## 4 Discussion  
This study explored the nonlinear dynamics of human gait and their correlation with the metabolic cost using ANNs. It assessed the predictive capabilities of ANNs compared to traditional MLR models across a spectrum of nonlinear gait measures. In the discussion of the comparative analysis of the ANN models, each tailored to a specific gait nonlinearity measure (LyE\({}_{\text{R}}\), DFA, ApEn, CD, SpEn, LyE\({}_{\text{W}}\)), we observe a spectrum of performance metrics that highlight the complexity of estimating the metabolic cost using models based on the nonlinearity of gait measures. The performance metrics in Table 2 and Supplementary Figures demonstrate the training, validation, and test splits and the hidden layer sizes for each ANN model. These models were meticulously developed to capture the intricate relationships between the nonlinear gait measures and metabolic cost effectively. The training performance across all models was robust, with high R\({}^{2}\)ANN values close to one, indicating an excellent fit [44]. Specifically, the LyE\({}_{\text{R}}\), DFA, and ApEn models showed superior performance, with low MSE and high R\({}^{2}\)ANN values. These measures could capture the inherent dynamics in the gait pattern, which correlate with the metabolic cost more effectively than other measures. The results in Table 1 of the k-fold cross-validation show consistent trends in the ANN R\({}_{\text{ANN}}\) and \(p\)-values, similarly to the 'Total' values calculated using combined data from the training, validation, and test phases. This approach offers a more robust model evaluation, particularly with a small sample size, as it provides more reliable metrics and reduces the risk of overfitting. Significant  
Figure 2: Partial Dependence Plots (PDPs), the graphical analysis of gait nonlinearity measures, and their prediction errors. This figure illustrates the relationship between various gait parameters—such as joint angles, velocities, moments, center of mass (COM) displacement in the sagittal plane, and ground reaction force (GRF) magnitudes in vertical and anterior–posterior directions—and their influence on the prediction of the metabolic cost. Subfigures represent the mean of nonlinearity measures, (**A**) the Lyapunov Exponent based on [PERSON]’s algorithm (LyE\({}_{\text{R}}\)), (**B**) Detrended Fluctuation Analysis (DFA), (**C**) the Approximate Entropy (ApEn), (**D**) the correlation dimension (CD), (**E**) the Sample Entropy (SpEn), and (**F**) the Lyapunov Exponent based on [PERSON]’s algorithm (LyE\({}_{\text{w}}\)), respectively. Blue bars (left vertical axis) indicate the measure values, while red bars (right vertical axis) show the corresponding prediction error of energy expenditure percentages, highlighting the impact of each gait parameter on the precision of metabolic cost estimation. \(p\)-values were observed across all model assessment phases (validation, test, and training), confirming the robust predictive capabilities of the models. For example, the \(\text{LyE}_{\text{R}}\) reflects the system's sensitivity to small perturbations, potentially indicating stability and control adjustments in gait that directly impact energy expenditure. DFA captures long-range correlations in gait patterns, representing sustained coordination efforts that may increase metabolic demands. The ApEn quantifies the complexity of gait variability, which could relate to the adaptability of gait mechanics in response to changing conditions. In contrast, while still yielding statistically significant results, models such as \(\text{LyE}_{\text{w}}\) presented lower \(\text{R}^{2}_{\text{ANN}}\) values, especially in the validation phase, which could indicate a less generalizable model.