Performance of a high-dimensional R/S method for Hurst exponent estimation

https://doi.org/10.1016/j.physa.2008.08.014Get rights and content

Abstract

An extension of the R/S method to estimate the Hurst exponent of high-dimensional fractals is proposed. The method’s performance was adequate when tested with synthetic surfaces having different preset Hurst exponent values and different array sizes. The two-dimensional R/S analysis is used to analyze three images from nature and experimental data, revealing interesting scaling behavior with physical meaning.

Introduction

Many observable signals drawn from complex systems can be characterized with concepts and methods from fractals theory. Some records of observable quantities are in the form of ordered sequences (e.g., time series) and their fractal properties are commonly studied by means of a scaling analysis of the underlying fluctuations. To this purpose, several methods have been proposed in recent decades, from the classical spectral Fourier analysis to modern methods, such as R/S analysis [1], wavelet transform module maxima (WTMM) [2] and detrended fluctuation analysis (DFA) [3]. Scaling analysis methods offer advantages and drawbacks for implementation and estimation accuracy. Fourier analysis is constrained to unbiased stationary sequences. The R/S analysis has a strong theoretical back up, its statistical limitations for scaling (Hurst) exponent estimation are well established [4], and it finds wide acceptability for application in diverse econometric fields. On the other hand, due to the simplicity in implementation, the DFA is now becoming a widely used method in physics and engineering. The WTMM method was shown to be more accurate than DFA and R/S analysis, although its implementation is complicated, particularly for data sets with limited extension. In this form, diverse modifications of the R/S analysis and the DFA method have been proposed to improve performance while maintaining an easy implementation and straightforward understanding. For instance, the detrended moving average (DMA) proposed in Ref. [5] is intended to obtain a more accurate estimate of the scaling exponent over a wide of scrutinized scales. The scaling analysis methods commented on above have found applicability in a wide variety of fields, such as the analysis of DNA sequences [3], financial markets [6], daily temperature records [7], ion channels [8], EEG physiological records [9], and many more. The attractiveness of these methods relies on their model-free implementation with relatively low computational burden. In this form, useful information on long-term correlations can be obtained as a preliminary step to derive stochastic models oriented to describe the dynamics of the underlying time sequence.

The performance of the scaling analysis methods for the characterization of fractality and correlations in higher-dimensional spaces has been recently addressed. The WTMM method has shown accurate results, especially for scalar and vector fields of three-dimensional (3D) turbulence [10], [11], although its limitations are again related to a complex implementation and the specification of commonly ad hoc base functions. The accuracy of the DFA method has been studied by Gu and Zhou [12], showing that an accurate Hurst exponent estimation for fractal and multifractal objects (e.g., images) can be obtained via a suitable selection of the detrending function. An extension of the DMA method has also been proposed [13], yielding a computationally efficient and simple procedure for the Hurst exponent estimation of synthetic and real images. The 2D height–height correlation function has also been explored to describe the scaling properties of fractured materials [14]. The existence of anomalous scaling exponents in natural images has been explored by Frenkel et al. [15] by means of Fourier principal components. The above results reveal that the analysis of natural and man-made images is of practical interest from both practical and scientific standpoints.

The performance of a high-dimensional R/S analysis that retains an easy implementation and offers a simple understanding of results is studied in this work. The method is tested using synthetic surfaces with known fractal and multifractal properties, and the numerical results are in good agreement with the corresponding DFA [12] and DMA [13] high-dimensional versions. The 2D version of the R/S method is also used to analyze three images from nature and experimental data, revealing interesting scaling behavior that is interpreted from physical grounds.

Section snippets

A high-dimensional R/S method

For convenience, a brief description of the one-dimensional R/S analysis is given as follows [1], [16]. The R/S statistics is the range of partial sums of deviations of sequences from its mean, rescaled by its standard deviation. So, for a given N-dimensional vector XN=(xi) consider an M-dimensional sample sub-vector YM=(yi), where M=sN, and s(0,1). Then, the R/S statistics is estimated by computing the sub-sample mean y¯s=1Mk=1Myk, the sequence from partial summations zi=k=1i(yky¯s), the

Examples of image analysis

In this section, three real images are used to illustrate the performance of the two-dimensional R/S analysis method. The examples correspond to a typical scanning electron microscope (SEM) picture of a 300 Series stainless steel sample surface, a hurricane satellite image and a Pollock drip painting (see Fig. 4). The images, originally in any image format (e.g., jpg) were converted into 256 grey level numerical matrices (level 1 for black and level 256 for white) by means of the MATLAB®signal

Conclusions

In summary, the performance of a higher-dimensional extension of the R/S analysis method was evaluated with a synthetic surface with different preset Hurst exponents and different array sizes. The results showed that the precision of the Hurst exponent estimation is improved when the array size is increased, and that larger estimation errors are found for arrays with significant correlations. The two-dimensional method was applied to the analysis of three real images from experiments, nature

References (22)

  • P. Kestener et al.

    Three-dimensional wavelet-based multifractal method: The need for revisiting the multifractal description of turbulence dissipation data

    Phys. Rev. Lett.

    (2003)
  • Cited by (68)

    • Hurst exponent dynamics of S&P 500 returns: Implications for market efficiency, long memory, multifractality and financial crises predictability by application of a nonlinear dynamics analysis framework

      2023, Chaos, Solitons and Fractals
      Citation Excerpt :

      Notwithstanding, Al-Yahyaee et al. [53] present all markets under analysis to reveal Hurst exponents above 0.5 and, hence, conclusive evidence against the EMH. Different time horizons may be referred to as scales, whose properties, namely scaling properties, are measurable via the Hurst exponent [50,56]. Thus, scaling properties and self-similarity are notated synonymously within (multi)fractal dynamics [56].

    • Fractal and multifractal analysis on fused silica glass formed by bound abrasive grain mediated grinding using diamond grits

      2022, Journal of Non-Crystalline Solids
      Citation Excerpt :

      It helps in the calculation of the fractal dimension in terms of the holder exponent to a particular coordinate at a particular interval. There are many methods for the calculation of the fractal dimension which are error-free like the DFA (Detrended fluctuation analysis), MFDFA (Multifractal detrended fluctuation analysis), R/S method, etc. [8–10]. We in this paper make use of the 2D-MFDFA method for an improved calculation of the multifractality shown by the various images of the samples of fused silica formed by the polishing using grits of various sizes as well as materials.

    • Multifractal and cross-correlation analysis on mitochondrial genome sequences using chaos game representation

      2021, Mitochondrion
      Citation Excerpt :

      CGR was also applied to protein sequences and structures (Fiser et al., 1994). Several methods such as structure–function, R/S analysis, average wavelet coefficient method, WTMM, detrended moving average analysis and its variants, detrended fluctuation analysis, and its variants, etc., were used to characterize the fractal nature and correlation behaviour in one dimensional and two-dimensional data sets (Hurst, 1951; Muzy et al., 1991; Peng et al., 1994; Simonsen et al., 1998; Guihong et al., 2001; Kantelhardt et al., 2002; Alessio et al., 2002; Patrick, 2002; Manimaran et al., 2005; Jose et al., 2006; Jose et al., 2008; Manimaran et al., 2008; Manimaran et al., 2009; Gu and Zhou, 2010; Zhou et al., 2013; Alpatov et al., 2013; Wang et al., 2015). Later, various multifractal cross-correlation analysis methods were introduced to explore the multifractal nature in the power-law cross-correlations between non-stationary time series (Podobnik and Stanley, 2008; Zhou, 2008; Podobnik et al., 2009; Jiang and Zhou, 2011; Kristoufek, 2011; Xie et al., 2015; Qian et al., 2015).

    View all citing articles on Scopus
    1

    Also at Facultad de Ingenieria, UNAM, Mexico.

    View full text