## 4 Results
### Using the Model  
To ascertain how to use the instability sample detector (BBRBM-ANN) and the bubble detector (GGRBM-BRF), we propose monitoring the probability of instability and bubbles on a daily basis. The BBRBM is presented with samples (images) of discretized volatility and price variations. The GGRBM is presented with volatilities, price changes, slopes, and cumulative returns. In this way, the trained ANN and BRF models can identify what kind of sample is involved. An example of this is presented in Figure 9, where we have plotted the S&P500 index between 1995 and 2012, encompassing the 2000 dot-com crisis and the 2008 Subprime crisis, and between 2013 to June 2024, encompassing the COVID outbreak in 2020 and the stock market decline in 2022. The probability of instability and bubble predictions are observed along with the index.  
We suggest identifying two basic scenarios: (1) The instability zone--the probability of instability is high, while the probability of a bubble is low. It responds to the abrupt fall of the price or index characterized by high volatility and negative index changes. (2) The bubble zone--the probability of instability is low, while the probability of a bubble increases. In this case, these are samples preceding the critical point where high returns have accumulated, and the price increase rates (slopes) accelerate. The tipping point or regime change may be imminent.  
Considering the above scenarios, we can see by looking at the graph in Figure 9a that for the dot-com crisis, starting in 1997, the model begins to warn of an increase in bubble probability above 0.5, reaching a peak close to 1.0 in late 1998 and early 2000. The decline in the index started in mid-2000. Thereafter, the bubble probability falls sharply, and the instability detector begins to warn of probabilities above 0.5, particularly in early 2001 and mid-2002. As the BRF has been trained with more crises, we have a better chance of a better bubble representation, observed in Figure 9c,e. After that, the system seems calm, as the bubble and instability probabilities remain very close to zero. However, bubble probabilities rose at the beginning of 2007, although not alarmingly. But, in the second half of 2007, bubble probabilities were clearly above 0.5, also embellished by some probabilities above 0.5 for instability. In this case, it can be observed that the instability probabilities were very close to 1 in the middle of the crises of 2008 and 2009 when the market hit bottom. These observations indicate that simultaneously monitoring instability and bubble probabilities is more informative than looking at only one.  
Similarly, looking at Figure 9b,d,f reveals how the models behave with samples that the detectors have not previously seen. Note that in the COVID-19 outbreak, the bubble and instability probabilities (especially the latter) are high. In other words, the system recognizes instability but simultaneously has a greater than 0.5 probability of a bubble. It is worth mentioning that, unlike the other crises, the pandemic is exogenous to the market, so it would be tough to pretend that the model trained only with composite index variables could warn of such an episode. In any case, the system reasonably recognizes the instability caused by the stock market decline in 2022, which is preceded by a high probability of a bubble.  
Finally, by mid-2024, the system clearly warns of a high probability of a bubble while, at the same time, indicating a very low probability of instability. Although we cannot accurately predict the timing of an impending tipping point, given the results, it is at least worth mentioning that investors and regulators should be very cautious about this situation.