Fitting an FFM via NLS in practice assumes that a unique optimal solution exists and can be found by the algorithm applied. However, this idealistic scenario may not hold for two reasons: 1) the absolute minimum may not be unique; and 2) local minima, saddle points, and/or plateau features may exist that cause problems for certain algorithms. If there exist different parameter sets in the domain that share the same global minimum under standard NLS, then there is a situation where parameters aren’t uniquely identified without additional constraints or regularisation terms. However, more likely is that problems with the typical FFM fitting process will stem from the existence of local minima, saddles, or plateau features that cause the algorithm to converge to a solution not equal to the global minimum. Local optima can provoke sensitivities in the fitting process for first and second-order algorithms that are by definition local optimisers. This manifests as sensitivity to initial parameter estimates (i.e., the starting point the algorithm initialises the search from). The extent of starting point sensitivity is largely unknown in the context of FFMs for common algorithms adopted and has not been studied directly. Given this concern, research reporting a single model solution derived from ‘one shot’ minimisation of NLS via typical first and second-order algorithms is fundamentally limited by possible uncertainty as to the suitability of fitted estimates as global minimisers. Therefore, the primary aim of this study was to investigate the sensitivity of a classical first-order search algorithm to selection of initial estimates when fitting a fitness-fatigue model (FFM) via nonlinear least-squares (NLS), and to subsequently assess the existence of local optima. A secondary aim of this study was to examine the implications of any findings in relation to previous research and provide considerations for future experimentation. The aims of the study were addressed through a computer experiment (in silico) approach that adopted a deterministic assumption the FFM completely specified athlete response. Under this assumption, two FFMs (standard, and fitness-delay) were simulated under a set of hypothetical model inputs and manually selected ‘true’ parameter values (for each FFM), generating a set of synthetic performance data. The two FFMs were refitted to the synthetic performance data without noise (and under the same model inputs) by the quasi-Newton L-BFGS-B algorithm in a repetitive fashion initiated from multiple starting points in the parameter space, attempting to at each search recover the true parameter values. Estimates obtained from this process were then further transformed into prediction errors quantifying in-sample model fit across the iterations and non-true solutions. Within the standard model scenarios, 69.1-70.3% of solutions found were the true parameters. In contrast, within the fitness-delay model scenarios, 17.6-17.9% of solutions found were the true parameters. A large number of unique non-true solutions were found for both the standard model (N=275-353) and the fitness-delay model (N=383-550) in this idealistic environment. Many of the non-true extrema found by the algorithm were local minima or saddles. Strong in-sample model fit was also observed across non-true solutions for both models. Collectively, these results indicate the typical NLS approach to fitting FFMs is harder for a hill-climbing algorithm to solve than previously recognised in the literature, particularly for models of higher complexity. The findings of this study add weight to the hypothesis that there exists substantial doubt in reported estimates across prior literature where local optimisers have been used or models more complex than the standard FFM applied, particularly when optimisation procedures reported have lacked the relevant detail to indicate that these issues have been considered. Future research should consider the use of global optimisation algorithms, hybrid approaches, or different perspectives (e.g., Bayesian optimisation).

The fitness-fatigue model (FFM) has been around for more than 40 years and is one of most prominent conceptual models within exercise science. Translation from a purely conceptual form into a mathematical structure reveals there is no single model, but instead a collective of models with common properties. The greatest potential use of FFMs is to predict future performance of athletes with sufficient accuracy to assist with training program design. However, despite a long history and consistent study, there has been limited uptake in practice. This is most likely due to a lack of accessible resources explaining key concepts and processes required to fit models, and insufficient knowledge of predictive validity. The current review provides a comprehensive overview of FFMs and discusses three key aspects of fitting models in practice: 1) training load quantification; 2) criterion performance selection; and 3) parameter estimation. As the majority of athletes engage in sports where performance is complex and determined by a range of physical, psychological and technical factors, it is argued that greater focus should be placed on FFMs being used to quantify components of fitness (e.g. strength, power) targeted by an athletes training rather than competitive outcomes. Additionally, contemporary approaches to training monitoring (e.g. barbell velocity and repetitions in reserve) are recommended as tools to generate high frequency “performance” measurements to fit FFMs. Necessary further developments require collaboration between researchers and practitioners with larger data sets to establish conditions where suitable predictions to future training can be obtained.

The standard fitness-fatigue model (FFM) is known to include several limitations described by the linearity assumption, the independence assumption, and the deterministic assumption. These limitations ensure that the modelled response to chronic training does not match the complexity observed in practice. The purpose of part II of this review series was to describe previous extensions to the standard FFM to address these limitations, providing key mathematical insights and resources to both explain technical elements and enable researchers and practitioners to fit these extended models to their own data. To address the linearity assumption of the standard FFM and the associated limitation that doubling the training load predicts twice the performance improvement, two distinct extensions are reviewed including the addition of a non-linear transform to training inputs and inclusion of non-linear terms within the system of differential equations. To address the independence assumption where the response to a training session is unaffected by previous sessions, a popular extension where fatigue is updated as an exponentially weighted moving average of previous training loads is reviewed. Finally, the review introduces the concept of state-space models where uncertainty in the estimates of fitness, fatigue and performance measurement can be directly modelled eliminating the unsuited deterministic assumption of the standard FFM. The review also highlights how state-space models can be further expanded to include features such as the Kalman filter where parameter estimates can be updated with incoming performance measurements to better predict and manipulate training to optimise performance. Collectively, the range of topics covered in this review series and the resources provided should enable researchers and practitioners to better investigate the extensive area of FFMs and determine in what contexts models can assist with training monitoring and prescription.