## 1 Introduction  
The analysis of temporal patterns in multi-dimensional datasets represents one of the most challenging problems in contemporary data science [1, 2]. As organizations increasingly collect vast amounts of time-varying data across multiple domains--from economic indicators and social metrics to environmental measurements--the need for robust methodologies to extract meaningful insights from these complex temporal structures has become paramount [3, 4]. Traditional analytical approaches often fall short when confronted with the dual challenges of high dimensionality and temporal complexity, necessitating innovative frameworks that can simultaneously address both structural and evolutionary characteristics of multi-dimensional time series [5, 6]. The field of temporal data analysis has witnessed significant advancement through the development of clustering-based methodologies that group similar patterns and reveal underlying structures in time-varying datasets [1; 7]. Time-series clustering, as an unsupervised learning paradigm, has proven particularly effective in identifying latent patterns, detecting anomalies, and facilitating comparative analysis across different entities or time periods [2; 8; 9]. However, existing approaches predominantly focus on continuous numerical representations, often overlooking the interpretative advantages that categorical transformations can provide in revealing evolutionary trajectories and facilitating cross-entity comparisons. The transformation of continuous time series into categorical representations has emerged as a powerful approach for enhancing interpretability while preserving essential pattern information [10; 11]. Categorical time-series analysis offers several distinct advantages: it reduces sensitivity to measurement noise, provides intuitive interpretations of temporal states, and enables the application of discrete similarity measures that can capture structural differences between evolutionary trajectories [12; 13]. This approach is particularly valuable in comparative policy analysis, where the focus shifts from precise numerical values to broader categorical transitions that reflect fundamental changes in system behavior or policy regimes. Central to the effectiveness of categorical temporal analysis is the choice of appropriate similarity measures that can quantify differences between discrete sequences. The Hamming distance, originally introduced by [PERSON] in 1950 for error detection in digital communications [14], has found extensive applications in diverse fields requiring the comparison of categorical sequences [15; 16]. Its simplicity, computational efficiency, and intuitive interpretation make it particularly well-suited for analyzing categorical time series, where the focus lies on identifying positions of difference rather than measuring numerical magnitudes. In the specific context of economic and policy analysis, the ability to track and compare temporal evolution patterns across different countries, sectors, or time periods has become increasingly important for understanding convergence, divergence, and crisis dynamics [17; 18]. Government debt trajectories, for instance, provide valuable insights into fiscal policy effectiveness, crisis responses, and long-term sustainability patterns. However, traditional econometric approaches often struggle to capture the categorical nature of policy regimes and the discrete transitions between different fiscal states that characterize real-world policy evolution. The visualization of multi-dimensional temporal patterns poses additional challenges that extend beyond mere data representation to encompass cognitive interpretation and decision-making support [19; 20]. Effective temporal visualization must balance comprehensiveness with clarity, providing sufficient detail for analytical insight while maintaining interpretability for policy makers and stakeholders.