## 5 Conclusions
This lack of consensus complicates direct comparisons, especially since many models define the target variable differently, ranging from predicting the precise onset of financial crashes (e.g., [53; 78]) to identifying broader structural instabilities using system-level indicators (e.g., [48]). Our model, in contrast, is trained to detect the presence of bubble conditions preceding well-known crises, as defined by specific volatility-return dynamics. Furthermore, while some models are theory-driven (e.g., LPPLS), our approach is fundamentally data-driven, leveraging neural architectures to learn informative representations from empirical patterns. Future work should systematically evaluate our model's performance against classical econometric models (e.g., logit, XGBoost, LPPLS) using standardized metrics such as AUC, precision, and recall. Additionally, the development of benchmark datasets and shared evaluation protocols for financial bubble detection is still required to facilitate robust and fair comparisons across methodologies. Our system does not replace other criteria, which may be more qualitative or involve traditional financial-economic models. On the contrary, we view our approach as a complement to market surveillance and monitoring responsibilities. In this way, our model can be continuously trained with new market information to enhance the financial decision-making of a wide range of market participants. Compared to other models discussed in the Introduction section, a downside of our approach is the time required to train the RBMs and supervised models. However, as indicated in Section 4.5, we gain the ability to generate synthetic samples and conduct a what-if analysis that enables us to take actions in advance of the emergence of risks and significant changes in the stock markets. The phenomenon of financial bubble formation and its finale exhibit similar patterns across all stock markets, not just in the case studied here with the S&P500 index. Therefore, our model can be trained with various other market stock indices. This characteristic allows for more robust modeling and better addressing of the problem related to limited samples of bubbles and crises for analysis. This work also focused on simple bubble predictors based on cumulative returns, index changes, and volatilities. However, more sophisticated stock market indicators that capture other underlying characteristics of bubbles, such as periodogram-based features and those based on networks and multifractality, could be included. This extra information would offer a more comprehensive and robust approach to bubble analysis, enhancing instability detection regardless of the data source or market for training the model. A direct implication of our work is the need for stock market agents to monitor investor sentiment evolution, and that models of early warning detection bubbles incorporate these variables that capture the behavioral factor into their empirical approach. Whilepredicting a bubble's exact tipping point is impossible, monitoring investor sentiment indicators is feasible, as they are associated with irrational investor behavior and potential price bubbles. Although these indicators do not predict exact turning points, they are invaluable for assessing risk levels and bubble price dynamics. Indicators such as the VIX, the Put/Call ratio, activity over IPOs (Initial Public Offerings) and SPACs (Special Purpose Acquisition Companies), and corporate valuation multiples are widely used to capture sentiment among investors. Similarly, the CCI (Consumer Level Confidence Index) is a forward-looking economic indicator that reflects future household consumption and savings. Also, indicators that capture uncertainty, such as the EPU (Economic Public Uncertainty Index) and the WUI (World Uncertainty Index), track uncertainty levels across the economy.