  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.
