## 2 Literature Review
While their work raises important concerns about the limitations of quantile-based categorization in certain contexts, it also highlights the interpretative advantages that categorical transformations can provide in enhancing pattern recognition and facilitating comparative analysis. Trajectory analysis represents a natural extension of time-series clustering to spatial-temporal domains, where entities move through both space and time. Ref. [28] introduced TRACLUS, a pioneering algorithm for clustering trajectories that has influenced numerous subsequent developments in spatial-temporal data mining. Their work establishes important foundations for understanding how traditional clustering concepts can be ex tended to handle the dual spatial-temporal nature of trajectory data. Recent surveys by [29; 30] provide comprehensive overviews of trajectory clustering analysis, categorizing existing methods into unsupervised, supervised, and semi-supervised approaches, with applications extending to criminological research [31]. These reviews highlight the growing sophistication of trajectory analysis methods and their applications across diverse domains including transportation, surveillance, and behavioral analysis. The healthcare domain has proven particularly fertile for trajectory analysis applications, with [32] developing temporal pattern mining frameworks for electronic health records. Their work demonstrates how trajectory-based approaches can be applied to discrete event sequences, providing valuable insights into disease progression patterns and treatment effectiveness. Ref. [33] extended trajectory analysis to multivariate clinical data, introducing recent temporal pattern mining frameworks for event detection in complex temporal datasets. Their approach combines temporal abstraction with pattern mining to identify predictive patterns in electronic health records, demonstrating the practical utility of trajectory-based approaches in real-world applications. The effectiveness of temporal clustering and trajectory analysis fundamentally depends on appropriate distance measures that can capture meaningful similarities between temporal sequences. [PERSON]. [14] originally introduced the Hamming distance for error detection in digital communications, but its applications have expanded dramatically across diverse fields requiring comparison of categorical sequences. [PERSON]. [15] provides a comprehensive survey of distance and similarity measures between probability density functions, establishing important theoretical foundations for understanding how different distance metrics capture various aspects of temporal similarity. Their work is particularly relevant for understanding the relationship between different distance measures and their appropriate applications in temporal data analysis. Recent work by [16] focuses specifically on model-based clustering of categorical data using Hamming distance, providing important insights into how this classic distance measure can be effectively applied to modern categorical clustering problems. Their research demonstrates the continued relevance of Hamming distance in contemporary data mining applications. The application of distance measures to time-series comparison has been comprehensively reviewed by [8], who provide practical guidance for selecting appropriate distance measures for temporal data analysis. Their work emphasizes the importance of understanding the properties and limitations of different distance measures to ensure appropriate selection for specific analytical tasks. The broader field of temporal data mining provides important context for understanding how temporal pattern discovery fits within the larger landscape of knowledge discovery in temporal databases. Ref. [3] provides a comprehensive review of time-series data mining, covering various approaches including similarity-based methods, feature-based approaches, and model-based techniques. Ref.