## 7 Results
### Assumptions  
To facilitate analysis and ensure the applicability of the prediction model, this study proposes the following assumptions:  
1. Continuity of Historical Trends: We assume that the patterns observed in Olympic medal distributions and athlete performance from 1896 to 2024 will continue through 2028. This assumption is supported by the consistent evolution of Olympic performance, driven by macro-level factors such as advancements in training methodologies, sports science, and global competition dynamics. For instance, the steady increase in medal counts for emerging nations (e.g., China's rise since 1984) suggests a predictable trajectory that can be modeled using historical data. However, this assumption acknowledges potential variations due to minor rule changes or the inclusion of new events, which are accounted for in the ARIMA model's forecasting of principal components.
2. Consistency in National Sports Investment: We assume that countries will maintain their levels of investment in sports infrastructure, athlete training, and talent development over the decade leading up to 2028. This assumption is critical for non-medal-winning nations (Group \(\alpha_{4}\)), where significant policy shifts could disrupt predictive accuracy. Historical data supports this assumption, as most countries exhibit stable sports funding patterns unless influenced by major economic or political upheavals, which are not anticipated based on current global trends. For example, nations like Kenya have shown consistent investment in athletics, leading to predictable medal outcomes.
3. Controllability of External Factors: The model assumes that the 2028 Los Angeles Olympics will proceed without significant disruptions, such as wars, pandemics, or major rule changes, which could alter event structures or participation. This assumption is necessary to ensure that predictions are based on a stable operational environment, consistent with the majority of Olympic Games in history since 1896. Historical disruptions (e.g., the cancellations of the 1916, 1940, and 1944 Games) are rare, and recent global stability supports the feasibility of this assumption. The model incorporates robustness checks, such as sensitivity analyses, to mitigate the impact of minor unforeseen events.
4. Historical Data Sufficiency and Exclusion of Unknown Athletes: We assume that historical Olympic data from 1896 to 2024 is sufficient to predict the medal performance of countries in 2028, and that the impact of unknown (i.e., emerging) athletes on national medal tallies is negligible. Given the study's focus on the overall performance of 234 countries, the models (MPXG and FMPM) rely on systemic factors such as participation trends, medal growth rates, and national investment, which are comprehensively captured in historical data. Although exceptional individual performances (e.g., [PERSON]'s emergence in 2008) can have a significant impact in rare cases, these occurrences are statistically infrequent and lack predictive precursorsin existing datasets, and are thus excluded from the model to maintain focus on systemic factors. Including such unpredictable events would introduce noise and reduce the model's generalizability.
5. Stability of Data Quality: We assume that the quality and structure of Olympic data available for 2028 will be consistent with historical datasets (1896-2024). This includes the accuracy of participation records, medal counts, and event categorizations, as well as the availability of comparable input metrics for model inputs. This assumption is supported by the International Olympic Committee's standardized data collection practices, which have improved significantly since the mid-20 th century. Any minor inconsistencies in future data are expected to be addressed through preprocessing techniques similar to those applied in this study, ensuring model reliability.