A forecasting model that is based on an analysis of critical transitions in complex systems was proposed by Professor Didier Sornette, presently at ETH Zurich. It's based on looking at the connections in what's called a Bethe lattice. For systems like the Beta lattice, which has discrete scale invariance, it only looks the same if we zoom in and out over a discrete set of nitrification. When a system has discrete scale invariance, we expect the distributions of its physical properties to be logged periodic power laws of the form that are shown here. A complex system that has discrete scale invariance, means that the system only looks the same if we zoom in and out over a discrete set of magnifications. When a system has discrete scale invariance, we expect the distribution of its physical properties to be logged periodic power laws of the form shown here. Professor Sornette originally developed his method which he called the log periodic power-law method to predict earthquakes. But he found that it actually did a better job of producing fluctuations in the stock market. So here we have Black Monday. Proceeding Black Monday in 1987, these are the prices of stocks and as you can see, every time there's a crash it's preceded by an increase in the stock. The amplitude, the size of these fluctuations grows, and the distance between them, the time between them shortens. That general phenomenon allows you to predict the behavior once the phenomenon is set in motion. When Professor Sornette fitted such price oscillations to his log periodic power-law, he found that the fit is typically pretty good and the predicted critical transition time was typically close to the actual crash. Another way to predict regime shifts is what's called the soup-of-groups model. In this model we have collection of N individuals, which can form groups of size S. At each time step, we randomly choose groups proportional to their size, and the group can either disintegrate or it can merge with some other group. It's a process of the formation efficient infusion of groups within the soup-of-groups. Now if we allow this soup-of-group system to evolve for a long time according to the master equation, then the distribution of cluster sizes will become the power-law shown. The key piece of information we should take away from this power-law, is that it's universal. That is, it does not depend upon the details of the model, however, it does depend upon the dimensionality of the model. When a cluster can merge with any other cluster in the SOG system, the effective dimensionality of the system is infinite and the critical exponent is 5 over 2. When a cluster can only merge with neighboring clusters, they're critical exponent then depends on the dimension of the system. If we have SOG clusters interacting on a two-dimensional plane, perhaps like tectonic elements on a fault plane, the critical exponent drops to two, and we can show that this very naturally explains the origin of the Gutenberg–Richter law of earthquakes. The soup-of-groups model can be used to predict regime shifts. The idea is simple. In the beginning we've got different sized clusters, all of which are distributed according to a power law, but as time goes forward, based upon the evolving dynamics of the soup-of-groups, you get the formation of a one giant cluster and the whole system begins to depart from the power law distribution of a Gutenberg-Richter law. At that point it's moving towards a critical point, the regime shift will occur when this large cluster, the giant cluster, breaks apart. When the cluster grows to giant proportions, it will have incorporated a significant fraction of the individual elements in the system. The remaining elements can therefore only form smaller clusters. As I've mentioned, the complex system becomes more likely to undergo a noise driven equilibrium transition before we reach the critical point. Therefore as the giant cluster grows in size, it's also more likely to undergo fragmentation. This can happen when the giant cluster is 50 percent the size of this soup-of-group system or 70 percent, or even 90 percent the size of the whole SOG system. We can't predict that because these events are strongly stochastic in nature. However, we can predict an event at the largest scale. That is when the giant cluster consumes all of the individual elements of the SOG system, and must therefore fragment at the very next timestep. In the figure on the left, we show the time between fragmentation events. When no giant cluster is present, the time between fragmentation events is roughly constant and equal to some equilibrium value. But when a giant cluster grows beyond the average size of clusters, the number of clusters will start falling, and thus the time between fragmentation events will start rising. This goes on until the time between fragmentation events reaches a maximum and the giant cluster fragments. From this point on, we see a slow recovery process where the time between fragmentation events increases from a minimum value to the equilibrium value. As we can see, this gives us a very long early warning period before the extreme event occurs. In the figure on the right, we show the integrated event size where we add the size of a fragmentation event to a cumulative total as time goes on. From this we can observe that the integrated events size grows linearly with time when there's no giant cluster, because it receives contributions from all cluster sizes. But when there's a giant cluster beginning to grow, the number of clusters decreases with time and the average size of these clusters also decreases. Therefore, the integrated event size grows more slowly, eventually, flattening out before the giant cluster collapses. By comparing this slowed integrated event size curve against the straight line expected from an equilibrium model, we find that we can predict the maximum size of the fragmentation event that may happen the next moment, because it's proportional to the size of the giant cluster. Finally, to predict when a fragmentation of the maximum size will happen, which is when the giant cluster will go to the maximum size, we develop a mean-field theory. To derive the mean-field theory for the giant cluster, let's assume that the giant cluster grows linearly with time. Next, we assume that the size distribution of the other clusters follows the equilibrium distribution at all times. For earthquakes, this equilibrium distribution would be the Gutenberg Richter law, for other complex systems, it would be something else. With these two assumptions, we can treat the integrated event size at a mean-field level to derive a closed formula, which has a linear term and two log terms. We can now fit empirical integrated event size curves to this function to determine the time at which the giant cluster would reach maximum size. In the slides to follow, we show how SOG forecasting can be applied to earthquakes and to stock market crashes. Our first example is the September 1999,Chi-Chi earthquake in Taiwan. This was a magnitude 7.3 earthquake that produced widespread devastation in Taiwan and also resulted in many fatalities. For this study, we used high resolution data from the Taiwan earthquake catalog. An earthquake catalog is simply a record of all the earthquakes that occurred within a given region, showing the times and dates of the earthquakes, epicenters, and the magnitude. The magnitude of an earthquake is the base 10 log of the ratio of the energy released by the earthquake compared to the energy released by a reference earthquake whose magnitude is zero. A magnitude zero earthquake is just perceptible by a person. A magnitude 5 earthquake release is 10 times more energy than a magnitude 4 earthquake, and earthquakes with magnitudes larger than six are considered large earthquakes. In general, there's no way we would miss a magnitude 6 earthquake. Even if the sensors fail to pick it up, individuals near the epicenter would feel it keenly and report it. Here's the earthquake, and we can see that as time is going forward and the magnitude of events is increasing and so is the rate at which variation occurs until finally, we hit the major earthquake, after which the system returns to its equilibrium level. In this slide, we see the integrated event size curve in blue falling below the green straight line that is expected from the Gutenberg Richter law as we approach the earthquake. Over the period from 1997 to 2000, the dates of all large earthquakes with magnitude greater than six are shown as red vertical lines. To see the early warning signals more clearly, we subtract the empirical integrated events size curve from the equilibrium straight line to get the figure shown here. In this figure, we can see very clearly how the integrated events size curve stayed close to the equilibrium straight line for 1997 and the first half of 1998, in spite of the large earthquakes within this time period. But from mid 1998 on, we find only one large earthquake before the Chi-Chi earthquake in September 1999. Effectively, we have a statistically significant early warning signal in the form of the strong deviation starting in early 1999. This means that if we had this information in 1999, we would have had up to six months of early warning for the Chi-Chi earthquake. As mentioned earlier, an early warning signal, no matter how strong is not a prediction. In order to create model, we fit the earthquake data, the temporal data, the time series data of preceding small earthquakes to the mean-field theory. When our time window is long enough to include the suppression signature, the mean-field theory predicted an earthquake of the maximum size and a date very close to the Chi-Chi earthquake. This is not a fluke because the predicted date remains very close to the Chi-Chi earthquake as the time window becomes longer and longer, including more and more of the suppression signature. The final prediction date, which was constant for four weeks before the earthquake, is slightly more than one week after the earthquake, so it's very close. By breaking up in areas slightly larger than Taiwan into cells and 0.5 degree latitude and longitude and measuring the suppression signature at each cell, we're also able to predict the epicenter of the Chi-Chi earthquake. Next, let's move on to our second example, which is on using the SOG model to predict the October 2008 market crash on the Singapore Exchange, also called the SGX. Next, let's move on to our second example, the use of the super groups model to predict the crash in the Singapore stock market, which occurred in October 2008. In this figure, we show the normalized stock price in green for the Bank of Singapore, DBS, between March 2006 and August 2011. On this scale, the crash of October 2008 can be seen very clearly. On the same figure, we also show in blue the integrated absolute returns of the Development Bank of Singapore. Well, the stock market, we assume that a fragmentation event is caused by a group of traders executing the same strategy to sell off the stock. Before a fragmentation event, we would see the price increase continuously or decrease continuously. We call such price changes runs, and use as a proxy of event sizes, the absolute returns of such runs. Here we see a linear growth of the integrated absolute returns from March 2006 to July 2007. We then see an unexpected acceleration in the growth of the integrated absolute returns. We'd started slowing down in March 2008 before the stock market crashed in October 2008. After the crash, the integrated absolute returns went through a series of acceleration episodes before relaxing back to a linear growth in October 2009. Already we can see that things are a little different. When we use the mean-field theory in blue to fit the returns in red, for a time window starting in April 2007 and ending at a different time shown before the October 2008 crash, we find that the predicted crash is always very close to the end of the time window. It's only when the end of our time window gets close to the October 2008 crash that the predicted crash starts to hover around the real crash date. After the end of the time window goes beyond the real crash date, the predicted crash starts to move with the end of time window again. To check that this is a robust feature, we tested different starting dates for the time window. For the DBS Bank and for different starting dates, the predicted crash date started tracking the real crash date for time windows ending in December 2007 and lost track of the real crash date for time windows ending August 2008. We then perform this market crash prediction for the 30 component stocks of the Straits Times Index, also known as the STI. The predicted crash dates can be different for different stocks. In fact, for some stocks, there are no signs of the tracking behavior shown in DBS. Therefore, in order to incorporate information from the 30 components stocks of the STI and not rely on one stock or another, we created a heatmap. A given date within the period of prediction has a redder color if it's predicted many times from different stocks with different starting and ending dates as the date of the market crash. Conversely, it becomes a deep blue color if it's predicted very few times to be the date of the market crash. By using all 30 component stocks for the prediction, we find that we can reliably predict the Chinese Correction of February 2007, the subprime crisis of July 2007, the Asian Correction of February 2009. This is shown on the left. There are, however, no strong predictions for the September 2008 Lehman Brothers bankruptcy in the October 2008 crashes. Besides giving when, where, and how a crash is, we'd also like our prediction method to be timely. That is, we don't want reliable predictions that are one day before the actual crashes, but reliable prediction sufficiently early that we can act. Therefore, we show on the right the heatmap where we sum the predictions overall components stocks, varying the end date of the prediction time window. When the end date of the prediction window is very far from a given market crash, none of the stocks predicted a crash close to the one of interest. But when the end date of the prediction window approaches the given market crash, some stocks will start to predict a crash close to the one of interest. The number of stocks predicting such a crash increases as the end date of the prediction window approaches the market crash and we start to see a hot streak on the heatmap. Looking at the brightest streak in the heatmap, which is for the February 2009 Asian Correction, we find it about half a year before it occurred. In other words, starting around the time of the October 2008 crashes, the soup-of-groups method could predict the market low at the February 2009 Asian Correction. The Singapore stock market started recovering after this event. In this lecture, we've looked at the universal phenomenology of regimes and regime shifts, focusing on the differences between equilibrium and critical transitions between stable states. We've used the Landau theory as a general theory to explain many of the features of regime shifts. In particular, we looked at two of the universal features of critical transitions, the first of which is critical fluctuations. These occur when fluctuations in the state of a complex system increase in magnitude as they approach the point of a transition. The reason why fluctuations in the state of a complex system grow stronger as it approaches the critical point is because of the flattening of the free energy landscape. The second class of early warning signals is associated with critical slowdown, whereby a complex system which has been displaced from its equilibrium position takes a long time to recover again because the flattened free energy landscape produces weaker restoring forces. In the last part of this lecture, we touched on the problem of quantitative forecasting, where we must be not only certain that a regime shift is going to happen, but also predict when it will happen, how large it will be, and where it will happen. We reviewed the LPPL method developed by Sornette from the ETH Zurich as the established method for doing this before going on to introduce a method that Professor Siew Ann Cheong developed. In this method, based on the soup-of-groups model, we look for precursory signatures consistent with a growing giant cluster for which a mean-field equation can be derived. Prediction can then be done by fitting the empirical data to the mean-field equation. We show the application of this method to two problems: the prediction of large earthquakes and the prediction of market crashes. Both gave encouraging results. As with all prediction methods for regime shifts and complex systems, more tests are necessary.
