## 1 Introduction
Categorical approaches offer particular advantages in this context, as they naturally align with human cognitive patterns of classification and comparison, facilitating more intuitive understanding of complex temporal dynamics. Despite the growing recognition of categorical time-series analysis and the widespread application of clustering methodologies in temporal data mining, a significant gap remains in the development of integrated frameworks that combine categorical transformation, clustering-based trajectory analysis, and distance-based similarity measurement into a unified analytical approach. Existing methodologies typically address these components in isolation, limiting their ability to provide comprehensive insights into the multi-faceted nature of temporal pattern evolution. This paper addresses these limitations by introducing the Hamming diversification index, a novel clustering-based metric specifically designed to analyze and visualize time-evolution patterns in multi-dimensional datasets. Our methodology integrates quantile-based categorical transformation with trajectory clustering and Hamming distance measurement to provide a parsimonious yet comprehensive framework for temporal pattern analysis. The approach transforms continuous time series into categorical sequences rep resenting distinct evolutionary states, constructs temporal trajectories that capture entity-specific evolution patterns, and employs both regular and modified Hamming distances to quantify trajectory similarities and differences. The theoretical contribution of our work extends beyond methodological innovation to encompass broader implications for temporal data analysis and comparative policy research. By demonstrating how categorical transformations can preserve essential pattern information while enhancing interpretability, we provide a foundation for more accessible and actionable temporal analysis frameworks. The integration of clustering concepts with distance-based similarity measurement offers new perspectives on how to quantify and visualize evolutionary diversity in complex temporal systems. To validate our methodology and demonstrate its practical utility, we apply the Hamming diversification index to analyze government debt trajectories across eight developed economies using the Comparative Political Data Set spanning 1960-2022. This application provides concrete evidence of the framework's ability to capture meaningful fiscal policy patterns, identify periods of convergence and divergence, and facilitate comparative analysis across different economic systems and time periods. The remainder of this paper is structured as follows. Section 2 provides a comprehensive literature review examining current approaches to temporal data analysis, categorical time series, and distance-based similarity measurement. Section 3 presents our synthetic dataset construction methodology used to demonstrate the framework's capabilities across diverse evolutionary patterns. Section 4 details the complete methodology, including statistical transformation procedures, categorical boundary determination, entity-specific trajectory analysis, and distance metric implementation. Section 5 applies our framework to real-world government debt analysis using the Comparative Political Data Set spanning 1960-2022, demonstrating practical utility in comparative economic research. Section 6 explores future research directions and potential extensions of the methodology. Finally, Section 7 provides concluding remarks and discusses the broader implications of our clustering-based approach to temporal pattern analysis. **Problem Statement**  
**Given:**  
* A set of entities \(E=\{\,e_{1},\,e_{2},\,\ldots,\,e_{n}\,\}\) observed over time periods \(T=\{t_{1},\,t_{2},\,\ldots,t_{m}\}\),
* Continuous-valued features \(F=\{f_{1},\,f_{2},\,\ldots,f_{k}\}\) for each entity-time pair \((e_{i},t_{j})\),
* Statistical quartile boundaries \(Q=\{\)LB, \(Q_{1},\,Q_{2},\,Q_{3},\,\text{UB}\}\) derived from the global distribution of feature values.