# A Novel Criticality Analysis Method for Assessing Obesity Treatment Efficacy
## 3 Methodology
### Gait Data Representation and Spatiotemporal Analysis Using Criticality Analysis
Examining the gait patterns across the 6-week period, in Figure 2 (Week 1), the healthy group displayed a tight circular cluster, indicating consistent gait cycles. In contrast, the obesity group showed more elongated, scattered orbits, reflecting a higher degree of variability in gait. As we progressed to data depicted in Figure 3 (Week 2), the healthy group continued to maintain a tight cluster, while the obesity group's orbits, though still dispersed, appeared somewhat more rounded, suggesting some improvement in gait coordination compared to that of Week 1. Figure 4 (Week 3) portrayed the healthy group with a very tight cluster, indicative of highly consistent gait, while the obesity group exhibited more elongated orbits with flatter tops, indicating instability in their gait pattern. In Figure 5 (Week 4), the healthy control cluster became somewhat looser, possibly due to accumulating fatigue. The obesity group's orbits remained uneven but showed a slightly improved level of coordination compared to Week 3. Figure 6 (Week 5) demonstrates the healthy control group's cluster becoming more dispersed, reflecting increasing gait variability. On the other hand, the obesity group's plots were highly scattered with jagged trajectories, suggesting a worsening of gait control. Finally, in Figure 7 (Week 6), both groups display more dispersed orbits than in previous weeks, indicating the potential impact of fatigue on gait consistency in both groups. The obesity group's orbits appeared slightly more rounded than in Week 5, hinting at some recovery in coordination, although the overall trend indicated challenges in maintaining consistent gait patterns. In this paper, we utilised a kernel SVM classifier to distinguish between the obesity and healthy groups based on phase plot data. The choice of kernel SVM is particularly well suited for this analysis due to the inherently nonlinear and complex nature of the data. Phase plot data, representing dynamic patterns of physiological processes, often exhibit intricate and nonlinear relationships. Traditional linear classifiers may struggle to capture the patterns present in such data. However, the kernel SVM is designed to address this challenge by mapping the data into a higher-dimensional space, where complex patterns  
Figure 2: Comparison of phase space plots of walk patterns for healthy control and obesity groups in the clinical gait experiment conducted in \(w_{1}\) is presented. Healthy control walk patterns are shown on the left while obesity walk patterns are shown on the right. Figures 8 and 9 serve as invaluable tools for assessing the dynamic nature of gait progression throughout the 6-week study period. These graphical representations offered a comprehensive view of the data by plotting the peak values extracted from each phase plot orbit as discrete data points for every week. Consequently, these visualisations effectively generated trajectories that unveiled nuanced alterations in gait patterns over time. Fundamentally, each data point within these trajectories captured the maximum step length achieved during a specific gait cycle, encapsulating the essence of gait performance. By plotting these peak values across the 6-week observation window, an intuitive visual perspective emerged on how maximal step length evolved across multiple visits. Furthermore, these peak values functioned as numerical metrics that concisely represented the range of variability, which is an informative measure quantifying the degree of variation in maximal step length from one week to the next. To provide a rigorous statistical summary of this variation over time, standard deviation (SD) of the peak values for each participant was calculated. This SD became a pivotal indicator, with a higher value signifying a greater degree of inconsistency in the maximal step length achieved across different weeks. Consequently, comparing SD values before and after the intervention yielded a quantitative assessment of whether gait improved (resulting in a lower SD) or deteriorated (resulting in a higher SD).