## 4 Discussion
On the contrary, \(\text{net}_{\text{CD}}\) and \(\text{net}_{\text{DFA}}\) both exhibit slight underprediction biases, although their errors are primarily close to zero, suggesting overall accurate models. \(\text{Net}_{\text{LyEw}}\) is notable for its errors symmetrically distributed about the zero-error line, highlighting an unbiased model. Lastly, net\({}_{\text{LyE}}\)'s histogram, while still reflecting balanced predictive capabilities, slightly leans towards overpredictions. In the error histograms for the MLR models (Supplementary Figures), none of the models display a symmetrical error distribution around the zero line. Although the LyE\({}_{\text{R}}\) model shows a slight trend toward symmetry near zero, it does not exhibit a distinct, bell-shaped distribution. This lack of symmetry across the error distributions suggests that the MLR models struggle to capture the underlying dynamics fully, particularly in cases with higher nonlinearity. These visual error distributions are critical in assessing and refining each model's ability to accurately predict metabolic costs, with a tighter clustering of errors around zero indicating a higher model accuracy. Tables 3 and 4 comprehensively overview the multiple linear regression analyses performed on various nonlinear gait measures to predict the metabolic cost. The tables summarize the regression analysis results for the six different nonlinear measures. The R\({}^{2}\)MLR values in these tables represent how much variance in the metabolic cost can be explained by the models created using these nonlinear measures. Specifically, the LyE\({}_{\text{Er}}\) and DFA demonstrate the highest explanatory power with R\({}^{2}\)MLR values of 0.72 and 0.63, respectively. This implies that these models can explain 72% and 63% of the variance in the metabolic cost, indicating a substantial connection between the predicted metabolic cost and the variables included in these models. Each row in the tables corresponds to a different gait parameter. The coefficients, denoted as estimates, indicate how a one-unit change in each variable would impact the metabolic cost, assuming other variables remain constant. The Standard Error (SE) gives a sense of the estimate's precision, and the \(p\)-values test the statistical significance of each coefficient. Regarding statistical significance, values with significant \(p\)-values (less than 0.05) are highlighted in bold, underscoring their strong influence on the predicted metabolic cost. Tables 3 and 4 show that the hip moment and both components of the GRF have significant coefficients, indicating that they are important predictors of the metabolic cost in the ApEn and SpEn models. In Tables 3 and 4, the significant negative coefficients for the hip moment and positive coefficients for the ankle velocity and GRF vertical component in the DFA model suggest that these factors are particularly influential in predicting the metabolic cost using this measure. Key predictors for the LyEw and LyER models were the hip and ankle velocity. Based on Tables 3 and 4, we can identify the most influential variables in the regression analyses for predicting the metabolic cost from various nonlinear gait measures, mainly focusing on the LyE\({}_{\text{R}}\) and DFA due to their higher explanatory power (R\({}^{2}\)MLR values of 0.72 and 0.63, respectively). In Tables 3 and 4, GRF\({}_{\text{Ver}}\) and GRF\({}_{\text{AP}}\) have significant \(p\)-values across the ApEn, CD, and SpEn models, indicating their strong influence on the metabolic cost. For instance, GRF\({}_{\text{Ver}}\) has a coefficient of 0.3785 with a \(p\)-value of 0.0004 in the ApEn model, and GRF\({}_{\text{AP}}\) has a coefficient of \(-\)0.4155 with a \(p\)-value of 0.0003, suggesting that these GRF components are crucial predictors within these models. In the DFA model, variables like the ankle velocity and ankle moment have significant \(p\)-values and substantial coefficients, indicating their impactful role.