Generalized q analysis of log-periodicity: applications to critical ruptures.

We introduce a generalization of the q analysis, which provides a nonparametric tool for the description and detection of log-periodic structures associated with discrete scale invariance. We use this generalized q analysis to construct a signature called the (H,q) derivative of discrete scale invariance, which we use to detect the log-periodicity in the cumulative energy release preceding the rupture of five pressure tanks made of composite carbon-matrix material. We investigate the significance level of the spectral Lomb periodogram of the optimal (H,q) derivative. We confirm and strengthen previous parametric results that the cumulative energy release exhibits log-periodicity before rupture. However, our tests to use this method as a scheme for the prediction of the critical value of the stress at rupture are not encouraging.