Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 485623, 7 pages

http://dx.doi.org/10.1155/2015/485623

## Correlation Properties of (Discrete) Fractional Gaussian Noise and Fractional Brownian Motion

EA 2991 Movement to Health, Euromov, University of Montpellier, 34090 Montpellier, France

Received 13 April 2015; Revised 9 July 2015; Accepted 2 August 2015

Academic Editor: Paul Bogdan

Copyright © 2015 Didier Delignières. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The fractional Gaussian noise/fractional Brownian motion framework (fGn/fBm) has been widely used for modeling and interpreting physiological and behavioral data. The concept of 1/*f* noise, reflecting a kind of optimal complexity in the underlying systems, is of central interest in this approach. It is generally considered that fGn and fBm represent a continuum, punctuated by the boundary of “ideal” 1/*f* noise. In the present paper, we focus on the correlation properties of discrete-time versions of these processes (dfGn and dfBm). We especially derive a new analytical expression of the autocorrelation function (ACF) of dfBm. We analyze the limit behavior of dfGn and dfBm when they approach their upper and lower limits, respectively. We show that, as *H* approaches 1, the ACF of dfGn tends towards 1 at all lags, suggesting that dfGn series tend towards straight line. Conversely, as *H* approaches 0, the ACF of dfBm tends towards 0 at all lags, suggesting that dfBm series tend towards white noise. These results reveal a severe breakdown of correlation properties around the 1/*f* boundary and challenge the idea of a smooth transition between dfGn and dfBm processes. We discuss the implications of these findings for the application of the dfGn/dfBm model to experimental series, in terms of theoretical interpretation and modeling.

#### 1. Introduction

During the last decades, there has been a considerable interest in the use of stochastic fractal models for interpreting physiological or behavioral data. These models have been applied to various processes, including in the physiological domain heart-beat variability [1, 2], brain activity [3, 4], respiratory fluctuations [5], or blood flow [6]. Studies on sensorimotor processes included isometric force production [7], visual search [8], finger tapping [9, 10], or bimanual coordination [11]. Recent research has motivated the use of fractal models in medical and rehabilitation devices, in order to conceive efficient noninvasive stimuli for living organisms [12].

The most popular formalization of this approach refers to the concepts of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn), initially introduced by Mandelbrot and Van Ness [13]. fBm and fGn represent two families of correlated stochastic processes, possessing powerful statistical properties, which seem able to provide relevant models for a wide range of empirical observations in various domains.

The success of the fGn/fBm concept is mainly related to the presence, in some parts the model, of series possessing long-range correlation properties. Long-range correlations are characterized by a very slow decay of the autocorrelation function and suggest that the system possesses a long-term, multiscale memory of its previous states that affects its current behavior. Long-range correlations have been popularized through the concept of noise, thought of as a narrow range in the fGn/fBm model where long-range correlations reach maximal values. -like fluctuations have been discovered in the behavior of a number of physiological and behavioral systems (see references above) and also in several physical and nonliving systems [14–16]. In living systems, fluctuations have been evidenced in the behavior of young, healthy, and perennial organisms, and aging or disease seems to be characterized by an alteration of this long-range correlated behavior [1, 17]. This ubiquity has represented a very intriguing phenomenon in various scientific domains and its understanding has been a major challenge during the last decades. Long-range correlations are supposed to reflect the complexity of the underlying system, and noise a kind of optimal complexity, a compromise between order and disorder [1, 2].

Most analysis methods of long-range correlated processes are based on the statistical properties of fGn and fBm. However, we think that a number of issues about these processes remain unclear, often leading to erroneous interpretations. In this paper, we propose a formal analysis of the correlational properties of fBm and fGn, focusing on the limit behavior of these processes when approaching their definition boundaries. Considering the state of the art about this model, our main aim is to derive an analytical expression of the autocorrelation function of the discrete version of fBm.

#### 2. The fBm/fGn Model

Fractional Brownian motion (fBm), denoted by , was initially introduced by Mandelbrot and Van Ness [13] as a continuous stochastic process ranging over all nonnegative real values. A fundamental property of fBm is that in such process variance is a power function of the time span over which it is computed: where is the Hurst exponent, which can take any real value within the interval .

In experimental and engineering applications, researchers often deal with sampled data, and such sampling leads to a discrete-time version of fBm, , , referred to as dfBm. We focus in this paper on this discrete version, which corresponds to most analyses performed in physiological and behavioral experiments. Discrete-time fractional Gaussian noise (dfGn), denoted by , is defined as the series of increments in a dfBm ().

By definition, a dfGn is the difference of a dfBm, and conversely the cumulative sum of dfGn gives a dfBm. Each dfBm series is then related to a specific dfGn, and both are characterized by the same exponent. dfBm are nonstationary processes, as suggested by (1), whereas dfGn series present stationary mean and variance over time.

For , corresponds to ordinary Brownian motion, its variance is proportional to series length (normal diffusion), and is a white noise process. For , is subdiffusive, and successive values in are negatively correlated (antipersistent). In contrast, for , is overdiffusive and successive values in are positively correlated (persistent).

dfBm and dfGn are characterized by some essential basic properties [18]. Consider a dfBm process , , and its corresponding dfGn , . is the sum of first (). Let be the variance of the dfGn process . For large , the expected value of dfBm is zero (), as well as the expected values of differences within dfBm (). dfBm is characterized by self-similarity properties, which can be expressed at different levels. First, and similar to the property expressed in (1) for fBm, the variance of a sample path of length of a dfBm is a power function of :where is the scaling coefficient, depending on both and .

Another self-similarity property characterizes the variance of a fixed lag difference between dfBm values:

The autocovariance function of a dfGn is given by [18, 19]yielding a simple expression for the autocorrelation function of dfGn:

#### 3. The Autocorrelation Function of dfBm

The main aim of this paper is to derive an expression of the autocorrelation function of fBm in the discrete-time case. A well-known expression of the expected covariance of a continuous-time fBm series between two times and was given by Beran [18]:

This autocovariance function depends explicitly on and , and not only on , fBm being nonstationary [20]. Here we aim at deriving an approximate expression of the autocorrelation in the discrete-time case, depending only on and series length.

The autocorrelation of lag of dfBm is given by the ratio between the corresponding covariance and the variance of the process:which could be approximated for large as

For simplifying (8), we need an exact expression of , as a function of and . Consider a sample path of of length . The variance of this sample path, considering the nonbiased estimator, can be expressed as

In the simplest case , we have

Developing the preceding equation, we get

And replacing by ,

All terms containing vanishing in the previous equation, we obtain

Using similar calculations for , , and , we get, respectively,

Replacing by and incorporating (4), we finally obtain suggesting the extension, for a sample path of length :

One can easily show that if (18) is true for , then it also works for . In passing, we obtain an exact expression for the diffusion coefficient (2):

Combining (8) and (18), we get an expression for the autocorrelation function of dfBm:

#### 4. Limit Behaviors of dfBm and dfGn

Equations (5) and (20) allow analyzing the limit behaviors of dfBm and dfGn, when reaches the limits of the interval . Considering the upper limit of the interval, (5) predicts that when tends towards 1, tends towards 1.0 for all . In other words, when tends towards 1, tends towards a straight line.

One can easily show that

When tends towards 0, tends towards 1. Then, for all lag ,suggesting that tends towards white noise as tends towards 0. Accordingly, if one determines, on the basis of (5), the limit values of the autocorrelation function of dfGn when tends toward 0 [17, page 281], one obtains , , and, for , . This autocorrelation function corresponds to that of a differenced white noise [21].

We present in Figures 1(a) and 1(b) the theoretical values of and , obtained from (5) and (20), respectively, for values ranging from 0 to 1 in both families. This figure shows a clear breakdown in the correlation structures from dfGn to dfBm. As expected, starts at for , and it reaches 1.0 for . is 0 for and grows asymptotically towards 1.0 as increases.