## 8 Discussion  
In early Olympic medal count forecasting studies, researchers first adopted economic production models. In 2004, [PERSON] employed a Cobb-Douglas function, using national population and GDP to predict each country's medal tally [3]. Subsequently, time series approaches emerged. In 2007, [PERSON] applied ARIMA to China's medal count trajectory but found that linear models were insensitive to nonlinear dynamics [22]. In 2008, [PERSON]'s team quantified home-field advantage (\(C=0.298\)) and population effects within a regression framework [23]. In 2010, [PERSON] and in 2012, [PERSON] introduced gray relational analysis and gray forecasting models, respectively, into analyses of specific sports and athletics events [24; 26]. In 2011, [PERSON] combined genetic algorithms with regression to enhance model adaptability [25]. More recently, [PERSON] et al. (2022) showed that random forests not only improved predictive accuracy but also, via feature-importance rankings, illuminated the socioeconomic drivers of Olympic success [28]. Despite these advances, traditional single-model methods still face limitations in handling model complexity, heterogeneous data sources, and \"first-medal\" predictions for historically non-medal-winning nations.  
This study achieves significant breakthroughs on both methodological and theoretical levels. First, we innovatively integrate K-means clustering with a three-layer fully connected neural network (FMPM), enabling, for the first time, probabilistic forecasts of inaugural Olympic medals for countries with no prior medals. Second, our MPXG framework organically combines factor analysis, ARIMA, and XGBoost--precisely fitting linear trends while flexibly capturing complex nonlinear relationships while yielding narrower confidence intervals than ARIMA or univariate regression approaches and allowing trend inferences for entire clusters of nations based on limited samples. Third, we employ Bayesian linear regression to convert host country and coaching effects from simple dummy variables into quantifiable marginal contribution coefficients, deepening the quantitative interpretation of these key factors. Together, these innovations not only enhance model interpretability and generalizability but also establish a cross-disciplinary paradigm for sports big data and event prediction research.  
Our research also offers substantial practical value, providing actionable decision support tools for national sports authorities and event organizers. By forecasting the probability of a nation's first medal, governing bodies can swiftly identify and support potential dark horse contenders. Combining medal trend analysis with clustering enables the precise targeting of each country's strengths under budget constraints, optimizing training programs and funding allocation. Finally, the quantified host country and coaching effect coefficients supply objective, data-driven criteria for host nation preparations and coaching staff selection, thereby comprehensively enhancing overall competitive performance.  
This research also makes significant contributions to interdisciplinary fields involving sports. It can intersect with public policy by guiding equitable resource allocation to underrepresented nations, promoting social inclusion through sports development. In sports economics, clustering and trend analysis tools can optimize event planning, ensuring sustainable economic benefits for host cities while minimizing environmental impact [3]. In the field of education, the methodology for quantifying coaching effects can be adapted to assess teacher performance, improving training programs and enhancing student outcomes. Additionally, predictive models can be applied to sports-related public health initiatives, such as forecasting participation trends to design community fitness programs, fostering a healthier society [40]. By connecting sports events with these domains, the research provides actionable, data-driven tools that enhance inclusivity, economic efficiency, and societal well-being.