## 4 Methodology
### Comprehensive Trajectory Metrics Analysis
#### 4.6.1 Trajectory Metrics: Theoretical Foundation and Implementation  
The Hamming diversification index framework that we introduce in this paper incorporates eight complementary metrics that capture distinct dimensions of categorical trajectory behavior. Each metric addresses specific characteristics of temporal evolution, enabling comprehensive characterization of entity-specific patterns and facilitating meaningful comparative analysis across different evolutionary dynamics. 1. Volatility: Volatility measures the categorical transition instability through the standard deviation of absolute changes between consecutive time periods. This metric captures the degree of unpredictability in categorical transitions, with higher values indicating more erratic movement patterns. \[\text{Volatility}(\sigma_{v})=\sqrt{\frac{1}{(n-1)}\sum_{i=2}^{n}(|c_{i}-c_{i-1}|- \mu)^{2}} \tag{7}\]  
where \(\mu=\frac{1}{n-1}\sum_{i=2}^{n}|c_{i}-c_{i-1}|\) represents the mean absolute categorical change, and \(c_{i}\) denotes the categorical value at time \(i\). This formulation captures both the magnitude and consistency of categorical transitions, providing insights into the stability of evolutionary patterns. 2. Number of Steps (Discrete Transitions): This metric quantifies the total number of discrete categorical transitions throughout the observation period, serving as an indicator of trajectory dynamism. \[\text{Steps}(S)=\sum_{i=2}^{n}\mathbf{1}[c_{i}\
eq c_{i-1}]\] (8) where \(\mathbf{1}[\cdot]\) represents the indicator function. Higher step counts indicate more frequent categorical changes, suggesting adaptive or responsive behavioral patterns, while lower counts suggest persistence within categorical states. 3. Trajectory Changes (Directional Reversals): This metric captures the number of directional reversals in categorical movement, identifying periods where upward trends reverse to downward trends or vice versa. It provides insights into policy regime instability and adaptive responses to changing conditions. \[\text{Trajectory Changes}(TC)=\sum_{i=3}^{n}\mathbf{1}[\text{sign}(c_{i}-c_{i-1}) \
eq\text{sign}(c_{i-1}-c_{i-2})]\] (9) This formulation identifies points where the direction of categorical movement changes, capturing oscillatory behavior and trend reversals that may indicate strategic pivots or external pressure responses. 4. Range Span (Categorical Breadth): Range span quantifies the breadth of categorical states experienced throughout the observation period, providing insights into the scope of evolutionary dynamics. \[\text{Range Span}(R)=\max\{c_{i}\}_{i=1}^{n}-\min\{c_{i}\}_{i=1}^{n}\] (10) Larger range spans indicate entities that traverse multiple categorical states, suggesting greater adaptability or exposure to diverse conditions, while smaller spans indicate constrained evolutionary patterns. 5. Trend Strength (Directional Consistency): Trend strength measures the overall directional consistency of categorical evolution through linear regression slope analysis, capturing long-term systematic patterns. \[\text{Trend Strength}(TS)=|\beta|,\quad\text{where }c_{i}=\alpha+\beta t_{i}+ \epsilon_{i}\] (11) The absolute value of the regression slope provides a scale-invariant measure of directional consistency, with higher values indicating stronger systematic trends regardless of direction. 6. Persistence (Categorical Stability): Persistence quantifies the proportion of time spent maintaining the same categorical state, providing insights into institutional or behavioral stability. \[\text{Persistence}(P)=\frac{\sum_{i=2}^{n}\mathbf{1}[c_{i}=c_{i-1}]}{n-1} \tag{12}\]  
Higher persistence values indicate stable categorical positioning, while lower values suggest frequent state transitions and dynamic adjustment patterns. 7. Extreme Visits (Boundary Exposure): This metric counts observations in extreme categorical boundaries (categories 0 and 5), indicating exposure to exceptional conditions.