# A Statistical Approach for Modeling Individual Vertical Walking Forces
## 4 Statistical Characterization of the Walking Parameters  
This section summarizes the procedure adopted to statistically characterize the walking parameters of the force models presented in Section 3.  
* For each TS, collect all the measured steps \(N_{step}\).
* For each step, evaluate the temporal and spatial parameters that characterize the force models of Section 3:
* For the proposed force model, the model parameters are: \(f_{s}\), \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\), \(\Delta t\), \(l\), \(\Phi_{1}\), \(\Phi_{2}\) and \(\theta\). In particular, the temporal feature \(\Delta t\) is easily evaluated measuring the period during which the analyzed step overlaps the following one. The step frequency \(f_{s}\) (equal to the frequency of the first harmonic), the amplitudes \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\) and the phases \(\Phi_{1}\), \(\Phi_{2}\) are evaluated based on a Fourier analysis of the measured step. Finally, the distance \(l\) and the direction \(\theta\) from the step to the next one are evaluated considering the mid-position of the foot. These parameters are collected in three matrices, one for each TS, of dimensions \(N_{step}\times 10\).
* For the simplified load model, the model parameters are: \(A_{0}\), \(T_{c}\), \(\Delta t\), \(l\) and \(\theta\). Besides the parameters previously described, the temporal parameter \(T_{c}\) is evaluated measuring the total duration of the foot force while the amplitude \(A_{0}\) of the constant footfall force is calculated so that the integrals of the experimental and simulated step forces over the step duration \(T_{c}\) are the same. These parameters are collected in three matrices, one for each TS, of dimensions \(N_{step}\times 5\).  
Figure 5: Simplified load model: example of three successive steps. The dashed lines represent the measured steps while the solid lines are the corresponding modeled steps.  
* For each TS, calculate the correlation matrices \(R\) starting from the parameter matrices defined in the previous step. Correlation matrices collect the pairwise linear correlation coefficient between each pair of walking parameters. Depending on the considered force model (proposed or simplified), the resulting matrices have dimensions \(10\times 10\) or \(5\times 5\).
* To devise general information, calculate the average correlation matrix \(\overline{R}\) over the three TSs: Equations (6) and (11) for the proposed and simplified load model, respectively.
* Based on the average correlation matrices, evaluate which parameters are correlated and which are not. For both force models, results show that all model parameters are correlated except for the step direction \(\theta\).
* Assume a normal distribution for the uncorrelated model parameters and a multivariate distribution \(p(S)\) for the correlated parameters: Equation (7) for the proposed model and Equation (12) for the simplified one.
* Fit the multivariate distributions adopting a Gaussian Mixture model [32]: Equation (10) for the proposed model and Equation (14) for the simplified one.
* Assess the accuracy of the calibrated GM model through the Kolmogorov-Smirnov test [34].