## 2 Materials and Methods
### Statistical Analyses
The \(p\)-values reported in Table 2 were derived from the MATLAB _\"corroeg\"_ function, which calculates the \(\text{R}^{2}\) and a two-tailed \(p\)-value for each dataset (training, validation, and testing) [10; 14; 43]. These \(p\)-values test the null hypothesis, in particular, that there is no linear relationship between the predicted and actual values [10; 14; 43]. The extremely low \(p\)-values observed reflect a statistically significant correlation, validating the model's predictive accuracy [10; 14; 43]. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **LyE\({}_{\text{R}}\)** & **DFA** & **ApEn** & **CD** & **SpEn** & **LyE\({}_{\text{w}}\)** \\ \hline
**Validation ratio (\%)** & 15 & 15 & 15 & 15 & 15 & 15 \\
**Test ratio (\%)** & 15 & 15 & 15 & 15 & 15 & 15 \\
**Training ratio (\%)** & 70 & 70 & 70 & 70 & 70 & 70 \\
**Hidden layer size (neurons)** & 8 & 19 & 14 & 3 & 18 & 2 \\ \hline
**MSE performance (W/kg)\({}^{2}\):** & & & & & & \\ - Validation & \(50\times 10^{-6}\) & \(40\times 10^{-6}\) & \(40\times 10^{-6}\) & \(60\times 10^{-6}\) & \(90\times 10^  
Based on the literature and similar studies [1, 4, 10, 14, 37, 38, 43], we ensured that the statistical methods applied in this study were appropriate for the structure of our dataset. These adjustments align with the recommendations provided in prior studies that analyzed similar datasets. Figure 1 illustrates the research process, including each step involved in developing and evaluating the MLR and ANN models for predicting the metabolic cost. This visual highlights key nonlinear measures identified for accurate metabolic cost prediction, as well as insights into the strengths and limitations of each model. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \(\mathbf{LyE_{R}}\) & \(\mathbf{DFA}\) & \(\mathbf{ApEn}\) & \(\mathbf{CD}\) & \(\mathbf{SpEn}\) & \(\mathbf{LyE_{w}}\) \\ \hline \(\boldsymbol{p}\)**-value:** & & & & & & \\ - Validation & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ - Test & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ - Training & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ - Total & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ -5-Fold & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 1: Flow diagram of the research development process for predicting the metabolic cost using multiple linear regression (MLR) and Artificial Neural Network (ANN) models, based on six gait  
nonlinearity measures: the Lyapunov Exponent based on [PERSON]'s algorithm (LyER), Detrended Fluctuation Analysis (DFA), the Approximate Entropy (ApEn), the correlation dimension (CD), the Sample Entropy (SpEn), and the Lyapunov Exponent based on [PERSON]'s algorithm (LyEW). The diagram outlines the sequential steps, from data collection and preparation through model design, cross-validation, and evaluation, and a comparative analysis of ANN and MLR models. Each variable represents specific gait parameters, including joint angles, velocities, moments, ground reaction forces (GRFs), and center of mass (COM) metrics. Key nonlinear measures for accurate metabolic cost prediction are emphasized, along with conclusions on the strengths and limitations of each model.