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We estimate the probability that random noise, of several plausible standard distributions, creates a false alarm that a periodicity (or log-periodicity) is found in a time series. The solution of this problem is already known for independent Gaussian distributed noise. We investigate more general situations with non-Gaussian correlated noises and present synthetic tests on the detectability and statistical significance of periodic components. A periodic component of a time series is usually detected by some sort of Fourier analysis. Here, we use the Lomb periodogram analysis, which is suitable and outperforms Fourier transforms for unevenly sampled time series. We examine the false-alarm probability of the largest spectral peak of the Lomb periodogram in the presence of power-law distributed noises, of short-range and of long-range fractional-Gaussian noises. Increasing heavy-tailness (respectively correlations describing persistence) tends to decrease (respectively increase) the false-alarm probability of finding a large spurious Lomb peak. Increasing anti-persistence tends to decrease the false-alarm probability. We also study the interplay between heavy-tailness and long-range correlations. In order to fully determine if a Lomb peak signals a genuine rather than a spurious periodicity, one should in principle characterize the Lomb peak height, its width and its relations to other peaks in the complete spectrum. As a step towards this full characterization, we construct the joint-distribution of the frequency position (relative to other peaks) and of the height of the highest peak of the power spectrum. We also provide the distributions of the ratio of the highest Lomb peak to the second highest one. Using the insight obtained by the present statistical study, we re-examine previously reported claims of "log-periodicity" and find that the credibility for log-periodicity in 2D-freely decaying turbulence is weakened while it is strengthened for fracture, for the ion-signature prior to the Kobe earthquake and for financial markets.

We extend the analysis of Ref. 16 showing statistically significant log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number (≈ 2500) of three-dimensional fully developed turbulence. First, we develop a simple variant of the canonical averaging method using a rephasing scheme between different samples based on pairwise correlations that confirms Zhou and Sornette's previous results. The second analysis uses a simpler local spectral approach and then performs averages over many local spectra. This yields stronger evidence of the existence of underlying log-periodic undulations, with the detection of more than 20 harmonics of a fundamental logarithmic frequency f = 1.434 ± 0.007 corresponding to the preferred scaling ratio γ = 2.008 ± 0.006.

We apply two nonparametric methods to further test the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The term "parametric" refers here to the use of the log-periodic power law formula to fit the data; in contrast, "nonparametric" refers to the use of general tools such as Fourier transform, and in the present case the Hilbert transform and the so-called (H, q)-analysis. The analysis using the (H, q)-derivative is applied to seven time series ending with the October 1987 crash, the October 1997 correction and the April 2000 crash of the Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq indices. The Hilbert transform is applied to two detrended price time series in terms of the ln(t_c-t) variable, where t_c is the time of the crash. Taking all results together, we find strong evidence for a universal fundamental log-frequency f=1.02±0.05 corresponding to the scaling ratio λ=2.67±0.12. These values are in very good agreement with those obtained in earlier works with different parametric techniques. This note is extracted from a long unpublished report with 58 figures available at , which extensively describes the evidence we have accumulated on these seven time series, in particular by presenting all relevant details so that the reader can judge for himself or herself the validity and robustness of the results.

The presence of log-periodic structures before and after stock market crashes is considered to be an imprint of an intrinsic discrete scale invariance (DSI) in this complex system. The fractal framework of the theory leaves open the possibility of observing self-similar log-periodic structures at different time scales. In the present work, we analyze the daily closures of four of the most important indices worldwide since 2000: the DAX for Germany and the NASDAQ-100, the S&P 500 and the Dow Jones for the United States. The qualitative behavior of these different markets is similar during the temporal frame studied. Evidence is found for decelerating log-periodic oscillations of duration about two years and starting in September 2000. Moreover, a nested sub-structure starting in May 2002 is revealed, bringing more evidence to support the hypothesis of self-similar, log-periodic behavior. Ongoing log-periodic oscillations are also revealed. A Lomb analysis over the aforementioned periods indicates a preferential scaling factor λ~2. Higher order harmonics are also present. The spectral pattern of the data has been found to be similar to that of a Weierstrass-type function, used as a prototype of a log-periodic fractal function.

Discrete scale invariance, which corresponds to a partial breaking of the scaling symmetry, is reflected in the existence of a hierarchy of characteristic scales l₀,l₀λ,l₀λ²,…, where λ is a preferred scaling ratio and l₀ a microscopic cut-off. Signatures of discrete scale invariance have recently been found in a variety of systems ranging from rupture, earthquakes, Laplacian growth phenomena, "animals" in percolation to financial market crashes. We believe it to be a quite general, albeit subtle phenomenon. Indeed, the practical problem in uncovering an underlying discrete scale invariance is that standard ensemble averaging procedures destroy it as if it was pure noise. This is due to the fact, that while λ only depends on the underlying physics, l₀ on the contrary is realization-dependent. Here, we adapt and implement a novel so-called "canonical" averaging scheme which re-sets the l₀ of different realizations to approximately the same value. The method is based on the determination of a realization-dependent effective critical point obtained from, e.g., a maximum susceptibility criterion. We demonstrate the method on diffusion limited aggregation and a model of rupture.

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.

The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonously according to their indices. This special geometry induces a drift of particles flowing from the small to the large scales. In this article we further explore the dynamical and statistical properties of this billiard. We derive from the ensemble average of the velocity a conductivity formula previously obtained by invoking the equality between phase-space contraction rate and the phenomenological entropy production rate. This formula is valid close to equilibrium. We also review other transport and ergodic properties of this billiard.

Log-periodic power laws often occur as signatures of impending criticality of hierarchical systems in the physical sciences. It has been proposed that similar signatures may be apparent in the price evolution of financial markets as bubbles and the associated crashes develop. The features of such market bubbles have been extensively studied over the past 20 years, and models derived from an initial discrete scale invariance assumption have been developed and tested against the wealth of financial data with varying degrees of success. In this paper, the equations that form the basis for the standard log-periodic power law model and its higher extensions are compared to a logistic model derived from the solution of the Schröder equation for the renormalization group with nonlinear scaling function. Results for the S&P 500 and Nikkei 225 indices studied previously in the literature are presented and compared to established models, including a discussion of the apparent frequency shifting observed in the S&P 500 index in the 1980s. In the particular case of the Nikkei 225 anti-bubble between 1990 and 2003, the logistic model appears to provide a better description of the large-scale observed features over the whole 13-year period, particularly near the end of the anti-bubble.

The characteristic feature of the discrete scale invariant (DSI) processes is the invariance of their finite dimensional distributions by dilation for certain scaling factor. DSI process with piecewise linear drift and stationary increments inside prescribed scale intervals is introduced and studied. To identify the structure of the process, first, we determine the scale intervals, their linear drifts and eliminate them. Then, a new method for the estimation of the Hurst parameter of such DSI processes is presented and applied to some period of the Dow Jones indices. This method is based on fixed number equally spaced samples inside successive scale intervals. We also present some efficient method for estimating Hurst parameter of self-similar processes with stationary increments. We compare the performance of this method with the celebrated FA, DFA and DMA on the simulated data of fractional Brownian motion (fBm).