# A Novel Criticality Analysis Method for Assessing Obesity Treatment Efficacy
## 3 Methodology
### Gait Data Representation and Spatiotemporal Analysis Using Criticality Analysis  
Criticality Analysis is a method used to represent complex multivariate data patterns in a simplified form, typically in the form of a phase plot portrait or manifold. This method involves analysing the data in multiple dimensions and identifying patterns or structures that are most critical to understanding the underlying dynamics of the system. The extracted features by DGAS were used as perturbation inputs to the CA model represented by Equations (4) and (7), respectively. This aided in gaining a deeper understanding of the underlying mechanisms and dynamics of the system under study. The visual representations of gait regulation and coordination between spatial and temporal domains are depicted in Figures 2-7 through phase plot orbits. Well-regulated gait was characterised by smooth and narrow orbits, whereas dysfunctional gait control was evident in irregular and variable orbits. These phase plots served as a means to distinguish between healthy and pathological gaits by evaluating the dynamics of spatiotemporal coordination. These phase plots demonstrated clear differences between the gait patterns of the healthy control and obesity groups over the 6-week period. Specifically, in the healthy control group, the phase plots exhibited a consistent pattern characterised by smooth and regular oscillations with steady amplitudes and frequencies. These findings were indicative of a well-maintained, rhythmic gait pattern that demonstrated excellent coordination and balance. Notably, the orbits in this group remained relatively narrow, which underscored the efficiency of their biomechanics and the minimal occurrence of side-to-side body motion. In contrast, the obesity group displayed phase plots that deviated from the healthy control group's pattern. These plots appeared more irregular and distorted, with variable amplitudes and wider orbits, suggesting a compromised sense of balance and increased lateral swaying \begin{table}
\begin{tabular}{l c c} \hline
**Gait Parameter** & **Measurement** & **Unit** \\ \hline \multirow{4}{*}{Temporal} & Step Time & (ms) \\  & Step Time (Left) & (ms) \\  & Step Time (Right) & (ms) \\  & Stride Time & (ms) \\  & Cadence & (steps/min) \\ \hline \multirow{4}{*}{Spatial} & Step Length (Left) & (m) \\  & Step Length (Right) & (m) \\  & Stride Length & (m) \\  & Velocity & (m/s) \\ \hline \multirow{4}{*}{Multi-dimensional} & Duty Factor Double Stance & (\%) \\  & Duty Factor Single Stance & (\%) \\  & Froude Number & (au) \\  & Walk Ratio & (mm/steps/min) \\ \hline \multirow{4}{*}{Phase Plot Analysis} & Beta Angle & (Degree (\({}^{\circ}\))) \\  & SDa & (au) \\ \cline{1-1}  & SDb & (au) \\ \cline{1-1}  & Ratio = SDa/SDb & (Dimensionless) \\ \hline \end{tabular}
\end{table} 
Table 1: Extracted Gait Features. during gait. Furthermore, these phase plots exhibited more abrupt changes in direction, indicating the need for sudden adjustments to maintain stability. These observations were reflective of a slower and more effortful gait, likely resulting from the additional weight burdening the joints and muscles in individuals with obesity. Moreover, both groups exhibited a common trend of declining gait consistency over the 6-week observation period, potentially attributable to the onset of fatigue effects. The healthy control group, despite its initial robust gait pattern, displayed a gradual decrease in consistency, reflecting the possibility of accumulating fatigue from repeated gait assessments. The obesity group, already experiencing challenges in maintaining gait regularity, showed a similar decline in consistency, accentuating the toll that prolonged observation sessions might take on their gait patterns. This convergence in declining gait consistency underscores the importance of considering potential fatigue factors in the interpretation of gait analysis results across different population groups.