## 4 Methodology
### Entity-Specific Trajectory Analysis  
We add a new column to our existing dataset to elucidate the cluster value taken by an object at a given point in time. This allows us to  
* Isolate each entity's temporal trajectory;
* Apply the categorical transformation uniformly across all entities;
* Preserve the temporal sequence for subsequent evolutionary analysis. Now that we have our cluster descriptions, we perform an ordinal mapping analysis to assign numerical values to these clusters, as it will help us in visualizing the temporal cluster transitions. The mapping of clusters is pivotal, as it  
* Preserves the ordinal relationship between categories;
* Enables quantitative comparison between categorical trajectories;
* Facilitates visualization and distance metric computation. To facilitate quantitative analysis, we discretized continuous values into six ordinal categories based on their position relative to statistical thresholds: the lower bound (LB), upper bound (UB), and the quartiles (\(Q_{1}\), \(Q_{2}\), \(Q_{3}\)). Each category corresponds to a specific range and is assigned a unique integer from 0 to 5 to preserve the natural ordering. Values below the lower bound are labeled as \"Below LB\" (0), while those between LB and \(Q_{1}\) are categorized as \"LB < val < \(Q_{1}\)\" (1), and so forth, up to values exceeding the upper bound, which are labeled as \"Above UB\" (5). This mapping enables the incorporation of distributional information into downstream models while maintaining interpretability. The ordinal mapping scheme for categorical transformation is defined in Table 2. The six-category system preserves natural ordering through integer assignment (0-5), where each category represents a specific position relative to statistical thresholds. This mapping enables quantitative analysis while maintaining interpretability: categories {0, 1} represent below-average performance, categories 2-3 indicate moderate levels, and categories 4-5 signify above-average performance relative to the global distribution. The systematic integer progression allows for meaningful mathematical operations while preserving the ordinal nature of the categorical boundaries, facilitating both distance computation and visual interpretation of temporal trajectories. The values of the clusters for all the objects across the time period are given in Table 3. The categorical transformation results for all entities across the temporal observation period are presented in Table 3. The transformed values demonstrate the methodology's ability to capture distinct temporal patterns through categorical sequences. Entity \(X\) shows progressive categorical advancement in \(f_{1}\) (\(1\to 3\)) and stepwise increases in \(f_{2}\), reflecting a systematic upward trajectory with structural transitions. Entity \(Y\) exhibits a categorical decline in \(f_{1}\) (\(4\to 1\)) with high volatility in \(f_{2}\), capturing deteriorating trends accompanied by unstable secondary dynamics. Entity \(Z\) displays oscillatory categorical patterns in both features, demonstrating non-linear temporal evolution with frequent regime changes. These categorical sequences serve as input for subsequent trajectory analysis and distance computation, preserving essential pattern information while enabling comparative analysis across entities with different scales and distributions. The methodology of hamming diversification index is demonstrated in Algorithm 1. ```
0: Continuous data matrix \(X\in\mathbb{R}^{n\times m\times k}\) (entities \(\times\) time \(\times\) features)
0: Categorical trajectories \(T\), distance matrix \(D\), diversification index HDI
1: Step 1: Statistical Transformation
2: Compute global quartiles: \(Q_{1}\), \(Q_{2}\), and \(Q_{3}\) from all observations in \(X\)
3: Compute IQR: IQR \(=Q_{3}-Q_{1}\)
4: Compute bounds: LB \(=Q_{1}-1.5\times\) IQR, and UB \(=Q_{3}+1.5\times\) IQR
5: Define categorical mapping function: \[\text{CategoryMap}(x)=