The first step to debunking the discontinuity phenomenon is identifying the singularities. To detect these patterns, and the consequent discontinuity, we need to utilize continuous sequential analysis with the least number of gaps or jumps. Time series can be defined in many ways, such as tick or time compression. We will want to use the method that results in the smoothest data series. In our examples here, we use time-based charts.
The second requirement is the use of a specialized index. This can be done through the combination of positively correlated time series; this often yields more robust forecasting power than individual time series alone. The use of isolated time series makes the predictive forecasting less reliable. Part of the reason is while we may have the right tools to detect the pattern, we have no control over when, where and how it emerges. A complex series includes more information and, therefore, more clues to discovering the pattern.
The third rule is that while the discontinuity phenomenon may be a complex problem difficult to solve, we know that it begins and ends with the formation of similar but inverse patterns in sequential time frames.
Detection of these patterns is difficult because it’s rare that both can be defined in constant numerical terms across time series; this is why we see them as self-similarities. One tool for detection is a pattern-recognition scan, such as the technology used in database searches, facial recognition, DNA sequencing, etc. Another method applies coinciding statistical tools to the complex index across time frames. If the condition is met across the time frames, it automatically grants the pattern a higher statistical degree of reliability.
The latter approach will be used here. The techniques themselves do not have to be exceptionally sophisticated. For instance, a moving average crossover in different sequential time frames automatically makes the patterns similar and concurrently makes the forecasting relationship much stronger. This is the case with the charts included in “Cross-time signals” (below) where two exponential moving averages are applied in four sequential time frames on the Volatility Index (VIX).