## 6 Discussion  
The results obtained demonstrate that the proposed approach, based on the integration between Random Forest and Markov chains, represents a significant step forward in flood risk prediction. The model showed high predictive performance, with an R\({}^{2}\) of 0.80 and a very low mean squared error (MSE) (0.014), even surpassing an LSTM model used as a reference. These results confirm the validity of the initial objective of the research: to develop a predictive system that was accurate, interpretable, and capable of modeling the temporal dynamics of risk. A distinctive element of our work is the use of the Markov chain to simulate the evolution of risk over time, as illustrated in Figure 16. This figure, entitled \"Simulated Flood Risk Fluctuations Over Time\", represents the temporal evolution of the simulated flood risk through the integration of the Random Forest model with a Markov chain. The simulation was generated starting from an initial state of low risk and iteratively applying the transition matrix calculated from the model predictions. Each time step represents one month, and the entire simulation covers a span of 11 years (132 months), offering a dynamic view of the evolution of risk over time. Analyzing the results, we observe that the \"High\" and \"Very High\" risk states are the most common, accounting for 28.57% and 25.56% of the time, respectively. This indicates a higher overall frequency of elevated flood risk states in the simulation, though no clear temporal trend is visually evident in the time series. In contrast, the \"Low\" risk status is the least common, representing only 20.30% of the observations. A transition matrix governs the transitions between the risk states. For instance, from the \"Low\" state, there is a 27.78% chance of moving to either the \"High\" or \"Very High\" state. This highlights that even when the risk is low, there is a significant possibility of a rapid increase in risk. The graph also illustrates monthly fluctuations among the different risk states, indicating considerable variability in flood risk every month. This variability suggests that the risk can change quickly from one month to the next, emphasizing the importance of continuous monitoring. The figure does not have forecasting purposes, but provides a probabilistic representation useful for strategic purposes, to support the planning of risk mitigation interventions. The time series shows high monthly variability, with frequent transitions between all four risk states (\"Low\", \"Moderate\", \"High\", and \"Very High\"). Although no net visual trend emerges, the aggregate analysis shows that the \"High\" and \"Very High\" states become predominant over time suggesting that, in the absence of interventions, the territory could be subject to an escalation of hydrogeological risk. The original contribution of this study is divided into three main aspects: (1) the synergistic integration between machine learning models and stochastic models, (2) the use of high-resolution LIDAR data for the generation of digital terrain models, and (3) the ability of the model to provide both spatial and temporal predictions of risk. In particular, the use of LIDAR data with altimetric accuracy of less than 15 cm has made it possible to obtain an extremely detailed topographic representation, superior to that offered by low-resolution digital terrain models commonly used in the literature. This significantly improved the quality of spatial risk classification. In addition, direct comparison with an LSTM model highlighted the higher accuracy of our approach, which resulted in superior predictive metrics with a simpler and more interpretable computational structure. Finally, temporal modeling using Markov chains added a dynamic dimension to the analysis, making it possible to simulate the evolution of risk over time and to support long-term strategic decisions. Figure 16: Simulated Flood Risk Fluctuations Over Time.