Curve fitting problem: torque - velocity relationship with polynomials and Boltzmann sigmoid functions

The aim of this study was to investigate the curve fitting and model selection problem of the torque-velocity relationship of elbow flexors and extensors in untrained females. The second goal was to determine the optimal models in different function classes and the best, among the optimal ones. Lastly, test the best models to predict the torque were tested.

Using the polynomials (second - fourth degree) and Boltzmann sigmoid functions, and a different presentation of data points (averages, a point cloud, etc.), we determined the optimal models by both error criteria: minimum residual sum of squares and minimum of the maximal absolute residue. To assess the best models, we applied Akaike and Bayesian information criteria, Hausdorff distance and the minimum of the smallest maximal absolute residue and the predictive torque-velocity relationships of the best models with torque values, calculated beyond the experimental velocity interval.

The application of different error and model selection criteria showed that the best models in the majority of cases were polynomials of fourth degree, with some exceptions from second and third degree. The criteria values for the optimal Boltzmann sigmoids were very close to those of the best polynomial models. However, the predicted torque-velocity relationships had physiological behavior only in Boltzmann's sigmoid functions, and their parameters had a clear interpretation.

The results obtained suggest that the Boltzmann sigmoid functions are suitable for modeling and predicting of the torque-velocity relationship of elbow flexors and extensors in untrained females, as compared to polynomials, and their curves are physiologically relevant.