Forecasting extreme events in collective dynamics: An analytic signal approach to detecting discrete scale invariance

Abstract

A challenging problem in physics concerns the possibility of forecasting rare but extreme phenomena such as large earthquakes, financial market crashes and material rupture. A promising line of research involves the early detection of precursory log-periodic oscillations to help forecast extreme events in collective phenomena where discrete scale invariance plays an important role. Here we investigate two distinct approaches towards the general problem of how to detect log-periodic oscillations in arbitrary time series without prior knowledge of the location of the moveable singularity. We first show that the problem has a definite solution in Fourier space; however, the technique involved requires an unrealistically large signal-to-noise ratio. Then we show that the quadrature signal obtained via analytic continuation onto the imaginary axis, using the Hilbert transform, necessarily retains the log-periodicities found in the original signal. This finding allows the development of a new method of detecting log-periodic oscillations that relies on calculation of the instantaneous phase of the analytic signal. We then illustrate the method by applying it to the stock market crash of 1987 and explore the important role played by positive feedback mechanisms in relation to economic bubble formation, financial crashes, log-periodic dynamics and analytic behavior. Finally, we discuss the relevance of these findings for parametric rather than nonparametric estimation of critical times.