## 5 Real-World Application: Government Debt Analysis
### 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  
Figure 3: Categorical debt trajectory analysis for eight countries (1960–2022), highlighting distinct fiscal patterns. Japan and the U.S. exhibit structural shifts toward higher debt categories, while European and fiscally conservative countries show varied trajectories reflecting their unique policy and economic contexts. 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 < Q1\" 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 \"Q2 < val < Q3\" to \"Above UB\" categories, corresponding to the financial crisis response and subsequent fiscal expansion. * **Moderate Debt Evolution:**
* **Germany**: Exhibits relatively stable categorical positioning in \"Q2 < val < Q3\" and \"Q3 < val < UB\" ranges, with temporary increases during reunification and financial crisis periods. * **France**: Displays gradual categorical progression from \"Q2 < val < Q3\" 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 \"Q2 < val < Q3\", demonstrating successful fiscal consolidation policies. * **Switzerland**: Consistently operates within \"LB < val < Q1\" to \"Q2 < val < Q3\" 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 Tables 4, 6 and 7 refers to:  
**Regular Hamming Distance H(X,Y):**  
\[d_{H}(x,y)=\sum_{i=1}^{n}\delta(x_{i},y_{i})\text{ where }\delta(x_{i},y_{i})= \begin{cases}1&\text{if }x_{i}\
eq y_{i}\\ 0&\text{if }x_{i}=y_{i}\end{cases}