## 2 Literature Review  
The field of temporal data analysis has witnessed remarkable evolution across multiple domains, driven by the exponential growth of time-varying datasets and the need for sophisticated analytical frameworks capable of extracting meaningful insights from complex temporal structures. This literature review examines the current state of research across four interconnected areas that form the foundation of our work: time-series clustering methodologies, categorical time-series analysis, trajectory-based temporal pattern discovery, and distance-based similarity measurement techniques. Time-series clustering has emerged as a fundamental paradigm in temporal data mining, with comprehensive surveys by [1,2] establishing the theoretical foundations and practical applications of clustering temporal data. The field has evolved from classical approaches focused on continuous numerical representations to sophisticated methodologies that address the challenges of high-dimensionality, irregular sampling, and varying temporal scales [7,21,22]. Recent advances in time-series clustering have been comprehensively reviewed by [8], who provide a user-friendly guide to distance measures for comparing time series in ecological applications, and [13], who introduce quantum-inspired clustering methods for time-series images, building upon foundational hierarchical clustering algorithms surveyed by [9,23]. These works highlight the growing diversity of application domains and the need for specialized clustering approaches that can handle domain-specific temporal patterns. The deep learning revolution has significantly impacted time-series clustering, with [24] providing a comprehensive review of deep time-series clustering methods. Their survey identifies state-of-the-art approaches and presents an outlook on the important field of deep time-series clustering from multiple perspectives, including architectural design, loss functions, and evaluation metrics. This evolution toward deep learning-based approaches reflects the field's adaptation to increasingly complex temporal datasets that require sophisticated feature learning capabilities. A recent bridging survey by [25] traces the evolution from classical approaches to neural network-based methods, providing a unified taxonomy that connects traditional clustering methods with emerging deep learning algorithms. This work emphasizes the importance of understanding both conventional and modern approaches to select appropriate methodologies for specific temporal analysis tasks. The analysis of categorical time series represents a specialized but crucial area within temporal data mining, addressing the unique challenges posed by discrete-valued temporal sequences. [PERSON]. [10] introduced CategoricalTimeSeries.jl, a comprehensive toolbox for categorical time-series analysis that provides essential tools for handling discrete temporal sequences. Their work establishes important foundations for analyzing time series where values represent qualitative states rather than continuous measurements. Ref. [11] developed model-based clustering approaches specifically for categorical time series, demonstrating how Bayesian methods can be adapted to handle the discrete nature of categorical temporal data. Their work provides important theoretical foundations for understanding how traditional clustering concepts can be extended to accommodate categorical temporal sequences. Recent work by [26] addresses the challenges of analyzing categorical time series in complex real-world applications, particularly focusing on atmospheric circulation patterns. Their statistical analysis demonstrates how categorical temporal sequences can reveal important patterns in large-scale natural phenomena, providing valuable insights into the temporal dynamics of complex systems. The transformation of continuous variables into categorical representations has been extensively studied in epidemiological research, with [27] providing a critical analysis of quartile-based categorization approaches.