2006/11/28 by G. M. Viswanathan, Viswanathan, G. M.
Economics, Econometrics and Finance · Environmental Science · Physics and Astronomy · #Complex Systems and Time Series Analysis #Data Analysis #Ecosystem dynamics and resilience #FOS: Economics and business #FOS: Physical sciences #Physics and Society (physics.soc-ph) #Statistical Finance (q-fin.ST) #Statistics and Probability (physics.data-an) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.physics/0611281
openalex publication_date 2006/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 I 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. I first show that the problem has a definite solution in Fourier space, however the technique involved requires an unrealistically large signal to noise ratio. I then 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. I illustrate the method by applying it to the well documented stock market crash of 1987. Finally, I discuss the relevance of these findings for parametric rather than nonparametric estimation of critical times.