This chapter contains sections titled: Introduction Statistics of Earthquakes Diagnostics of Tips Variation of the Boundaries of Region S On Further Monitoring of Tips

Socioeconomic and natural complex systems persistently generate extreme events also known as disasters, crises, or critical transitions. Here we analyze patterns of background activity preceding extreme events in four complex systems: economic recessions, surges in homicides in a megacity, magnetic storms, and strong earthquakes. We use as a starting point the indicators describing the system's behavior and identify changes in an indicator's trend. Those changes constitute our background events (BEs). We demonstrate a premonitory pattern common to all four systems considered: relatively large magnitude BEs become more frequent before extreme event. A premonitory change of scaling has been found in various models and observations. Here we demonstrate this change in scaling of uniformly defined BEs in four real complex systems, their enormous differences notwithstanding.

The estimation of seismic risk is made for three types of objects in the central Italy, considering three kinds of models: 1) - A(2I,g): the intensity of the Poisson's flow of earthquakes, M being the magnitude, g the liypocentre. 2) - I(g,g,M): giving the distribution on the surface for a single earthquake (g,M), g being the epicentre. 3) - x(g,I): giving the effect x of the shakings of intensity / , g being the position of the object. For actual decision-making additional computations may be necessary in order to estimate how our results are influenced by the errors in these models. However practical decision can be made on the basis of these data, because the experience shows that normally results are exagerated.

This study concerns the fields of economics and the dynamics of complex systems, specifically the process of recovery of American economy from recession. We identify a robust pattern of six macroeconomic indicators that appears within 6 months before the end of each American recession since 1960 and at no other time during these recessions. Its definition is formal and reproducible; as a precursor to the incipient recovery it is corroborated by sensitivity analysis (i.e. variation of its adjustable parameters) and application to out-of-sample data; noteworthy, it emerged before the end of the 2001 recession, which was not being considered while its definition developed. That pattern identified here appears through the whole time considered despite extraordinary changes in the economy. This reflects a well-known general feature of complex systems: they exhibit regular collective behavior patterns, transcending the complexity. Like many other complexity studies, identification of such patterns requires a robust holistic analysis; accordingly we used here a methodology combining pattern recognition of infrequent events and techniques developed in non-linear dynamics. This methodology inherits some of the features of the diffusion indicators of classical business cycle analysis, reformulating these features in the robust pattern recognition language. That includes formulation of the prediction problem per se, given the time series up to a moment t, to recognize whether this moment belongs or not to the last Delta months of recession. In terms of time series analysis our targets of prediction are extreme point events, and prediction is a discrete sequence of alarms; this is different from more traditional (Kolmogoff-Wiener) formulation, where prediction targets and predictors are continuous functions. That methodology is complementary to and compatible with other approaches to predictive understanding of recessions. The present study is a natural continuation of our previous one, aimed at predicting the start of a recession. We find that precursory trends of financial indicators are opposite during transition to a recession and recovery from it. To the contrary, precursory trends of economic indicators happen to have the same direction ( upward or downward) but are steeper during recovery.

This study concerns the fields of economics, labor relations, and the dynamics of complex systems. We consider a specific phenomenon in the dynamics of unemployment-episodes of a sharp increase in the unemployment rate, called here ''fast acceleration of unemployment'' (FAU). Our study is a ''technical'' analysis that is a heuristic search of phenomena preceding FAUs. We use the methodology of pattern recognition of infrequent events developed by the artificial intelligence school of Gelfand for a study of rare phenomena of highly complex origin, that, by their nature, limit the possibilities of using classical statistical or econometric methods. Our goal is to identify by an analysis of macroeconomic indicators a robust and rigidly defined prediction algorithm of the ''yes or no'' variety indicating at any time moment, whether a FAU should be expected or not within the subsequent months. Considering unemployment in France between 1962 and 1997, we have found a specific ''premonitory'' pattern of three macroeconomic indicators that may be used for algorithmic prediction of FAUs. Among seven FAUs identified within these years six are preceded within 12 months by this pattern that appears at no other time. The application of this algorithm to Germany, Italy and the USA yields similar results. Such predictability reflects the fact that the economy, like other complex systems, exhibits regular collective behavior patterns. The final test, as in any prediction research, should be advance prediction. The first such prediction, for the USA for early 2000, has been correct.