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Register a new userWavelet-based Estimator for the Hurst Parameters of Fractional Brownian Sheet
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Added by
Liang Wu
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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It is proposed a class of statistical estimators $hat H =(hat H_1, ldots, hat H_d)$ for the Hurst parameters $H=(H_1, ldots, H_d)$ of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when $H_i geq 1/2$, the estimators are efficient, and when $H_i < 1/2$, there are some bias.
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We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{alpha, beta}$ with Hurst parameter $(alpha, beta) in (0,1)^2$. When $0<alpha leq 1-frac{1}{2q}$ or $0<beta leq 1-frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $qgeq 2$, while for $1-frac{1}{2q}<alpha, beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.
In this paper, we show how concentration inequalities for Gaussian quadratic form can be used to propose exact confidence intervals of the Hurst index parametrizing a fractional Brownian motion. Both cases where the scaling parameter of the fractional Brownian motion is known or unknown are investigated. These intervals are obtained by observing a single discretized sample path of a fractional Brownian motion and without any assumption on the parameter $H$.
In this presentation, we introduce a new method for change point analysis on the Hurst index for a piecewise fractional Brownian motion. We first set the model and the statistical problem. The proposed method is a transposition of the FDpV (Filtered Derivative with p-value) method introduced for the detection of change points on the mean in Bertrand et al. (2011) to the case of changes on the Hurst index. The underlying statistics of the FDpV technology is a new statistic estimator for Hurst index, so-called Increment Bernoulli Statistic (IBS). Both FDpV and IBS are methods with linear time and memory complexity, with respect to the size of the series. Thus the resulting method for change point analysis on Hurst index reaches also a linear complexity.
We consider the problem of estimating the probability of an observed string drawn i.i.d. from an unknown distribution. The key feature of our study is that the length of the observed string is assumed to be of the same order as the size of the underlying alphabet. In this setting, many letters are unseen and the empirical distribution tends to overestimate the probability of the observed letters. To overcome this problem, the traditional approach to probability estimation is to use the classical Good-Turing estimator. We introduce a natural scaling model and use it to show that the Good-Turing sequence probability estimator is not consistent. We then introduce a novel sequence probability estimator that is indeed consistent under the natural scaling model.
Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst index $Hin(0,1)$. Special attention has been paid to the impact of Hurst index on the topological properties. It is found that the clustering coefficient $C$ decreases when $H$ increases. We also found that the mean length $L$ of the shortest paths increases exponentially with $H$ for fixed length $N$ of the original time series. In addition, $L$ increases linearly with respect to $N$ when $H$ is close to 1 and in a logarithmic form when $H$ is close to 0. Although the occurrence of different motifs changes with $H$, the motif rank pattern remains unchanged for different $H$. Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension $d_B$ decreases with $H$. Furthermore, the Pearson coefficients of the networks are positive and the degree-degree correlations increase with the degree, which indicate that the horizontal visibility graphs are assortative. With the increase of $H$, the Pearson coefficient decreases first and then increases, in which the turning point is around $H=0.6$. The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative.
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