Identification of the multiscale fractional Brownian motion with biomechanical applications

In certain applications, for instance biomechanics, turbulence, finance, or Internet traffic, it seems suitable to model the data by a generalization of a fractional Brownian motion for which the Hurst parameter $H$ is depending on the frequency as a piece-wise constant function. These processes are called multiscale fractional Brownian motions. In this contribution, we provide a statistical study of the multiscale fractional Brownian motions. We develop a method based on wavelet analysis. By using this method, we find initially the frequency changes, then we estimate the different parameters and afterwards we test the goodness-of-fit. Lastly, we give the numerical algorithm. Biomechanical data are then studied with these new tools

Wavelet analysis of the multivariate fractional Brownian motion

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behavior of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral density is also considered in a second part. Its existence is proved and its evaluation is performed using a von Bahr-Essen like representation of the function \sign(t) |t|^\alpha. The behavior of the cross-spectral density of the wavelet field at the zero frequency is also developed and confirms the results provided by the asymptotic analysis of the correlation

Wavelet-Based Entropy Measures to Characterize Two-Dimensional Fractional Brownian Fields

The aim of this work was to extend the results of Perez et al. (Physica A (2006), 365 (2), 282–288) to the two-dimensional (2D) fractional Brownian field. In particular, we defined Shannon entropy using the wavelet spectrum from which the Hurst exponent is estimated by the regression of the logarithm of the square coefficients over the levels of resolutions. Using the same methodology. we also defined two other entropies in 2D: Tsallis and the Rényi entropies. A simulation study was performed for showing the ability of the method to characterize 2D (in this case, α = 2) self-similar processes

Wavelet entropy and fractional Brownian motion time series

We study the functional link between the Hurst parameter and the Normalized Total Wavelet Entropy when analyzing fractional Brownian motion (fBm) time series--these series are synthetically generated. Both quantifiers are mainly used to identify fractional Brownian motion processes (Fractals 12 (2004) 223). The aim of this work is understand the differences in the information obtained from them, if any.Comment: 10 pages, 2 figures, submitted to Physica A for considering its publicatio

Expectiles for subordinated Gaussian processes with applications

In this paper, we introduce a new class of estimators of the Hurst exponent of the fractional Brownian motion (fBm) process. These estimators are based on sample expectiles of discrete variations of a sample path of the fBm process. In order to derive the statistical properties of the proposed estimators, we establish asymptotic results for sample expectiles of subordinated stationary Gaussian processes with unit variance and correlation function satisfying $\rho(i)\sim \kappa|i|^{-\alpha}$ (\kappa\in \RR) with $\alpha>0$. Via a simulation study, we demonstrate the relevance of the expectile-based estimation method and show that the suggested estimators are more robust to data rounding than their sample quantile-based counterparts

Intermittent process analysis with scattering moments

Scattering moments provide nonparametric models of random processes with stationary increments. They are expected values of random variables computed with a nonexpansive operator, obtained by iteratively applying wavelet transforms and modulus nonlinearities, which preserves the variance. First- and second-order scattering moments are shown to characterize intermittency and self-similarity properties of multiscale processes. Scattering moments of Poisson processes, fractional Brownian motions, L\'{e}vy processes and multifractal random walks are shown to have characteristic decay. The Generalized Method of Simulated Moments is applied to scattering moments to estimate data generating models. Numerical applications are shown on financial time-series and on energy dissipation of turbulent flows.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1276 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study

We analyze, within the wavelet theory framework, the wandering over a screen of the centroid of a laser beam after it has propagated through a time-changing laboratory-generated turbulence. Following a previous work (Fractals 12 (2004) 223) two quantifiers are used, the Hurst parameter, $H$, and the Normalized Total Wavelet Entropy, $\text{NTWS}$. The temporal evolution of both quantifiers, obtained from the laser spot data stream is studied and compared. This allows us to extract information of the stochastic process associated to the turbulence dynamics.Comment: 11 pages, 3 figures, accepted to be published in Physica

Singularity and similarity detection for signals using the wavelet transform

The wavelet transform and related techniques are used to analyze singular and fractal signals. The normalized wavelet scalogram is introduced to detect singularities including jumps, cusps and other sharply changing points. The wavelet auto-covariance is applied to estimate the self-similarity exponent for statistical self-affine signals

A Multiscale Guide to Brownian Motion

We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains

Scaling detection in time series: diffusion entropy analysis

The methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance. This means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved. We illustrate a method of statistical analysis based on the Shannon entropy of the diffusion process generated by the time series, called Diffusion Entropy Analysis (DEA). We adopt artificial Gauss and L\'{e}vy time series, as prototypes of ordinary and anomalus statistics, respectively, and we analyse them with the DEA and four ordinary methods of analysis, some of which are very popular. We show that the DEA determines the correct scaling exponent even when the statistical properties, as well as the dynamic properties, are anomalous. The other four methods produce correct results in the Gauss case but fail to detect the correct scaling in the case of L\'{e}vy statistics.Comment: 21 pages,10 figures, 1 tabl