Singular Gaussian Measures in Detection Theory

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Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal 1/f1/f noise

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level, results in an important difference between the mean and the typical values of the counting function. This can be further used to determine the typical threshold $x_m$ of extreme values in the pattern which turns out to be given by $x_m^{(typ)}=2-c\ln{\ln{M}}/\ln{M}$ with $c=3/2$. Such observation provides a rather compelling explanation of the mechanism behind universality of $c$. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum $p_{max}$ of intensity is to be given by $-\ln{p_{max}} = \alpha_{-}\ln{M} + \frac{3}{2f'(\alpha_{-})}\ln{\ln{M}} + O(1)$, where $f(\alpha)$ is the corresponding singularity spectrum vanishing at $\alpha=\alpha_{-}>0$. For the $1/f$ noise we also derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.Comment: 28 pages; 7 figures, published version with a few misprints corrected, editing done and references adde

Multifractality in the stock market: price increments versus waiting times

By applying the multifractal detrended fluctuation analysis to the high-frequency tick-by-tick data from Deutsche B\"orse both in the price and in the time domains, we investigate multifractal properties of the time series of logarithmic price increments and inter-trade intervals of time. We show that both quantities reveal multiscaling and that this result holds across different stocks. The origin of the multifractal character of the corresponding dynamics is, among others, the long-range correlations in price increments and in inter-trade time intervals as well as the non-Gaussian distributions of the fluctuations. Since the transaction-to-transaction price increments do not strongly depend on or are almost independent of the inter-trade waiting times, both can be sources of the observed multifractal behaviour of the fixed-delay returns and volatility. The results presented also allow one to evaluate the applicability of the Multifractal Model of Asset Returns in the case of tick-by-tick data.Comment: Physica A, in prin

A scale-space approach with wavelets to singularity estimation

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, "the structural intensity", that computes the "density" of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed

Geometric methods for estimation of structured covariances

We consider problems of estimation of structured covariance matrices, and in particular of matrices with a Toeplitz structure. We follow a geometric viewpoint that is based on some suitable notion of distance. To this end, we overview and compare several alternatives metrics and divergence measures. We advocate a specific one which represents the Wasserstein distance between the corresponding Gaussians distributions and show that it coincides with the so-called Bures/Hellinger distance between covariance matrices as well. Most importantly, besides the physically appealing interpretation, computation of the metric requires solving a linear matrix inequality (LMI). As a consequence, computations scale nicely for problems involving large covariance matrices, and linear prior constraints on the covariance structure are easy to handle. We compare this transportation/Bures/Hellinger metric with the maximum likelihood and the Burg methods as to their performance with regard to estimation of power spectra with spectral lines on a representative case study from the literature.Comment: 12 pages, 3 figure

Landmark-Based Registration of Curves via the Continuous Wavelet Transform

This paper is concerned with the problem of the alignment of multiple sets of curves. We analyze two real examples arising from the biomedical area for which we need to test whether there are any statistically significant differences between two subsets of subjects. To synchronize a set of curves, we propose a new nonparametric landmark-based registration method based on the alignment of the structural intensity of the zero-crossings of a wavelet transform. The structural intensity is a multiscale technique recently proposed by Bigot (2003, 2005) which highlights the main features of a signal observed with noise. We conduct a simulation study to compare our landmark-based registration approach with some existing methods for curve alignment. For the two real examples, we compare the registered curves with FANOVA techniques, and a detailed analysis of the warping functions is provided