Hamming Diversification Index: A New Clustering-Based Metric to Understand and Visualize Time Evolution of Patterns in Multi-Dimensional Datasets

Abstract

One of the most challenging problems in data analysis is visualizing patterns and extracting insights from multi-dimensional datasets that vary over time. The complexity of data and variations in the correlations between different features adds further difficulty to the analysis. In this paper, we provide a framework to analyze the temporal dynamics of such datasets. We use machine learning clustering techniques and examine the time evolution of data patterns by constructing the corresponding cluster trajectories. These trajectories allow us to visualize the patterns and the changing nature of correlations over time. The similarity and correlations of features are reflected in common cluster membership, whereas the historical dynamics are described by a trajectory in the corresponding (cluster, time) space. This allows an effective visualization of multi-dimensional data over time. We introduce several statistical metrics to measure duration, volatility, and inertia of changes in patterns. Using the Hamming distance of trajectories over multiple time periods, we propose a novel metric, the Hamming diversification index, to measure the spread between trajectories. The novel metric is easy to compute, has a simple machine learning implementation, and provides additional insights into the temporal dynamics of data. This parsimonious diversification index can be used to examine changes in pattern similarities over aggregated time periods. We demonstrate the efficacy of our approach by analyzing a complex multi-year dataset of multiple worldwide economic indicators.

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 Richard Hamming 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. 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 quartile-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 representing 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 $E=\{e_1,e_2,\dots ,e_n\}$ observed over time periods $T=\{t_1,t_2,\dots ,t_m\}$,
• Continuous-valued features $F=\{f_1,f_2,\dots ,f_k\}$ for each entity-time pair $(e_i,t_j)$,
• Statistical quartile boundaries $Q=\{\mathrm{LB},Q_1,Q_2,Q_3,\mathrm{UB}\}$ derived from the global distribution of feature values.

Find:

• Categorical trajectory sequences $T_i=(c_{i,1},c_{i,2},\dots ,c_{i,m})$ for each entity $e_i$, where $c_{i,j}$ is a categorical label (e.g., quartile bin) derived from the feature value at time $t_j$,
• A diversification index D that measures temporal pattern heterogeneity across all entities,
• Distance metrics that quantify trajectory similarities while preserving the ordinal structure of categorical values.

Objective: Develop a parsimonious framework that transforms continuous temporal data into interpretable categorical sequences, and quantifies evolutionary diversity using modified distance measures that account for the ordinal structure of categorical labels.

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. Ref. [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. 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 extended 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. Ref. [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. Ref. [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. [34] introduced frameworks for mining recent temporal patterns for event detection in multivariate time-series data, demonstrating how temporal pattern mining can be effectively applied to complex real-world problems. Their work establishes important connections between pattern mining and predictive modeling in temporal domains. Recent advances in temporal pattern mining have focused on addressing the challenges of pattern explosion and spurious pattern identification. Ref. [32] developed minimal predictive temporal pattern frameworks that address these challenges by focusing on patterns that provide unique predictive information beyond their subpatterns. The visualization of multi-dimensional temporal patterns has become increasingly important as datasets grow in complexity and dimensionality. Ref. [19] provides a comprehensive taxonomy and survey of dynamic graph visualization techniques, establishing important foundations for understanding how temporal relationships can be effectively visualized. Ref. [20] contributes fundamental principles of visualization analysis and design that are particularly relevant to temporal data visualization. Their work provides important theoretical foundations for understanding how visual representations can enhance temporal pattern interpretation and facilitate decision-making in complex temporal analysis tasks. Recent work by [35] addresses the specific challenges of visualizing temporal patterns in high-dimensional data, proposing formal extensions to existing dimensionality reduction methods that explicitly incorporate temporal progression. Their approach demonstrates how traditional visualization techniques can be enhanced to better reveal temporal dynamics in complex datasets.

Despite the substantial progress across these interconnected research areas, several important gaps remain that our work addresses:

• Integration of Categorical Transformation with Clustering-Based Trajectory Analysis: While categorical time-series analysis and trajectory clustering have been developed independently, limited work has explored their integration into unified analytical frameworks. Existing approaches typically address these components in isolation, limiting their ability to provide comprehensive insights into temporal pattern evolution.
• Distance Measures for Categorical Temporal Trajectories: Although Hamming distance has been widely applied to categorical sequence comparison, its application to categorical temporal trajectories derived from continuous data through statistical transformation has received limited attention. The development of modified distance measures that account for the ordinal nature of quartile-based categories represents an important methodological gap.
• Parsimonious Diversification Metrics: Current approaches to measuring temporal pattern diversity often rely on complex statistical measures or require extensive computational resources. The development of simple, interpretable metrics that can effectively capture temporal diversification patterns while remaining accessible to practitioners represents an important practical need.
• Policy-Relevant Temporal Analysis: While temporal pattern analysis has been extensively applied in various domains, limited work has focused specifically on developing frameworks that are suitable for policy analysis and comparative economic research. The integration of interpretable categorical transformations with robust distance measures represents an important application gap. Our research addresses these gaps by introducing the Hamming diversification index, a novel clustering-based metric that integrates quartile-based categorical transformation with trajectory analysis and both regular and modified Hamming distance measurement. This integrated approach provides a parsimonious yet comprehensive framework for temporal pattern analysis that is particularly well-suited for policy-relevant applications while maintaining strong theoretical foundations in the established literature of temporal data mining, clustering analysis, and distance-based similarity measurement.

In the next section (Section 3), we demonstrate a simple example.

3. Dataset Construction

We construct a sample dataset, keeping in mind the myriad characteristics that we want our cluster trajectories to project. Our aim is to provide values to features in a way where the temporal movement of clusters encompasses all possible variations across distinct metrices, including volatility, variability, and distance.

Therefore, we curated a dataset that has the below characteristics:

• A temporal dimension spanning the values 1 through 12.
• Three entities (X, Y, and Z) that are part of the ordinal column in our dataset.
• Two continuous variables, $f_1$ and $f_2$, representing the metric of interest.

We have 3 objects that are assigned values in the way described below:

• Country X exhibits a monotonically increasing trend with some variability for the feature $f_1$ and a sudden increase in value for $f_2$.
• Country Y demonstrates a monotonically decreasing trend for $f_1$ and an oscillating trend for $f_2$.
• Country Z displays a vacillating trend for $f_1$ and a U-shaped change in direction for the feature $f_2$.

This diversity in patterns enables the subsequent analysis to demonstrate the methodology’s capacity to capture different evolutionary trajectories. Our synthetic dataset is designed to exhibit diverse patterns across the following three formally defined metrics [36]:

• Volatility (${\sigma }_v$): Measures categorical transition instability:

  $${\sigma }_v=\sqrt{{\displaystyle \frac{1}{(n-1)}}\sum _{i=2}^n\left(\right|c_i-c_{i-1}{|-\mu )}^2}$$

  (1)

  where $\mu$ is the mean absolute categorical change and $c_i$ represents categorical value at time i.
• Variability (V): Quantifies range of categorical states experienced:

  $$V=max\left(c_i\right)-min\left(c_i\right)\mathrm{for}i\in \{1,2,\dots ,n\}$$

  (2)
• Distance (d): Trajectory dissimilarity between entities using modified Hamming distance [16]:

$$d(T_i,T_j)=\sum _{t=1}^m|c_{i,t}-c_{j,t}|$$

(3)

These metrics enable comprehensive evaluation of temporal pattern diversity across distinct evolutionary trajectories, supporting robust comparative analysis of multi-dimensional dataset dynamics [33].

Table 1. Synthetic dataset for methodology demonstration.

Years	X		Y		Z
	${\mathit{f}}_1$	${\mathit{f}}_2$	${\mathit{f}}_1$	${\mathit{f}}_2$	${\mathit{f}}_1$	${\mathit{f}}_2$
1	1	3	24	8	8	12
2	2	3	22	10	15	11
3	3	4	20	6	3	10
4	4	4	18	12	12	9
5	5	7	16	4	7	8
6	6	9	14	14	19	8
7	7	14	12	2	5	8
8	8	14	10	16	10	9
9	9	15	8	1	2	10
10	10	15	6	18	17	11
11	11	16	4	3	6	12
12	12	16	2	20	14	13

presents the synthetic dataset constructed to demonstrate the methodology’s capacity to capture diverse temporal evolution patterns. The dataset comprises three entities $(X,Y,Z)$ observed over 12 time periods across two continuous variables ($f_1,f_2$).

Entity X exhibits a monotonic increase in $f_1$ with a sudden escalation in $f_2$, demonstrating systematic progression with structural breaks. Entity Y shows a monotonic decrease in $f_1$ alongside oscillatory behavior in $f_2$, reflecting declining trends with high-frequency volatility. Entity Z displays vacillating patterns in $f_1$ and a U-shaped evolution in $f_2$, capturing non-linear temporal dynamics.

This controlled diversity enables a comprehensive evaluation of the clustering-based trajectory analysis framework across distinct evolutionary dynamics, ensuring robust testing of the methodology’s discriminatory power.

4. Methodology

4.1. Statistical Transformation and Categorization

The methodology employs statistical quartile analysis to establish categorical boundaries that add meaning to our clusters [37]. The data distribution of the three objects is given by the boxplot provided in Figure 1. The distributional characteristics of the synthetic dataset are illustrated through boxplot analysis for features $f_1$ and $f_2$ across entities X, Y, and Z in Figure 1. The visualization reveals distinct distributional patterns that serve as the basis for quartile-based categorical boundary determination.

Entity X shows lower variability, with values concentrated in the lower quartiles for both features, reflecting stable and systematic progression. Entity Y exhibits a wider distribution with higher median values, particularly for $f_1$, indicating greater variability around elevated baseline levels. Entity Z demonstrates intermediate characteristics with moderate spread, capturing balanced temporal dynamics between the extremes represented by entities X and Y.

These distributional differences establish the statistical foundation for the subsequent categorical transformation process, ensuring that the derived quartile boundaries capture meaningful variation across distinct evolutionary patterns.

The clusters are created in a way where the end points represent two adjacent values in the boxplot. The clusters signify the magnitude of the absolute value of the feature as it ranges from Below Lower Bound to Above Upper Bound.

This transformation function

• Converts continuous numerical values into ordinal categories.
• Creates a standardized framework for comparative analysis.

The categorical labels provide a parsimonious yet significant understanding of the representation provided by the various categories extrapolated from the boxplot.

4.2. 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. Ordinal mapping for categorical transformation.

Category Value	Category Quartile
0	Below LB
1	$\mathrm{Low} \mathrm{Bound} <\mathrm{value} <Q_1$
2	$Q_1<\mathrm{value}<Q_2$
3	$Q_2$ < val < $Q_3$
4	$Q_3$ < val < UB
5	Above UB

. 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. Categorical transformation results.

Years	X		Y		Z
	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{F}}_1$	${\mathit{F}}_2$
1	1	1	4	2	2	3
2	1	1	4	3	4	3
3	1	1	4	1	1	3
4	1	1	4	3	3	2
5	2	2	4	1	2	2
6	2	2	4	4	4	2
7	2	4	3	1	2	2
8	2	4	3	4	3	2
9	3	4	2	1	1	3
10	3	4	2	4	4	3
11	3	4	1	1	2	3
12	3	4	1	4	4	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 → 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 → 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.

Algorithm 1: Hamming Diversification Index

Require:  Continuous data matrix $X\in {\mathbb{R}}^{n\times m\times k}$ (entities × time × features) Ensure:  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: $\mathrm{IQR}=Q_3-Q_1$ 4: Compute bounds: $\mathrm{LB}=Q_1-1.5\times \mathrm{IQR}$, and $\mathrm{UB}=Q_3+1.5\times \mathrm{IQR}$ 5: Define categorical mapping function: $$\begin{array}{c}\hfill \mathrm{CategoryMap}\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x<\mathrm{LB}1\hfill \\ 1\hfill & \mathrm{if}\mathrm{LB}\le x<Q_{1}2\hfill \\ 2\hfill & \mathrm{if}{Q}_1\le x<Q_{2}3\hfill \\ 3\hfill & \mathrm{if}{Q}_2\le x<Q_{3}4\hfill \\ 4\hfill & \mathrm{if}{Q}_3\le x<\mathrm{UB}5\hfill \\ 5\hfill & \mathrm{if}x\ge \mathrm{UB}\hfill \end{array}\right.\end{array}$$ (4) 6: Step 2: Categorical Assignment 7: for $i=1$ to n do 8:     for $t=1$ to m do 9:         for $f=1$ to k do 10:            $c_{i,t,f}\leftarrow \mathrm{CategoryMap}\left(x_{i,t,f}\right)$ 11:         end for 12:     end for 13: end for 14: Step 3: Trajectory Construction 15: for $i=1$ to n do 16:     if $k=1$ then 17:         $T_i\leftarrow (c_{i,1},c_{i,2},\dots ,c_{i,m})$ 18:     else 19:         $T_i\leftarrow \mathrm{FeatureAggregation}{\left({\left\{c_{i,t,f}\right\}}_{f=1}^k\right)}_{t=1}^m$ e.g., weighted average 20:     end if 21: end for 22: Step 4: Distance Computation 23: Initialize distance matrix $D\in {\mathbb{R}}^{n\times n}$ 24: for $i=1$ to n do 25:     for $j=1$ to n do 26:         Standard Hamming: $d_H(T_i,T_j)={\sum }_{t=1}^m\mathbf{1}[c_{i,t}\ne c_{j,t}]$ 27:         Modified Hamming: $d_{MH}(T_i,T_j)={\sum }_{t=1}^m|c_{i,t}-c_{j,t}|$ 28:         $D_{i,j}\leftarrow d_{MH}(T_i,T_j)$ 29:     end for 30: end for 31: Step 5: Diversification Index Calculation 32: $\mathrm{HDI}={\displaystyle \frac{1}{n(n-1)}}{\sum }_{i=1}^n{\sum }_{j\ne i}{d}_{MH}(T_i,T_j)$ 33: Complexity: $\mathcal{O}(n^{2}m+nmk)$ where n = entities, m = time periods, k = features

4.3. Trajectory Visualization

Finally, we visualize the cluster movements of the objects in our dataset across time.

The visualization approach

• Presents categorical trajectories in their temporal sequence;
• Enables direct visual comparison of evolutionary patterns;
• Translates numerical ordinal values back to meaningful categorical labels for interpretation.

The temporal trajectory evolution across categorical states for features $f_1$ and $f_2$ is demonstrated in Figure 2. The visualization validates the framework’s capacity to capture and differentiate complex temporal patterns through categorical state transitions.

The left panel reveals divergent $f_1$ trajectories: Entity X exhibits monotonic upward progression from category 1 → 3, reflecting systematic categorical advancement over time. Entity Y shows a systematic decline from category 4 → 1, indicating sustained categorical deterioration. Entity Z oscillates between categories 1–4, demonstrating high-frequency categorical transitions with no clear directional trend.

The right panel illustrates $f_2$ dynamics: Entity X demonstrates stepwise categorical advancement with distinct transition points; Entity Y shows high-frequency oscillations between extreme categories; and Entity Z maintains relative stability around category 3 with moderate fluctuations.

This categorical trajectory visualization enables direct comparison of evolutionary patterns while maintaining interpretability for policy analysis and comparative assessment.

Figure 2 demonstrates the variation of the two objects across clusters when compartmentalized using the three features that are part of our dataset. When the objects are clustered using feature $f_1$, we observe an increasing trend for X, a decreasing trend for Z, and a more arbitrary trend for Y. When clustered using $f_2$, we observe a significant shift in the cluster position for X, and a vacillating trend between extremities for Y. For our calculated feature, we see object X not showing major deviations from its trends across $f_1$ and $f_2$ while Y and Z have seen a sharp rise in volatility in their oscillations.

4.4. Distance Metric Implementation

Our broader objective through this cluster-based approach is to assess the dissimilarity in the trends vis-à-vis the category assigned to objects over a period of time. For this, we are using regular Hamming distance and modified Hamming distance using differences in cluster values since the clusters have an idiosyncratic meaning. The modified Hamming is computed by taking the absolute value of the difference in the cluster positions and adding them.

The standard Hamming distance treats all categorical differences as equal (binary: different = 1, same = 0). Our modification incorporates ordinal structure by weighting differences based on categorical proximity:

• Standard Hamming Distance:

$$d_h(x,y)=\sum _{i=1}^n\delta (x_i,y_i)\mathrm{where}\delta (x_i,y_i)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{x}_i\ne y_i\hfill \\ 0\hfill & \mathrm{if}{x}_i=y_i\hfill \end{array}\right.$$

• Modified (Weighted) Hamming Distance:

$$d_{mh}(x,y)=\sum _{i=1}^n|x_i-y_i|$$

The modification replaces binary difference indicators with absolute categorical distances, recognizing that moving from category $1\to 2$ represents a smaller change than $1\to 5$. This provides a more nuanced similarity assessment for ordinal categorical trajectories.

If all weights $w_i=1$, then the weighted Hamming distance coincides with the Manhattan distance ($L_1$-norm). The weights allow us the flexibility to assign different importance to different features. We will use the notation $H(X,Y)$ to denote the regular Hamming distance and $H_W(X,Y)$ to denote the weighted Hamming distance.

The

Table 4. Pairwise comparison of regular and modified Hamming distance.

Index	${\mathit{f}}_1$	${\mathit{f}}_2$
$H(X,Y)$	1.00	0.67
$H(Y,Z)$	0.75	0.92
$H(Z,X)$	0.75	0.83
$H_W(X,Y)$	1.00	0.71
$H_W(Y,Z)$	0.67	0.67
$H_W(Z,X)$	0.58	0.63

compares regular and modified Hamming distance measures across feature pairs for all entity combinations. Regular Hamming distance captures binary differences in categorical positions, while modified Hamming distance incorporates ordinal proximity through weighted differences. The results reveal systematic patterns: Entity X demonstrates maximum dissimilarity from others, reflecting its unique evolutionary trajectory. The Y vs./Z comparison shows notable divergence between metrics (0.75 vs. 0.67 for $f_1$), indicating that while these entities occupy different categorical states, their ordinal proximity reduces weighted distance measures. These findings validate the utility of modified Hamming distance for categorical trajectory analysis.

Country X exhibits a unique developmental pathway, with transition patterns markedly distinct from established norms. This suggests an alternative adaptation mechanism possibly driven by different internal constraints or external pressures than those affecting countries Y and Z. The pronounced discrepancy between regular and modified metrics for the $Y-Z$ relationship reveals a deceptive similarity: while these systems occupy comparable states across the observation period, their transition timing differs substantially. This asynchronicity in state transitions represents a crucial consideration for policy intervention design, as it indicates differing response latencies to external stimuli.

The global average global Hamming distance is calculated by taking the cumulative average of the Hamming distances calculated for all the objects.

This aggregate Hamming distance measure

• Provides a global assessment of dissimilarity across the entire system;
• Enables comparison between different systems or time periods;
• Can serve as a summary statistic for complex multi-entity evolutionary dynamics.

The

Table 5. Comparison of regular and modified Hamming for the features.

Index	${\mathit{f}}_1$	${\mathit{f}}_2$
Regular Hamming	0.83	0.81
Modified Hamming	0.75	0.67

summarizes aggregate distance measures across the entire system. Regular Hamming distances (0.83 for $f_1$, 0.81 for $f_2$) indicate substantial overall heterogeneity in categorical trajectories, while modified Hamming distances (0.75 for $f_1$, 0.67 for $f_2$) reflect the ordinal structure’s moderating effect on perceived dissimilarity. The consistent reduction in modified distances suggests that categorical proximity provides meaningful information beyond binary state differences, supporting the theoretical foundation for ordinal-aware distance measures in temporal trajectory analysis.

Distance Metric Theoretical Foundation:

The modified Hamming distance preserves ordinal structure through mathematical formalization that extends classical categorical similarity measures [15,16]:

Regular Hamming Distance:

$$d_H(x,y)=\sum _{i=1}^n\delta (x_i,y_i)\mathrm{where}\delta (x_i,y_i)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{x}_i\ne y_i\hfill \\ 0\hfill & \mathrm{if}{x}_i=y_i\hfill \end{array}\right.$$

(5)

Modified Hamming Distance (Ordinal-Aware):

$$d_{MH}(x,y)=\sum _{i=1}^n|x_i-y_i|$$

(6)

Theoretical Properties:

• Non-negativity: $d_{MH}(x,y)\ge 0$;
• Symmetry: $d_{MH}(x,y)=d_{MH}(y,x)$;
• Triangle inequality: $d_{MH}(x,z)\le d_{MH}(x,y)+d_{MH}(y,z)$;
• Ordinal preservation: $d_{MH}$ captures magnitude of categorical differences.

This formalization ensures mathematical rigor while maintaining interpretative clarity for policy applications, addressing the limitations of binary categorical distance measures that treat all category differences as equivalent [38].

The average Hamming distance values show how on the calculated feature the objects tend to be more similar that the independent features.

4.5. Methodological Significance and Implications

The methodology demonstrates statistical robustness through

• Utilization of quartile-based boundaries, which are resistant to outliers;
• Transformation of continuous variables into categorical classifications, reducing sensitivity to measurement noise;
• Employment of non-parametric distance metrics, avoiding assumptions about data distribution.

The approach enhances interpretative accessibility by

• Converting abstract numerical values into semantically meaningful categories;
• Providing visual representations of categorical trajectories;
• Quantifying similarities and differences through intuitive distance metrics.

The prototype implementation reveals several conceptual insights:

• Pattern Recognition: The methodology effectively distinguishes between different evolutionary patterns (monotonic increase, monotonic decrease, and random fluctuation).
• Temporal Coherence: The approach identifies temporal coherence (or lack thereof) between different entities’ trajectories.
• Dimensional Reduction: By transforming continuous variables into categorical classifications, the methodology achieves dimensional reduction while preserving essential pattern information.
• System Dynamics: The aggregate distance measures provide insights into overall system dynamics and the degree of heterogeneity in evolutionary trajectories.

Country-Specific Weight Implementation: For each country pair $(i,j)$, the weighted distance becomes

$$d_{WH}(T_i,T_j)=\sum _{t=1963}^{2022}{w}_t·|c_{i,t}-c_{j,t}|$$

where $c_{i,t}$ represents the categorical debt level for country i in year t, and $w_t$ represents the temporal weight assigned to year t. This formulation ensures that the weighting scheme is applied consistently across all country comparisons while allowing for different temporal emphasis based on analytical objectives.

4.6. Comprehensive Trajectory Metrics Analysis

Having established the categorical transformation and visualization framework, we now demonstrate the complete analytical capabilities of the Hamming diversification index through systematic calculation of trajectory metrics that quantify distinct aspects of temporal evolution patterns. These metrics provide the foundation for comparative analysis across entities and serve as the building blocks for the diversification index calculation.

4.6.1. Trajectory Metrics: Theoretical Foundation and Implementation

The Hamming diversification index framework that we introduce in this paper incorporates eight complementary metrics that capture distinct dimensions of categorical trajectory behavior. Each metric addresses specific characteristics of temporal evolution, enabling comprehensive characterization of entity-specific patterns and facilitating meaningful comparative analysis across different evolutionary dynamics.

• Volatility: Volatility measures the categorical transition instability through the standard deviation of absolute changes between consecutive time periods. This metric captures the degree of unpredictability in categorical transitions, with higher values indicating more erratic movement patterns.

  $$\mathrm{Volatility}\left({\sigma }_v\right)=\sqrt{{\displaystyle \frac{1}{(n-1)}}\sum _{i=2}^n\left(\right|c_i-c_{i-1}{|-\mu )}^2}$$

  (7)

  where $\mu ={\displaystyle \frac{1}{n-1}}{\sum }_{i=2}^n|c_i-c_{i-1}|$ represents the mean absolute categorical change, and $c_i$ denotes the categorical value at time i. This formulation captures both the magnitude and consistency of categorical transitions, providing insights into the stability of evolutionary patterns.
• Number of Steps (Discrete Transitions): This metric quantifies the total number of discrete categorical transitions throughout the observation period, serving as an indicator of trajectory dynamism.

  $$\mathrm{Steps}\left(S\right)=\sum _{i=2}^n\mathbf{1}[c_i\ne c_{i-1}]$$

  (8)

  where $\mathbf{1}[·]$ represents the indicator function. Higher step counts indicate more frequent categorical changes, suggesting adaptive or responsive behavioral patterns, while lower counts suggest persistence within categorical states.
• Trajectory Changes (Directional Reversals): This metric captures the number of directional reversals in categorical movement, identifying periods where upward trends reverse to downward trends or vice versa. It provides insights into policy regime instability and adaptive responses to changing conditions.

  $$\mathrm{Trajectory}\mathrm{Changes}\left(TC\right)=\sum _{i=3}^n\mathbf{1}[\mathrm{sign}(c_i-c_{i-1})\ne \mathrm{sign}(c_{i-1}-c_{i-2})]$$

  (9)
  This formulation identifies points where the direction of categorical movement changes, capturing oscillatory behavior and trend reversals that may indicate strategic pivots or external pressure responses.
• Range Span (Categorical Breadth): Range span quantifies the breadth of categorical states experienced throughout the observation period, providing insights into the scope of evolutionary dynamics.

  $$\mathrm{Range}\mathrm{Span}\left(R\right)=max{\left\{c_i\right\}}_{i=1}^n-min{\left\{c_i\right\}}_{i=1}^n$$

  (10)
  Larger range spans indicate entities that traverse multiple categorical states, suggesting greater adaptability or exposure to diverse conditions, while smaller spans indicate constrained evolutionary patterns.
• Trend Strength (Directional Consistency): Trend strength measures the overall directional consistency of categorical evolution through linear regression slope analysis, capturing long-term systematic patterns.

  $$\mathrm{Trend}\mathrm{Strength}\left(TS\right)=\left|\beta \right|,\mathrm{where}{c}_i=\alpha +\beta t_i+{ϵ}_i$$

  (11)
  The absolute value of the regression slope provides a scale-invariant measure of directional consistency, with higher values indicating stronger systematic trends regardless of direction.
• Persistence (Categorical Stability): Persistence quantifies the proportion of time spent maintaining the same categorical state, providing insights into institutional or behavioral stability.

  $$\mathrm{Persistence}\left(P\right)={\displaystyle \frac{{\sum }_{i=2}^n\mathbf{1}[c_i=c_{i-1}]}{n-1}}$$

  (12)
  Higher persistence values indicate stable categorical positioning, while lower values suggest frequent state transitions and dynamic adjustment patterns.
• Extreme Visits (Boundary Exposure): This metric counts observations in extreme categorical boundaries (categories 0 and 5), indicating exposure to exceptional conditions.

  $$\mathrm{Extreme}\mathrm{Visits}\left(EV\right)=\sum _{i=1}^n(\mathbf{1}[c_i=0]+\mathbf{1}[c_i=5])$$

  (13)
  Higher extreme visit counts suggest exposure to crisis conditions or exceptional performance periods, providing insights into risk exposure and exceptional state management.

4.6.2. Empirical Results: Simple Example Analysis

Applying these comprehensive metrics to our synthetic dataset reveals distinct evolutionary patterns that demonstrate the framework’s discriminatory power.

Table 6. Comprehensive yrajectory metrics: simple example analysis.

Entity	Feature	Volatility	Steps	Trajectory Changes	Range Span	Trend Strength	Persistence
X	$f_1$	0.3857	2	0	2	0.2238	0.8182
	$f_2$	0.6166	2	0	3	0.3636	0.8182
Y	$f_1$	0.4454	3	0	3	0.3077	0.7273
	$f_2$	0.6556	11	10	3	0.0455	0.0909
Z	$f_1$	0.5774	9	6	3	0.0350	0.1818
	$f_2$	0.4454	5	4	1	0.0350	0.5455

presents the complete trajectory metrics analysis for entities X, Y, and Z across features $f_1$ and $f_2$.

Let us examine X, Y and Z in more detail:

• Entity X—Systematic Progression Pattern

Entity X demonstrates highly consistent evolutionary patterns across both features, characterized by exceptional persistence (0.8182) and minimal trajectory changes (0 for both features). The moderate trend strength (0.2238 for $f_1$, 0.3636 for $f_2$) combined with low step counts (2 for both features) indicates systematic, gradual progression through categorical states. The absence of extreme visits and moderate range spans (2–3) suggest stable institutional frameworks that accommodate sustained advancement without exposure to crisis conditions.

• Entity Y—Dual-Pattern Evolution

Entity Y exhibits contrasting behavioral patterns across features, demonstrating the framework’s ability to capture multi-dimensional complexity. For $f_1$, Entity Y shows moderate stability (persistence: 0.7273) with systematic decline (trend strength: 0.3077), reflecting controlled adjustment processes. However, $f_2$ reveals extraordinary volatility (0.6556) with extensive categorical transitions (11 steps, 10 trajectory changes) and minimal persistence (0.0909), indicating highly responsive or reactive behavior patterns. This dual-pattern evolution suggests sophisticated adaptive mechanisms that maintain stability in core dimensions while allowing flexibility in secondary characteristics.

• Entity Z—High-Frequency Adjustment Pattern

Entity Z demonstrates the most dynamic evolutionary pattern, characterized by moderate to high volatility (0.5774 for $f_1$, 0.4454 for $f_2$) and frequent categorical transitions (9 steps for $f_1$, 5 for $f_2$). The multiple trajectory changes (6 for $f_1$, 4 for $f_2$) combined with low persistence (0.1818 for $f_1$, 0.5455 for $f_2$) indicate active adjustment strategies with frequent directional reversals. The constrained range span for $f_2$ (1) suggests effective boundary management despite high adjustment frequency.

The proposed volatility metrics provide a simple visualization of changes in pattern similarity.

We believe that it is easier to capture the changing nature of patterns with the proposed methodology as opposed to standard statistical techniques.

4.6.3. Methodological Implications and Validation

The comprehensive metrics analysis demonstrates several key capabilities of the Hamming diversification index framework:

• Pattern Discrimination

The framework successfully distinguishes between systematic progression (Entity X), dual-pattern evolution (Entity Y), and high-frequency adjustment (Entity Z) patterns, providing nuanced characterization of different evolutionary strategies.

• Multi-Dimensional Analysis

The feature-specific metric variations (particularly Entity Y’s contrasting $f_1$ and $f_2$ patterns) demonstrate the framework’s capacity to capture complex, multi-dimensional evolutionary dynamics that would be obscured by univariate approaches.

• Ordinal Structure Preservation: The consistent differences between regular and modified Hamming distances validate the importance of incorporating ordinal proximity in categorical trajectory analysis, providing more accurate similarity assessments.
• Scalability and Interpretability:
  All metrics remain interpretable within [0,1] bounds or provide meaningful count-based measures, ensuring accessibility for policy analysis while maintaining analytical rigor.

This comprehensive analysis establishes the foundation for applying the Hamming diversification index to real-world datasets, where similar metric patterns can provide insights into institutional performance, policy effectiveness, and comparative evolutionary dynamics across diverse entities and time periods.

5. Real-World Application: Government Debt Analysis

5.1. Dataset Description

To demonstrate the practical efficacy of our Hamming diversification index methodology, we apply our framework to the Comparative Political Data Set (CPDS) spanning from 1960 to 2022. This comprehensive dataset, curated by [39], provides a robust foundation for analyzing temporal patterns in economic indicators across multiple countries over an extended period.

The CPDS dataset contains

• Temporal Coverage: 63 years of annual observations (1960–2022);
• Geographic Scope: 36 developed economies;
• Feature Dimensions: 335 variables covering political, economic, and social indicators;
• Primary Variable: Government debt as percentage of GDP (debt_hist).

Government debt trajectories, for instance, provide valuable insights into fiscal policy effectiveness, crisis responses, and long-term sustainability patterns [17,40]. 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 [18]. For our analysis, we selected eight representative countries that demonstrate diverse fiscal trajectories and economic philosophies, enabling comprehensive evaluation across different debt-to-GDP categories as identified in the empirical literature [17,41]:

High-Debt Trajectory Countries:

• United States: Post-2008 financial crisis debt accumulation, with debt-to-GDP ratio reaching approximately 133% by recent estimates [17], reflecting extensive stimulus policies and structural fiscal challenges;
• Japan: Sustained exceptionally high debt levels exceeding 250% of GDP, primarily driven by demographic challenges and prolonged economic stagnation following the 1990s asset price bubble collapse [42];
• United Kingdom: Variable debt patterns significantly influenced by economic cycles, financial sector developments, and Brexit-related uncertainties [41];
• France: Gradual debt increases reflecting comprehensive welfare state policies and counter-cyclical fiscal responses, consistent with continental European fiscal approaches [18];
• Germany: Moderate debt levels maintained through institutional fiscal discipline, particularly following reunification adjustments and European fiscal coordination mechanisms.

Fiscally Conservative Countries:

• Denmark: Consistent fiscal discipline achieved through structural reforms and adherence to European stability frameworks, maintaining debt levels well below the 60% GDP threshold established by the Stability and Growth Pact;
• Switzerland: Traditional low-debt approach supported by constitutional fiscal rules and decentralized fiscal governance structures, with debt-to-GDP ratio around 32% [41];
• Norway: Resource-wealth management through sovereign wealth fund mechanisms, enabling counter-cyclical fiscal policies while maintaining relatively low net debt positions despite gross debt levels around 55% of GDP [43].

This balanced selection enables comprehensive evaluation of our methodology across different economic models and fiscal philosophies, ensuring robust testing of the Hamming diversification index across diverse evolutionary trajectories. The country selection reflects the empirical literature on fiscal policy variations, incorporating both high-debt economies experiencing fiscal stress and conservative fiscal regimes maintaining sustainable debt dynamics [17,18].

5.2. Methodology Adaptation to Real-World Data

The transition from our synthetic demonstration dataset to real-world economic data required several methodological adaptations while preserving the core theoretical framework.

Unlike the controlled synthetic environment, real-world data presented several challenges:

• Missing Values: Handled through forward-fill imputation using available debt values;
• Temporal Alignment: Standardized annual observations across all countries;
• Scale Normalization: Maintained original debt-to-GDP percentages for interpretability.

We adapted our feature construction approach by focusing exclusively on the primary debt indicator ($f_1$), eliminating the complexity of multi-feature weighted averages used in the synthetic example. This simplification allows for

• Direct interpretation of categorical transitions in terms of fiscal policy changes.
• Clear identification of debt crisis periods and fiscal consolidation efforts.
• Transparent comparison across countries with different economic structures.

The categorical transformation employed global quartile boundaries calculated across all countries and time periods:

$$\begin{array}{cc}\hfill \mathrm{Lower}\mathrm{Bound}& =Q_1-1.5\times IQR=-21.15\%\hfill \\ \hfill Q_1& =22.73\%\hfill \\ \hfill Q_2\left(\mathrm{Median}\right)& =44.16\%\hfill \\ \hfill Q_3& =66.80\%\hfill \\ \hfill \mathrm{Upper}\mathrm{Bound}& =Q_3+1.5\times IQR=132.92\%\hfill \end{array}$$

(14)

This global approach ensures consistent categorical interpretation across all entities, enabling meaningful cross-country comparisons while preserving the relative position of each country within the broader fiscal landscape.

Each country’s fiscal trajectory was constructed as a sequence of categorical assignments over the 63-year observation period. The trajectory $T_i$ for country i is defined as follows:

$$T_i=(c_{i,1960},c_{i,1961},\dots ,c_{i,2022})$$

(15)

where $c_{i,t}\in \{0,1,2,3,4,5\}$ represents the categorical assignment for country i in year t.

5.3. Implementation and Results

The categorical transformation of government debt trajectories reveals distinct fiscal evolution patterns across the eight selected countries. Figure 3 presents individual subplot analysis for each country, demonstrating the temporal progression through categorical debt levels from 1960 to 2022.

The Figure 3 presents individual categorical trajectory analysis for eight representative countries spanning 1960–2022, demonstrating diverse fiscal evolution patterns. Japan exhibits the most dramatic transformation, transitioning from low-debt categories (1–2) in the 1980s to consistently high-debt categories (5) post-2000, reflecting prolonged economic stagnation and demographic pressures. The United States shows a clear structural break circa 2008, shifting from moderate categories (3–4) to high categories (5), corresponding to financial crisis response. European economies display varied patterns: Germany maintains stable moderate positioning with temporary increases, France shows gradual progression toward higher categories, while the United Kingdom exhibits significant volatility during 1970s–1990s before stabilizing. Fiscally conservative countries demonstrate distinct patterns: Denmark maintains predominantly low categories with occasional moderate periods, Switzerland consistently operates in low-to-moderate ranges, while Norway shows the highest volatility among conservative countries, reflecting oil revenue management strategies. These categorical trajectories effectively capture fundamental differences in long-term fiscal policy approaches across diverse economic systems.

The visualization reveals several key trajectory patterns:

Sustained High-Debt Trajectories:

• Japan: Demonstrates the most dramatic categorical transition, moving from “LB < val < $Q_1$” in the 1980s to consistently “Above UB” post-2000, reflecting the prolonged impact of demographic challenges and economic stagnation.
• United States: Shows a clear structural break around 2008, transitioning from “$Q_2$ < val < $Q_3$” to “Above UB” categories, corresponding to the financial crisis response and subsequent fiscal expansion.

Moderate Debt Evolution:

• Germany: Exhibits relatively stable categorical positioning in “$Q_2$ < val < $Q_3$” and “Q3 < val < UB” ranges, with temporary increases during reunification and financial crisis periods.
• France: Displays gradual categorical progression from “$Q_2$ < val < $Q_3$” to “Above UB” over the observation period, reflecting consistent welfare state fiscal policies.
• United Kingdom: Shows significant categorical volatility, particularly during the 1970s–1990s period, with eventual stabilization in higher categories.

Fiscally Conservative Patterns:

• Denmark: Maintains predominantly low categorical positions with periods in “$Q_2$ < val < $Q_3$”, demonstrating successful fiscal consolidation policies.
• Switzerland: Consistently operates within “LB < val < $Q_1$” to “$Q_2$ < val < $Q_3$” categories, exemplifying traditional fiscal conservatism.
• Norway: Exhibits the most volatile categorical pattern among conservative countries, reflecting oil revenue management policies and counter-cyclical fiscal approaches.

In our empirical analysis, the distinction between H(X,Y) and HW(X,Y) reported in Table 4, Table 6 and

Table 7. Pairwise comparison of Hamming and modified Hamming distances for countries.

Country Pair	$\mathit{H}(\mathit{X},\mathit{Y})$	${\mathit{H}}_{\mathit{W}}(\mathit{X},\mathit{Y})$
USA-Germany	0.6349	0.3730
USA-Japan	0.8254	0.5397
USA-United Kingdom	0.5079	0.3175
USA-France	0.6667	0.3333
USA-Denmark	0.8095	0.5238
USA-Switzerland	0.6825	0.5794
USA-Norway	0.7705	0.6393
Germany-Japan	0.7143	0.5000
Germany-United Kingdom	0.6032	0.4683
Germany-France	0.5397	0.3889
Germany-Denmark	0.5873	0.3889
Germany-Switzerland	0.5714	0.3968
Germany-Norway	0.8033	0.5656
Japan-United Kingdom	0.9048	0.8095
Japan-France	0.7619	0.5397
Japan-Denmark	0.4921	0.4921
Japan-Switzerland	0.7778	0.8492
Japan-Norway	0.8852	1.0164
United Kingdom-France	0.7143	0.4286
United Kingdom-Denmark	0.9365	0.7778
United Kingdom-Switzerland	0.6825	0.6111
United Kingdom-Norway	0.7213	0.5902
France-Denmark	0.8413	0.6984
France-Switzerland	0.8571	0.6746
France-Norway	0.8525	0.7541
Denmark-Switzerland	0.6667	0.4841
Denmark-Norway	0.8033	0.6885
Switzerland-Norway	0.5082	0.3197

refers to:

Regular Hamming Distance H(X,Y):

$$d_H(x,y)=\sum _{i=1}^n\delta (x_i,y_i)\mathrm{where}\delta (x_i,y_i)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{x}_i\ne y_i\hfill \\ 0\hfill & \mathrm{if}{x}_i=y_i\hfill \end{array}\right.$$

Modified Hamming Distance HW(X,Y):

$$d_{MH}(x,y)=\sum _{i=1}^n|x_i-y_i|$$

The “W” notation indicates our ordinal-aware modification that captures the magnitude of categorical differences, not temporal weighting. All time periods receive equal consideration in both measures.

The weighted notation HW(X,Y) in our tables refers specifically to the ordinal-aware distance calculation that weighs categorical differences by their numerical distance rather than treating them as binary. No temporal weighting scheme is applied—each year from 1960 to 2022 contributes equally to the trajectory comparison.

The comprehensive pairwise analysis across all 28 possible country combinations yields detailed insights into trajectory similarities and divergences. Table 7 presents the complete matrix of regular and modified Hamming distances.

The Table 7 presents comprehensive pairwise Hamming distance analysis across all 28 possible country combinations, revealing trajectory similarities and divergences in fiscal policy evolution. The analysis identifies distinct relationship clusters: Switzerland–Norway demonstrate highest similarity (Regular: 0.5082, Modified: 0.3197), suggesting convergent fiscal approaches despite different resource endowments. Conversely, United Kingdom–Denmark exhibit maximum divergence (Regular: 0.9365, Modified: 0.7778), reflecting fundamentally different fiscal philosophies. European economies show moderate convergence patterns, with Germany–France (0.5397) and Germany–Denmark (0.5873) indicating some policy coordination effects. Japan’s unique trajectory results in high distances with most countries, particularly Norway (1.0164 modified distance), confirming its exceptional fiscal evolution following economic bubble collapse. The systematic differences between regular and modified distances highlight the importance of ordinal proximity in categorical trajectory comparison.

The aggregate analysis reveals substantial heterogeneity in fiscal trajectory patterns across the selected countries.

Table 8. Summary statistics for trajectory dissimilarity analysis.

Metric	Value
Total Countries Analyzed	8
Total Pairwise Comparisons	28
Coverage	100%
Aggregate $H(X,Y)$	0.7186
Aggregate $H_W(X,Y)$	0.5624
$H{(X,Y)}_{min}$	0.4921
$H{(X,Y)}_{max}$	0.9365
$H{(X,Y)}_{std}$	0.1258
$H_W{(X,Y)}_{min}$	0.3175
$H_W{(X,Y)}_{max}$	1.0164
$H_W{(X,Y)}_{std}$	0.1720

summarizes the comprehensive distance metrics.

The Table 8 provides summary statistics for trajectory dissimilarity analysis across the eight-country sample. The high aggregate distances (Regular: 0.7186, Modified: 0.5624) confirm substantial heterogeneity in fiscal trajectory evolution, validating the discriminatory power of the Hamming diversification index. The wide range of distance values (Regular: 0.4921–0.9365, Modified: 0.3175–1.0164) demonstrates the framework’s sensitivity to different evolutionary patterns. Standard deviations (Regular: 0.1258, Modified: 0.1720) indicate moderate dispersion around mean distances, suggesting balanced representation of both similar and divergent trajectory pairs. Complete coverage (100%) across all pairwise comparisons ensures comprehensive assessment of inter-country fiscal policy relationships, providing robust foundation for comparative economic analysis.

The distance analysis reveals several important patterns:

• Most Similar Fiscal Trajectories: Switzerland and Norway exhibit the highest similarity (Regular: 0.5082, Modified: 0.3197), despite Norway’s oil wealth, suggesting convergent fiscal management approaches among resource-rich and traditionally conservative economies.
• Greatest Trajectory Divergence: The United Kingdom and Denmark demonstrate the most divergent patterns (Regular: 0.9365, Modified: 0.7778), reflecting fundamentally different approaches to fiscal policy and economic management over the observation period.
• Moderate Convergence Clusters: Germany demonstrates relatively similar trajectories to France (0.5397 regular distance) and Denmark (0.5873 regular distance), suggesting some convergence in European fiscal approaches despite different starting positions.
• Exceptional Trajectory Patterns: Japan’s unique fiscal evolution results in high distances with most countries, particularly Norway (1.0164 modified distance), reflecting the singular nature of Japan’s prolonged debt accumulation following its economic bubble collapse. The high aggregate distances (0.7186 regular, 0.5624 modified) confirm substantial heterogeneity in fiscal trajectory evolution, validating the discriminatory power of the Hamming diversification index in capturing meaningful differences in long-term fiscal policy patterns across developed economies.

5.4. Individual Country Trajectory Metrics Analysis

The Hamming diversification index framework incorporates eight complementary metrics to provide comprehensive characterization of categorical trajectory patterns. These metrics capture distinct aspects of temporal evolution, enabling detailed comparative analysis of fiscal policy dynamics across countries. Each trajectory metric quantifies specific characteristics of categorical evolution patterns:

Formal Trajectory Metrics Definitions:

Each trajectory metric quantifies specific characteristics of categorical evolution patterns, enabling detailed comparative analysis of temporal dynamics [33,36]:

• Volatility:

  $${\sigma }_v=\sqrt{{\displaystyle \frac{1}{(n-1)}}{\sum }_{i=2}^n\left(\right|c_i-c_{i-1}{|-\mu )}^2}$$

  (16)

  where $\mu ={\displaystyle \frac{1}{n-1}}{\sum }_{i=2}^n|c_i-c_{i-1}|$ represents mean absolute categorical change.
• Number of Steps:

  $$S=\sum _{i=2}^n\mathbf{1}[c_i\ne c_{i-1}]$$

  (17)

  where $\mathbf{1}[·]$ is the indicator function counting discrete categorical transitions.
• Trajectory Changes:

  $$TC=\sum _{i=3}^n\mathbf{1}[\mathrm{sign}(c_i-c_{i-1})\ne \mathrm{sign}(c_{i-1}-c_{i-2})]$$

  (18)

  quantifying directional reversals in categorical movement, capturing policy regime instability.
• Range Span:

  $$R=max{\left\{c_i\right\}}_{i=1}^n-min{\left\{c_i\right\}}_{i=1}^n$$

  (19)
• Trend Strength:

  $$TS=\left|\beta \right|\mathrm{where}\beta \mathrm{is}\mathrm{the}\mathrm{slope}\mathrm{from}:c_i=\alpha +\beta t_i+{\epsilon }_i$$

  (20)
• Persistence:

  $$P={\displaystyle \frac{{\sum }_{i=2}^n\mathbf{1}[c_i=c_{i-1}]}{n-1}}$$

  (21)
• Extreme Visits:

  $$EV=\sum _{i=1}^n(\mathbf{1}[c_i=0]+\mathbf{1}[c_i=5])$$

  (22)

  counting observations in boundary categories (Below LB or Above UB), indicating exposure to exceptional fiscal conditions.

These formalized metrics provide comprehensive characterization of categorical trajectory patterns, enabling systematic comparison of temporal evolution dynamics across different entities and time periods [32].

Table 9. Individual country trajectory metrics: comprehensive fiscal evolution analysis.

Country	Volatility	Steps	Trajectory Changes	Range Span	Trend Strength	Persistence	Extreme Visits	Sequence Length
USA	0.2457	4	1	3.0	0.0347	0.9355	3	63
Germany	0.3165	7	4	3.0	0.0609	0.8871	0	63
Japan	0.2957	6	2	4.0	0.0860	0.9032	24	63
United Kingdom	0.5264	17	11	4.0	0.0194	0.7258	3	63
France	0.3165	7	1	4.0	0.0460	0.8871	7	63
Denmark	0.3951	12	6	3.0	0.0355	0.8065	0	63
Switzerland	0.3954	8	5	2.0	0.0189	0.8710	0	63
Norway	0.5323	29	23	2.0	0.0048	0.5167	0	61

presents comprehensive trajectory metrics for all eight countries, revealing distinct fiscal evolution patterns that align with established economic narratives and policy frameworks.

The trajectory metrics reveal four distinct fiscal evolution patterns that correspond to established economic paradigms and policy frameworks:

• Stable High-Debt Trajectories (USA, Japan): The United States demonstrates exceptional categorical persistence (0.9355) with minimal trajectory changes (1), reflecting stable institutional frameworks that accommodate sustained high debt levels. The low volatility (0.2457) combined with moderate trend strength (0.0347) indicates systematic debt accumulation following the 2008 financial crisis without frequent policy reversals. Japan exhibits the most dramatic long-term transformation, with the highest trend strength (0.0860) and maximum range span (4.0), capturing its transition from low-debt to extraordinarily high-debt categories. The exceptional extreme visits (24) confirm Japan’s unique position in the “Above UB” category, reflecting prolonged economic stagnation and demographic pressures requiring sustained fiscal expansion.
• Moderate Volatility European Economies (Germany, France): Germany shows balanced trajectory characteristics with moderate volatility (0.3165) and multiple trajectory changes (4), indicating responsive fiscal policy within institutional constraints. The absence of extreme visits demonstrates successful adherence to European fiscal frameworks while maintaining flexibility for economic stabilization. France exhibits similar volatility patterns but with minimal trajectory changes (1), suggesting gradual, consistent fiscal expansion reflecting comprehensive welfare state policies. The moderate extreme visits (7) indicate occasional exposure to high-debt categories while maintaining overall fiscal stability.
• High-Volatility Economies (United Kingdom, Norway): The United Kingdom demonstrates the highest categorical volatility (0.5264) with extensive trajectory changes (11), reflecting the impacts of economic cycles, financial sector developments, and Brexit-related uncertainties. The low persistence (0.7258) and high number of steps (17) confirm frequent categorical transitions, indicating reactive fiscal policy responses to changing economic conditions. Norway exhibits extraordinary instability despite fiscal conservatism, with the highest number of steps (29) and trajectory changes (23), reflecting sophisticated counter-cyclical fiscal management enabled by oil revenue. The lowest persistence (0.5167) indicates active fiscal policy adjustment, yet the limited range span (2.0) and zero extreme visits demonstrate effective constraint within sustainable bounds.
• Fiscally Conservative Economies (Denmark, Switzerland): Denmark displays moderate volatility (0.3951) with balanced trajectory changes (6), indicating successful fiscal consolidation policies within European frameworks. The absence of extreme visits and moderate range span (3.0) demonstrate adherence to fiscal rules while maintaining policy flexibility. Switzerland exhibits the most constrained fiscal trajectory with the smallest range span (2.0) and zero extreme visits, exemplifying traditional fiscal conservatism. The moderate number of steps (8) and trajectory changes (5) reflect deliberate, measured fiscal adjustments within constitutional fiscal constraints.

The above metrics analysis reveals systematic differences in fiscal policy approaches that align with institutional frameworks, economic structures, and policy traditions. Countries with strong institutional constraints (Switzerland, Denmark) demonstrate limited range spans and zero extreme visits, while economies with flexible fiscal frameworks (USA, Japan) show persistent high-debt patterns with fewer trajectory changes. The volatility-persistence trade-off demonstrates that countries employing active counter-cyclical policies (Norway, UK) sacrifice categorical stability for responsive fiscal management, while countries with stable institutional frameworks (USA, Japan) maintain high persistence despite elevated debt levels.

These findings validate the discriminatory power of the Hamming diversification index in capturing meaningful differences in long-term fiscal policy evolution, providing a robust foundation for comparative economic analysis and policy evaluation frameworks.

6. Future Scope

The Hamming diversification index methodology presented in this research establishes a foundational framework for categorical temporal trajectory analysis, yet numerous avenues for theoretical advancement and empirical extension warrant systematic exploration. The following research directions represent both natural extensions of our core methodology and novel applications that could substantially expand the analytical scope and practical utility of clustering-based temporal pattern analysis. While our quartile-based categorization approach demonstrates robust performance across diverse temporal patterns, future research should investigate alternative statistical transformation methods that may better capture domain-specific temporal dynamics. The development of adaptive categorization schemes that dynamically adjust boundary conditions based on local temporal characteristics represents a particularly promising direction. Such approaches could incorporate rolling statistical windows, regime-change detection algorithms, or hierarchical clustering methods to establish more nuanced categorical boundaries that evolve with underlying data structures. Furthermore, the integration of domain-specific knowledge into categorical boundary determination merits systematic investigation, where statistical boundaries are augmented with contextually relevant thresholds derived from theoretical frameworks or practical experience, particularly valuable in policy-oriented applications where predetermined categorical thresholds carry substantive meaning for decision-making processes.

The current implementation employs univariate trajectory analysis with equal weighting across temporal periods, suggesting substantial opportunities for methodological advancement through sophisticated multidimensional distance metrics that can simultaneously analyze multiple correlated time series while preserving the interpretative advantages of categorical transformation. The development of tensor-based Hamming distance measures could enable comprehensive analysis of high-dimensional temporal systems while maintaining computational efficiency and interpretative clarity. Additionally, the incorporation of temporal weighting schemes that assign differential importance to recent versus historical observations represents a crucial extension for applications requiring adaptive pattern recognition, implementing exponential decay functions, regime-specific weighting, or importance-weighted distance measures that emphasize particular temporal periods based on analytical objectives or domain-specific considerations. The deterministic nature of our current categorical assignment approach, while providing computational efficiency and interpretative clarity, may inadequately represent inherent uncertainty in temporal pattern classification.

Future research should develop probabilistic extensions that incorporate uncertainty quantification into categorical trajectory analysis through Bayesian approaches to categorical boundary determination, fuzzy logic implementations of categorical assignment, and Monte Carlo methods for uncertainty propagation. The development of confidence intervals for Hamming distance calculations, incorporating both measurement uncertainty and categorical boundary uncertainty, would provide more nuanced similarity assessments that acknowledge the inherent limitations of categorical transformation processes, proving particularly valuable in applications involving noisy data, incomplete observations, or inherently stochastic processes.

The demonstrated success of the Hamming diversification index on moderate-scale datasets suggests substantial potential for application to high-dimensional temporal systems involving hundreds or thousands of entities observed over extended periods. Future research should investigate computational optimization strategies, parallel processing implementations, and distributed computing approaches that can scale the methodology to contemporary big data environments while preserving analytical fidelity. The development of approximate distance computation algorithms, hierarchical clustering approaches for large-scale trajectory analysis, and streaming analytics implementations for real-time pattern recognition represent particularly promising directions for computational advancement that could enable application to massive temporal datasets in fields such as financial markets analysis, environmental monitoring, and social media dynamics, extending classical time-series methodologies [44].

The integration of our clustering-based trajectory analysis framework with contemporary machine learning methodologies offers substantial opportunities for enhanced pattern discovery and predictive modeling, complementing established forecasting frameworks [45]. Future research should explore the incorporation of deep learning architectures specifically designed for categorical sequence analysis, the development of attention mechanisms for temporal pattern recognition, and the implementation of reinforcement learning approaches for adaptive categorical boundary optimization, building upon established pattern recognition frameworks [46]. The development of automated pattern discovery algorithms that can identify previously unknown trajectory types, detect temporal anomalies, and predict future categorical transitions would substantially enhance the practical utility of the framework through unsupervised learning techniques, evolutionary algorithms, or neural architecture search methods to optimize categorical transformation and distance computation procedures for specific applications.

The successful application to government debt analysis demonstrates the framework’s potential for broader policy research applications across diverse policy domains including environmental regulation, social welfare systems, healthcare policy, and international relations where clustering methods have proven valuable for identifying situational patterns [47]. The development of domain-specific categorical schemes, policy-relevant distance metrics, and comparative analytical frameworks could establish the Hamming diversification index as a standard tool for longitudinal policy analysis. Particularly promising applications include cross-national comparative studies of policy convergence and divergence, analysis of policy diffusion mechanisms, evaluation of intervention effectiveness across different contexts, and investigation of long-term policy sustainability patterns, where the framework’s ability to capture discrete policy regime transitions while maintaining comparative analytical capability makes it especially suitable for research questions involving policy learning, institutional adaptation, and governance effectiveness. The categorical transformation approach demonstrates particular promise for financial markets analysis, where discrete regime changes, crisis dynamics, and comparative performance analysis represent central analytical challenges.

Future research should investigate applications to stock market volatility analysis, currency stability assessment, commodity price dynamics, and systemic risk evaluation through the development of finance-specific categorical boundaries based on volatility thresholds, return distributions, or risk measures that could provide novel insights into market behavior and institutional performance. The framework’s capacity to handle multiple entities over extended periods makes it particularly suitable for portfolio analysis, sector comparison studies, international financial integration research, central bank policy analysis, sovereign risk assessment, and macroeconomic convergence studies that could provide valuable insights for both academic research and practical policy formulation. The methodology’s demonstrated ability to capture complex temporal patterns suggests substantial potential for environmental and climate science applications including temperature trend analysis, precipitation pattern comparison, biodiversity dynamics, and pollution monitoring. The development of environmentally relevant categorical schemes based on ecological thresholds, tipping points, or sustainability criteria could provide novel approaches to environmental monitoring and assessment. Particularly promising applications include comparative analysis of climate adaptation strategies, evaluation of environmental policy effectiveness across different regions, assessment of ecosystem resilience patterns, and investigation of human-environment interaction dynamics, where the framework’s ability to handle long-term temporal patterns makes it especially suitable for climate change research requiring comparison of adaptation trajectories across different geographical or political contexts.

While our synthetic and real-world applications provide initial validation of the methodology’s effectiveness, future research should develop comprehensive validation frameworks that systematically assess performance across diverse temporal pattern types, data characteristics, and analytical objectives. The development of standardized benchmark datasets, comparative evaluation metrics, and robustness assessment protocols would substantially enhance the credibility and adoption potential of the methodology. Future validation studies should investigate sensitivity to parameter choices, robustness to missing data, performance under different distributional assumptions, and comparative effectiveness relative to alternative temporal pattern analysis methods, while developing automated validation procedures that can assess methodology appropriateness for specific applications to enhance practical utility while reducing implementation barriers. The establishment of systematic comparison frameworks that evaluate the Hamming diversification index relative to alternative temporal analysis methods across diverse application domains represents a crucial research priority, encompassing traditional time-series analysis methods, contemporary machine learning approaches, and domain-specific analytical techniques to establish clear understanding of comparative advantages and limitations.

7. Conclusions

This research introduces the Hamming diversification index, a novel metric for analyzing time-evolving patterns in multi-dimensional datasets through categorical transformation and trajectory-based similarity assessment. The methodology integrates quartile-based transformations, clustering, and a modified Hamming distance adapted for ordinal categories to enable interpretable, comparative, and computationally efficient temporal analysis. By converting continuous values into categorical states based on global distributions, the framework facilitates comparison across entities with differing scales or distributions, making it particularly useful for policy applications. Empirical validation using synthetic data and government debt trajectories across eight developed economies illustrates the method’s ability to identify meaningful patterns, such as fiscal convergence and divergence. The approach demonstrates robustness to outliers, flexibility across domains, and accessibility for non-technical stakeholders—essential for policy-oriented research. Limitations include potential information loss from categorical transformation and the need for more adaptive boundary setting in diverse contexts. Nevertheless, the Hamming diversification index offers a valuable, theoretically grounded tool for temporal pattern analysis, balancing analytical rigor with interpretative clarity, and holds promise for broader adoption in comparative social science and applied economic research.