individual time trial performance¹⁷. On-ice and off-ice training loads were used as the explanatory variable as described in the Appendix 1 from Imbach, Perrey, et al.¹².

### The 2016 dataset

Fifteen (15) national elite Short-track speed skaters voluntary participated to a previous study¹⁸. One of the skaters was injured and removed from the study, leaving 14 athletes in. Athletes had the same coach for both datasets. Out of these 14 athletes, 2 were also represented in the 2019 data set.

A total of 948 training and 755 performance observations were collected, distributed as 67.7±5.6 training observations and 53.9±5.4 performances per athlete. The period of study included 13 weeks from July to October. During this period, the skaters sustained a mean weekly training volume of 25±5 h.

Following a 15-minutes standardised warm-up, participants performed weekly flying starts for a one-lap maximal run. Four photocells were settled for recording skating times. The dependent variable further modelled was the individual maximal acceleration (from speed derivation); hence a higher performance value was sought. On-ice and off-ice training sessions were used as the explanatory variables, as described by Méline et al.¹⁸ and consistently with the 2019 dataset.

## Models

We considered the Banister's fitness-fatigue model (FFM) from a stochastic perspective, in which the predicted performance of athlete s at time t: $y_s(t)$, was considered as the expectation from a gaussian distribution:

$$y_s(t) \sim N(\hat{y}_s(t), \sigma_s^2) \qquad (1)$$

of variance $\sigma_s^2$ and expectation $\hat{y}_s(t)$ a function of the daily training doses or workloads $w_s(i)^1$:

$$\hat{y}_{FFM, s}(t) = a_{0,s} + k_{G,s} \sum_{i=1}^{t-1} w_s(i) e^{\frac{-(t-i)}{\tau_{G,s}}} - k_{H,s} \sum_{i=1}^{t-1} w_s(i) e^{\frac{-(t-i)}{\tau_{H,s}}}$$

In the above equation, $a_0$ denotes the intercept, $k_G$ and $k_H$ represent the linear coefficients for fitness and fatigue, respectively, and $\tau_G$ and $\tau_H$ the time exponents for fitness and fatigue, respectively. All these parameters are written with an s subscript that denotes the athlete identity, as the parameters are estimated at the individual level.
The FFM being relatively complex, we decomposed it into a simpler, potentially more parsimonious version. This simpler version only encompassed a fitness component:

$$ \hat{y}_{FOM,s}(t) = a_{0,s} + k_{G,s} \sum_{i=1}^{t-1} w_s(i) e^{\frac{-(t-i)}{\tau_{G,s}}} $$

This is referred to as the fitness-only model (FOM) in this study.

## Model fitting

### Prior knowledge on the time exponents

One of the main interests of the FFM is its biological interpretability. Exponential decays can be interpreted from their half-life (i.e. the amount of time needed for half the effect of an input to vanish). The time exponent value $\tau$ leading to a half-life $h$ is:

$$ e^{\frac{-x}{\tau}} = 0.5 \iff \tau = \frac{h}{\ln(2)} $$

In this study, we investigated the option to provide information to $\tau_G$ and $\tau_H$ either by fixing them in a frequentist framework or using bayesian priors. Priors were based on background, prior knowledge that an athlete may recover half the fatigue of a training session in about the length of a weekend (i.e. 2 days) and may lose half of the fitness from a training session in about a month (i.e. 30 days). Therefore, when informing the time exponents, we provided the target value of $\tau_G = 43.28$ for fitness and $\tau_H = 2.89$ for fatigue.

### Frequentist model fitting

The frequentist model fitting refers to the estimation of $\theta$ by minimizing the standard deviation $\sigma$, which is equivalent to maximizing the likelihood under Gaussian dispersion assumption (Eq. 1). The minimization was performed using the Powell algorithm from the scipy library¹⁹.

The frequentist fitting process consisted in 10 independent runs from which we retained runs that converged and minimized the residual standard deviation. Since the model was ill-conditioned, as we explored in depth throughout this study, the parameters estimation became unstable even under constraints optimization (i.e. boundaries) for convergence (Table 1).

Alternatively, we fixed $\tau_G$ and $\tau_H$ (see Sect. 2.3.1) before estimating the other parameters, hence reducing the system to a linear model.

### Bayesian model fitting

Models were also fitted in a bayesian framework. We used proper, mildly informative priors for $a_0 \sim N(0, 5^2)$; $k_G = -|X|$ (for the 2019 dataset) (or $k_G = +|X|$ for the 2016 dataset), with $X \sim N(0, 5^2)$; idem for $k_H$; and a flatter prior for $\sigma = |X|$ with $X \sim N(0, 25^2)$. Half-normal distribution is part of the half-Student distribution family, which are generally advised for variance parameters priors²⁰.

We compared two strategies for the time exponent estimations. On the one hand, we used non-informative priors $\tau_G \sim \Gamma(a=1, b=0.5)$ which is equivalent to a $\chi^2$ distribution with 2 degrees of freedom; same for $\tau_H$. On the other hand, we fitted the models with informative priors for the time exponents: $\tau_G \sim \Gamma(a=2*43.28, b=2)$ and $\tau_H \sim \Gamma(a=2*2.89, b=2)$. The average of a gamma distribution defined by its shape (a) and rate (b) is a/b, hence these priors are consistent with the background knowledge target as developed in 2.3.1. We emphasize that these targets have been designed without information from the data. The rate parameter $b=2$ was set high but not extreme.

Bayesian fitting was performed using Markov Chain Monte Carlo (MCMC), with 10 independent Markov chains for each model fitting. Independent Markov chains were initiated from random initial values uniformly sampled within the intervals described in Sect. 2.3.2. Each chain had $4 \times 10^4$ iterations warm-up, $1 \times 10^4$ iterations sampling, and a 10-iteration thinning so they ended up $10^3$ iterations long. Computation were performed using the Python implementation of Stan²¹.

## Cross-validation

The predictive ability of the models was assessed by the mean of cross-validation. Since we deal with time series, the data were time-ordered and split accordingly in subsets in respect of hidden-value block cross-validation with an incremental window¹²,²². We used minimal values of 35 days and 15 days for training and test sets, respectively. Due to potential dependencies between training and test data, a 5-day gap between each training

<table><caption>**Table 1.** Boundaries used to constraint the frequentist parameter estimation.</caption><thead><tr><td>Parameter</td><td>Min</td><td>Max</td></tr></thead><tbody><tr><td>a<sub>0</sub></td><td>-10<sup>4</sup></td><td>10<sup>4</sup></td></tr><tr><td>k<sub>G</sub> and k<sub>H</sub> (2019 dataset)</td><td>-1</td><td>0</td></tr><tr><td>k<sub>G</sub> and k<sub>H</sub> (2016 dataset)</td><td>0</td><td>1</td></tr><tr><td>τ<sub>G</sub> and τ<sub>H</sub></td><td>0</td><td>100</td></tr></tbody></table>
and test subsets was considered to ensure unbiased model predictions. This parameterization yielded 2 folds for the 2019 dataset and 3 folds for the 2016 dataset as described in Table 2.

We used root mean square error (RMSE) between predicted and observed performance to quantify the predictive ability of the models. RMSE of the training data, i.e. the residual deviation of the model $\sigma$, was also employed to measure the goodness of fit of a model to its training data.

# Results

## Estimation of the fitness time exponents

*Informativeness for the fitness time exponent*

The Fitness-Only Model (FOM) was fitted within:

* A frequentist framework where all parameters are estimated;

* A frequentist framework with the fitness time exponent $\tau_G$ being fixed to 43.28 (see Sect. 2.3.1 for details);

* A Bayesian framework using a non-informative prior for $\tau_G$;

* A Bayesian framework with an informative prior for $\tau_G$.

The prior had a strong influence on the posterior for $\tau_G$, suggesting that it carried more information than the data itself for the estimation of this parameter (see Fig. 1a for a representative athlete, Appendix B for all other athletes from the 2019 dataset). Accordingly, following an informative prior, the posterior distribution embraced almost perfectly the prior distribution (Fig. 1b, Appendix B).

Fixing $\tau_G$ or providing its information in the prior yielded to close parameter estimates, hence close predictive performances (Fig. 2). On the other hand, using a Bayesian framework with non-informative prior for $\tau_G$ or estimating $\tau_G$ in a frequentist framework resulted in poorer predictive performances (Fig. 2). The differences between approaches using information (with Bayesian informative priors or fixed parameters) versus estimating $\tau_G$ only from the data (with Bayesian genuine priors or frequentist estimation of the parameters) were close to significance, as seen in Table 3.

*Grid search for an optimal fitness time exponent*

In the previous section, we observed that providing information for $\tau_G$ was beneficial to the predictive ability of the model. This questions the appropriate value to which $\tau_G$ should be informed. We addressed this question by the mean of a grid search. The grid ranged from one day to 100 days with a step of 1 day for the fitness half-life. The time exponent parameter $\tau_G$ was computed according to each fitness half-life, fixed to this value, and all other parameters ($a_0, k_G, \sigma^2$) were estimated. The fitness (Fig. 3a) and the predictive ability (Fig. 3b) of the model were then assessed.

No optimum was found for the likelihood in the investigated interval for 6 out of the 7 athletes. Besides, the higher half-life for $\tau_G$ led to a higher likelihood for these 6 athletes. Only athlete 5 reported an optimum, with a maximum of likelihood given for a fitness half-life = 43 ($\Leftrightarrow \tau_G = 62.04$, see Fig. 3a). The performance in cross-validation were even more idiosyncratic, and no clear trend could be drawn. For 4 combinations of athlete and fold, an optimum appeared clearly close to the half-life suggested for $\tau_G$ or used in the Bayesian prior (fitness half-life = 30 $\Leftrightarrow \tau_G = 43.28$). An optimum also emerged for athlete 2 - fold 2, but with a lower value (fitness half-life = 16 $\Leftrightarrow \tau_G = 23.08$). For the 9 other combinations of athletes and folds (i.e. 64.3% of the combinations), no optimal value was found over the studied interval (see Fig. 3b).

## Estimation of the fatigue time exponents

*Informativeness for the fatigue time exponent*

The identifiability of $\tau_G$ and $\tau_H$ was investigated in a Bayesian framework with informative priors (see Fig. 4 for a representative athlete and Appendix B for all athletes involved in the 2019 dataset).

One to four Markov chains converged out of 10 for any athlete, and convergence always occurred within the peak density of the prior for both $\tau_G$ and $\tau_H$ (Fig. 4a, b, c and d, respectively; see Appendix B for all athletes). The convergence of $\tau_G$ and $\tau_H$ was always associated with the convergence for all other parameters ($a_0, k_G, k_H, \sigma$).

On the other hand, the chains that did not converge exhibited a strong autocorrelation within chains for all parameters and inconsistency between chains for most parameters. In the unconverging chains, $\tau_G$ and $\tau_H$ displayed a mirror behaviour: they took a close value for a given chain at a given iteration (Fig. 4a and c, and Appendix C). The mirror posture was also observed for $k_G$ and $k_H$ that were almost identical for a given non-converged chain at a given iteration (Fig. 5, and Appendix B). Note that on Fig. 5 (and Appendix C), the sample

<table><caption>**Table 2.** Cross-validation folds description for each of the 2 datasets involved in the study.</caption><thead><tr><th rowspan="2">Dataset</th><th rowspan="2"></th><th colspan="2">2019 dataset</th><th colspan="3">2016 dataset</th></tr><tr><th>Fold 1</th><th>Fold 2</th><th>Fold 1</th><th>Fold 2</th><th>Fold 3</th></tr></thead><tbody><tr><td rowspan="3">Train set</td><td>Number of days</td><td>46</td><td>61</td><td>42</td><td>57</td><td>72</td></tr><tr><td>Number of performance observations</td><td>≥15</td><td>≥25</td><td>≥17</td><td>≥23</td><td>≥33</td></tr><tr><td>Number of training sessions</td><td>≥21</td><td>≥36</td><td>≥22</td><td>≥31</td><td>≥42</td></tr><tr><td rowspan="2">Test sets</td><td>Number of performance observations</td><td>6–11</td><td></td><td>5–12</td><td></td><td></td></tr><tr><td>Number of training sessions</td><td>9–12</td><td></td><td>7–13</td><td></td><td></td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Markov chains of τ_G estimated from the model including fitness only for athlete 1, using informative prior for τ_G (**a**, **b**) or a flat prior for τ_G (**c**, **d**). Plots (**a**) and (**c**) show the raw Markov chains, each color represents one of 10 chains, the x-axis is the iteration number, and the y-axis are the τ_G values. Plot (**b**) (respectively (**d**) represent the compilation of the 10 chains represented in (**a**) (respectively (**c**) in a blue histogram whose legend is on the left; and the prior is represented in yellow as a density function whose legend is on the right. The red dotted lines in (**b**, **d**) is the average posterior; the plain lines are a 95% confidence interval.</figcaption></figure>

dispersion for $k_G$ and $k_H$ for the converged chain(s) is too small to be noticeable at the given scale, and only one chain converged for this athlete.

For two athletes (athletes 3 and 5, see Appendix B), $k_H$ exhibited instability that was expressed in the converged chains: it wandered to extreme values before coming back to the maximal density area of the chain. These expeditions caused local autocorrelation for $k_H$ and were locally associated with close-to-0 samples for $τ_H$.

For all athlete, bunching the 10 chains (i.e. chains that converged in the peak density of the prior and chains that did not converge at all) led to poor model properties (Appendix B). These results were even worse when using flat priors or a frequentist framework for estimating $τ_G$ and $τ_H$ (results not shown). Since non-convergence of Markov chains discredit the models, we fixed $τ_G$ and $τ_H$ for the rest of this investigation.

### Grid search for an optimal fatigue time exponent

As proceeded with $τ_G$, a grid search for $τ_H$ was computed under constraints. First, $τ_G$ was fixed to 43.28 (↔ fitness half-life = 30 days). Second, the upper bound of the grid search for $τ_H$ was the value to which $τ_G$ had been fixed. Therefore, the fatigue half-life grid ranged from 1 day to 29 days with a step of 1 day. Since RMSE reached extremes values when the half-life approaches the limit, we only presented the grid search from day = 1 to day = 22, for figure readability reasons (Fig. 6).

Maxima of likelihood emerged for $τ_H$ in the investigated range and for all athletes. These maxima were specific to each athlete and comprised between fatigue half-life = 2 days (↔ $τ_H$ = 2.89) to fatigue half-life = 6 days (↔ $τ_H$ = 8.66) (Fig. 6a). These maxima of likelihood were not associated with better predictive abilities, excepted for fold 1 of athlete 8 and fold 1 of athlete 4 (Fig. 6b).
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** RMSE in cross-validation depending on the way the fitness-only model was fitted. The boxplot corresponds to the 7 athletes for each fold of the cross-validation. “Frequentist”: all parameters, including $τ_G$, were directly estimated by minimization of the RMSE in the training set. “Fixed”: is identical to “Frequentist” with the exception that $τ_G$ was fixed to 43.28 (corresponding to a half-life of 30 days). “Flat” and “Informative” correspond to bayesian fitting with a gamma distribution for $τ_G$. “Flat”: the prior for $τ_G$ was $Γ(a = 1, b = 0.5)$. “Informative”: the prior for $τ_G$ was $Γ(a = 2 \times 43.28, b = 2)$.</figcaption></figure>

<table><caption>**Table 3.** Uncorrected p-value of the difference between the RMSE in cross-validation depending on the way the fitness-only model was fitted. Repetitions correspond to the 7 athletes and 2 folds of the cross-validation. The p-values were obtained with a two-sided student mean comparison. Significance threshold after Bonferroni correction: $α = 5%/6 = 0.0083$.</caption><thead><tr><th></th><th>Frequentist</th><th>Fixed</th><th>Flat</th></tr></thead><tbody><tr><td>Fixed</td><td>0.0186</td><td></td><td></td></tr><tr><td>Flat</td><td>0.3224</td><td>0.0131</td><td></td></tr><tr><td>Informative</td><td>0.0118</td><td>0.0267</td><td>0.0084</td></tr></tbody></table>

### Performance of the fitness-fatigue model

Adding the fatigue component provided to the model the ability to capture the short-lasting variations, notably the summer break effect over the modelled performance (Appendix A). Indeed, during this 2-week break, the FOM predicted a loss of performance whereas the FFM predicted a gain of performance, which turned out to be more accurate. The benefit of including a fatigue component in the model led to a reduction of 0.017 in the training set RMSE (Cohen's $d = 0.585$, one-sided Student mean comparison of paired data: p-value < 0.001).

Unfortunately, this better goodness of fit did not translate into a better predictive ability. The predictive ability is slightly, though not significantly, decreased when providing a fatigue component with an addition of 0.001 to the RMSE (Cohen's $d = 0.022$, p-value = 0.57).

### Confirmation on a new independent dataset

#### Informativeness for the fatigue time exponent

Appendix C displays FOM and FFM outputs from the independent 2016 dataset. Note that the 2016 and 2019 datasets share 2 common athletes, which make them somewhat related. However, they are considered independent due to the time gap between them and, more generally, because the performance criteria differ.

Both models were fitted in a Bayesian framework with informative priors for $τ_G$ and $τ_H$. Among the 14 athletes
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Quality of the fitness model fitted with fixed values for the time exponent $τ_G$. The time exponent is computed from the resulting half-life for fitness, provided as the x-axis and varying from 1 to 365. Plot (a) shows the quality of fit in the training processus, assessed using the likelihood. Plot (b) shows the predictive ability assessed as the RMSE for each fold (fold 1: plain line and fold 2: hashed line). Each athlete is represented with a different color detailed in the legend. Likelihood and RMSE in prediction are returned as percentage of the difference between their maximum value and their minimum value on the y-axis. The vertical black line at x=30 days correspond to the value that was used to fix $τ_G$ throughout this paper or to inform it in the bayesian prior.</figcaption></figure>

included in the 2016 dataset, 6 did not show any convergent Markov chains for the FFM (athletes F5, H2, H5, H6, H11 and H13, see Appendix C). The other height athletes had between 1 and 3 converged chains out of 10. The conclusions are identical to the ones drawn from the 2019 data (Figs. 5 and 6, Appendix B):

* The time exponents ($τ_G$ and $τ_H$) only converged in the peak density of the prior;

* $k_H$ was very instable in converged chains, frequently wandering at extreme values;

* Non-convergent chains for $τ_G$ and $τ_H$ did not converge for the other parameters neither;

* Non-convergent chains exhibited a mirroring pattern with nearly identical $τ_G$ and $τ_H$, and $k_G$ and $k_H$ for a given chain at a given iteration.

The latter aspect was associated with strong correlations between the fitness and fatigue states which resulted in a poor quality of fit to the training data. This situation was particularly problematic for the 5 athletes for whom no chain converged at all, as in this situation, the fitness and fatigue components were exactly equal at each time point.

On the other hand, FOM (Appendix C) showed convergence for all 10 chains and for all athletes. Convergence happened for $τ_G$ within its densest prior segment, or near to it. The greatest difference between the prior and the posterior was observed for athlete H10 (see Appendix C), suggesting the expression of a slight information from the data likelihood for this athlete. All other parameters ($a_0$, $k_G$, $σ$) converged for all athletes and Markov chains.

### Performance of the fitness-fatigue model

The gain in performance when adding a fatigue component using the 2016 dataset yielded to the same results than using the 2019 dataset. Indeed, the goodness of fit to the training data was slightly but significantly increased, with a reduction of $2.8 \times 10^{-4}$ in RMSE (Cohen's $d = 0.056$, p-value < 0.01, one-sided Student mean comparison of paired data). Again, this gain in goodness of fit to the training data did not translate into a significant gain in predictive ability. The RMSE in cross-validation was reduced by $8.8 \times 10^{-5}$, this difference was not-significant (Cohen's $d = 0.006$, p-value = 0.41).

## Discussion

In this study, we investigated the estimability of the Fitness-Fatigue Model (FFM) parameters for sports performance modelling. We concluded that fitness and fatigue parameters are antagonistic in the estimation process, therefore non-identifiable, and that the fatigue component does not improve the predictive ability of the model. The results were supported by an analysis conducted over two independent datasets.

Using a Fitness-Only Model (FOM), we showed that the optimal value for the time exponent was not estimable from the likelihood. The Fig. 1 (c-d) exemplified this pattern: using a flat prior for the time exponent in the FOM led to estimates with no physiological sense at all. Adding biologically meaningful information to the fitness dynamic, by fixing its time exponent or by means of a bayesian prior, led to the model with the best
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Markov chains of $\tau_G$ (**a**, **b**) and $\tau_H$ (**c**, **d**) estimated from the model with fitness and fatigue for athlete 1, using informative priors for $\tau_G \sim \Gamma(a=2 \times 43.28, b=2)$ and $\tau_H \sim \Gamma(a=2 \times 2.89, b=2)$. Plots **a** and **c** show the raw Markov chains, each color represents one of 10 chains, the x-axis is the iteration number and the y-axis are the $\tau_G$ (**a**) and $\tau_H$ (**c**) values. Plot (**b**) (respectively (**d**) ) represent the compilation of the 10 chains represented in (**a**) (respectively (**c**) ) in a blue histogram whose legend is on the left; and the prior is represented in yellow as a density function whose legend is on the right.</figcaption></figure>

predictive ability in cross-validation. Adding a fatigue component implies two challenges: the estimation of the fatigue time exponent itself and dealing with an antagonism between the fitness and the fatigue components during the estimation process. The diagnosis of the Markov chains used to fit the FFM in a Bayesian framework illustrated how this antagonism prevented the model from converging. Noteworthy, maxima of likelihood were easily identifiable for the fatigue time exponent when the fitness time exponent had been fixed, but never when both parameters were simultaneously estimated. Consequently, augmenting the model complexity with supplementary state variables (i.e. fatigue) does not ensure enhanced model performance, as parsimony appears to prevail in time-invariant FFMs$^{23}$.

The FFM has already been extensively criticized in the literature$^{13,24,25}$. In particular, Hellard et al.$^{13}$ highlighted the fitness-fatigue model ill-conditioning by estimating the asymptotic correlation matrix of parameters. The authors showed a high correlation between $k_G$ and $k_H$ ($\text{cor}(k_G, k_H) = 0.91$) and $\tau_G$ and $\tau_H$ ($\text{cor}(\tau_G, \tau_H) = 0.99$) and concluded that the model was not reliable. We reproduced and illustrated this result from a Bayesian perspective using Markov chains. The extra step undertook with our paper, aside confirming the model ill-conditioning on two new independent datasets, was to show that adding information with a prior was not sufficient to overcome this issue.

Though the fatigue component captured an appreciable amount of variance in the training data once the fatigue time exponent had been fixed, it did not contribute to the predictive ability of the model in cross-validation. Overfitting is characterized by an increased model complexity leading to a gain in the goodness of fit that does not translate into a better predictive ability. From this definition, overfitting issues can only be tackled with a test dataset unconsidered for model training. However, the FFM and derivative models are scarcely
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Raw Markov chains of $k_G$ (a) and $k_H$ (b) estimated from the model with fitness and fatigue, for athlete 1, using informative priors for $\tau_G$ and $\tau_H$. Each color represents one of 10 chains, the x-axis is the iteration number and the y-axis are the $k_G$ (a) and $k_H$ (b) values. $k_G$ and $k_H$ both followed a flat priors $k = -|X|$ with $X \sim N(0, 5^2)$. Note that only the yellow Markov chain converged.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Quality of the fitness-fatigue model fitted with fixed values for the time exponent $\tau_H$. The time exponent is computed from the resulting half-life for fatigue, provided as the x-axis and varying from 1 to 29 (a) and 1 to 22 (b). Plot (a) shows the quality of fit in the training processus, assessed using the likelihood. Plot (b) shows the predictive ability assessed as the RMSE for each fold (fold 1: plain line and fold 2: hashed line). Each athlete is represented with a different color detailed in the legend. Likelihood and RMSE in prediction are returned as percentage of the difference between their maximum value and their minimum value on the y-axis. The vertical black line corresponds to x=2 days, i.e. the half-life we informed elsewhere in this paper.</figcaption></figure>

challenged in cross-validated studies¹²,²⁶,²⁷. For this reason, we partially agree with Peng et al.¹⁶ who conclude that a Bayesian approach overcomes the main challenges of a FFM. If adding information in the prior indeed helped the parameters estimation process in some cases, it has not been shown to make the fatigue component relevant with regards to the predictive ability, neither in our study nor anywhere in the literature as far as we are aware of. Models are meant to represent the real world, the lack of statistical relevance of a component questions its biological relevance. Fundamentally, it is not trivial that the fatigue as modelled in the FFM really mimics a biological phenomenon, since humans are driven by sophisticated physiological processes in interaction²⁸,²⁹.

Not all issues with the FFM transposed to the FOM. The Markov chains of the FOM converged when provided some information. A few combinations of athletes and folds showed an optimal value for $\tau_G$ in terms of predictive ability, that was close to our prior conception of a half-life of 30 days for a gain in fitness (Fig. 3b). This suggests that in these few situations, the fitness time component may have captured a biological phenomenon
with which our prior idea of the magnitude was consistent. The data likely misses the information for this result to be reproducible to each athlete and to each fold since we worked with relatively short time series (less than 3 months including a 2-week summer break). Nevertheless, the FOM presents strong assets: (i) it is not affected by the structural ill-conditioning due to the antagonism between fitness and fatigue components, as discussed so far; (ii) it still allows for tackling the asynchronicity between trainings and performance, which is fundamental in the sports performance applications; (iii) it allows pure prediction on the basis of a training program; (iv) the fitness value may be directly interpreted by athletes, coaches and sport professionals as an actual performance potential at a given time for a given athlete. The points (iii) and (iv) are obviously conditional on the predictive ability of the model that may vary from one situation to the other.

However, caution should be taken since only the long-lasting fitness, not the short-lasting fatigue, are captured by the FOM. For instance, the FOM fails to capture the punctual recovery effect of breaks period, as it has been illustrated in our study and due to its simple formulation, FOM cannot capture overtraining or maladaptation. Another limitation is that we did not investigate alternative ways to model the short-lasting variation, that may likely be better captured by other metrics than the training load itself. Furthermore, training loads were quantified deterministically, based on the head coach's experience with top-level athletes and tailored to session modalities (i.e. on-ice and off-ice). While such prior information can provide valuable insights for the model from his experience, it also introduces the potential for bias in the model's estimates.

Initiating the model state variables with relevant values has a potential implication in the inference²⁶. Null state variables come with the assumption of modelling performances of untrained athletes, which is usually not realistic. Consequently, we investigated simulations of previous training session (up to 2 years priors to the observed datasets) to start from non-null fitness and fatigue states. However, it did not lead to a reliable improvement of the predictive ability of the model, neither for the FOM, neither for the FFM model (data not shown).

To conclude, the FFM presents major dysfunctions that prevents its use for predictive purposes. These results are consistent with the conclusions that other authors have already reached and published. One may be tempted to use FFM to support the design of athletic training programs; these conclusions warn us against this practice. Adding prior information, for instance with a Bayesian approach, seems beneficial to the FOM. This simple model may be the starting point for further improvement, that considers other sources of information and/or an alternative transfer function from training to short-term effects on sport performance.

## Practical considerations

* The Fitness-Fatigue Model (FFM) is a widely used model for predicting sport performance. Its key advantage lies in its ability to forecast athletes' performance based solely on a training program, which can be preliminarily designed by their coach.

* This study demonstrates that the FFM has significant statistical limitations. Further research is necessary to develop a more reliable model formulation, particularly to address the challenges of performance prediction when designing training programs.

* Using a Bayesian framework, we found that some parameters could not be estimated solely from the available data. However, the model demonstrated a strong capacity to incorporate prior information. The formulation incorporating prior knowledge about fitness dynamics outperformed all others in this study.

* The fatigue component in the FFM is considered as antagonistic to the fitness component. To address this issue, alternative formulations of fatigue or new metrics with greater physiological relevance should be explored.

## Data availability

Correspondence and requests for materials should be addressed to F.I.

Received: 30 July 2024; Accepted: 24 January 2025

Published online: 29 January 2025

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## Acknowledgements

We are grateful to the Fédération Française des Sports de Glace, the Institut National du Sport, de l'Expertise et de la Performance and to Dr. Anthony MJ Sanchez and Robert Solsona (Laboratoire Européen Performance Santé Altitude, University of Perpignan Via Domitia) for collaboration and sharing data sets.

## Author contributions

Conceptualisation, A.M., F.I.; methodology and investigation, A.M., Y.W., O.B., F.I.; data curation, T.M., Y.W.; recruitment, T.M.; resource development, A.M., Y.W., O.B.; formal analysis, A.M.; writing original draft preparation, A.M., F.I.; writing-review and editing, A.M., Y.W., O.B., F.I.; supervision, F.I.; project administration, F.I. All authors have read and agreed to the published version of the manuscript.

## Declarations

### Competing interests

The authors declare no competing interests.

### Additional information

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1038/s41598-025-88153-7.

**Correspondence** and requests for materials should be addressed to F.I.

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