The spectra and periodograms of anti-correlated discrete fractional Gaussian noise

Abstract

Discrete fractional Gaussian noise (dFGN) has been proposed as a model for interpreting a wide variety of physiological data. The form of actual spectra of dFGN for frequencies near zero varies as f^1−2H, where 0<H<1 is the Hurst coefficient; however, this form for the spectra need not be a good approximation at other frequencies. When H approaches zero, dFGN spectra exhibit the 1−2H power-law behavior only over a range of low frequencies that is vanishingly small. When dealing with a time series of finite length drawn from a dFGN process with unknown H, practitioners must deal with estimated spectra in lieu of actual spectra. The most basic spectral estimator is the periodogram. The expected value of the periodogram for dFGN with small H also exhibits non-power-law behavior. At the lowest Fourier frequencies associated with a time series of N values sampled from a dFGN process, the expected value of the periodogram for H approaching zero varies as f⁰ rather than f^1−2H. For finite N and small H, the expected value of the periodogram can in fact exhibit a local power-law behavior with a spectral exponent of 1−2H at only two distinct frequencies.

Introduction

Recently, there has been considerable interest in the use of stochastic fractal models to help interpret physiological data [1]. Two models that have been investigated extensively in this context are fractional Brownian motion (FBM) and fractional Gaussian noise (FGN) [2]. These two models are related to one another, but the connection between them and how they are used in practical applications brings up some subtle issues that have not been fully appreciated amongst practitioners and that are the focus of the present manuscript.

Our starting point is FBM, which we denote here as B_H(t), 0⩽t⩽∞. FBM is a continuous parameter stochastic process that depends upon a parameter H, the Hurst coefficient, where 0<H<1. The qualifier ‘continuous parameter’ refers to the fact that the independent variable, t, ranges over all non-negative real values. In practical applications, we must deal with sampled data, which leads us to consider B_H(t) at just the integers $\text{t=0,1,2,…}\text{.}$ This sampling leads to a discrete parameter version of FBM, which we refer to as dFBM. For clarity, we henceforth refer to the continuous parameter version of FBM as cFBM.

Flandrin [3] shows that, even though cFBM is a non-stationary process, it has a well-defined power spectrum S(f) that obeys a power law exactly over all frequencies; i.e., S(f) is proportional to f^−1−2H for −∞<f<∞. Flandrin also considers a form of a derivative of cFBM, namely, a process defined as $\text{lim}{}_{\mathrm{\delta \to 0}}\text{((B}{}_{\mathrm{H}}\text{(t+δ)−B}{}_{\mathrm{H}}\text{(t))/δ)}$. He shows that this derivative process also has a spectrum that obeys a power law exactly over all frequencies and is given by S(f)∼f^1−2H. The concept of FGN as formulated by Mandelbrot and Van Ness is related to this derivative process in that, rather than letting δ decrease to zero, we fix it at unity. This leads us to what we will refer to as continuous FGN (cFGN) and discrete FGN (dFGN). cFGN is defined as X(t)=B_H(t+1)−B_H(t), where 0⩽t⩽∞, resulting in a spectrum given by the product of the squared gain function for a first difference filter and the spectrum of cFBM, i.e., $\text{S(f)∼4}\text{sin}{}^2\text{(πf)f}{}^{\mathrm{-1-2H}}$, which approaches a power law only as f→0. dFGN is obtained by restricting t to the non-negative integers. Note that, in addition to being the sampled version of cFGN, the process dFGN can be regarded as the first difference of dFBM. We also note that, due to the sampling, the spectra for dFBM and dFGN are even periodic functions with a period of unity, so the frequencies of interest satisfy −1/2⩽f⩽1/2, and their respective spectra also approach power laws only as f→0.

Both dFGN and dFBM have found application in many areas of science and have been applied to discretely sampled time series of many natural processes. The fact that cFBM and its derivative have spectra that exactly obey a power law over −∞<f<∞ has sometimes mistakenly been taken to hold both for the spectra of dFBM and dFGN over −1/2⩽f⩽1/2 and for estimates of these spectra given by the periodogram. Sometimes the power-law expression is cited directly, and it is implied or assumed that the expected estimated spectra as computed by the periodogram for dFGN are a power law everywhere [4], [5], [6]. The idea that the spectra of cFGN are a power law everywhere has been transferred to power-law behavior of dFGN spectral estimates, and the caveat that the actual spectra for dFGN are only a power law in the limit as f→0 appears to be unheeded. Churilla et al. [4] posit separate power-law behavior for the low frequencies and the high frequencies of the periodogram and derive separate Hurst coefficients for both by fitting each band of frequencies with a different power law based on the power-law formulation. The spectral synthesis method of Peitgen and Saupe [7] and that modified by Bassingthwaighte and Raymond [8] generated dFGN by an inverse Fourier transform of the power-law spectral coefficients after phase randomization and multiplying the amplitudes by random Gaussian numbers. The expected value of periodograms from this kind of process will be a power law, and does not have the same spectrum as a dFGN. Pilgram and Kaplan [5] used a spectral synthesis method whose basis is the power-law spectrum, but studied only spectra for H⩾0.5 and therefore did not observe the incorrect spectral representation at the low frequencies though their method does also represent the highest frequencies incorrectly.

If we plot S(f)=f^1−2H versus f on a log–log scale, we see a line with a slope of 1−2H. To determine frequencies over which S(f) for dFGN varies as f^1−2H, we define a local spectral exponent

$$\text{e(f,δ)=}\text{log}\text{(S(f+δ))−}\text{log}\text{(S(f−δ))}\text{log}\text{(f+δ)−}\text{log}\text{(f−δ)}\text{,}$$

where 0<δ<f. As δ→0, $\text{f→0,}\text{e(f,δ)→1−2H}$. We also define a local Hurst coefficient

$$\text{H(f,δ)=(1−e(f,δ))/2}\text{.}$$

By plotting H(f,δ) as a function of f with δ set to a small number, it is possible to see at what frequencies this function is approximately equal to the Hurst coefficient, showing the region where S(f) varies as f^1−2H to a good approximation. It is not widely appreciated how very low the frequency must become when H is also small for the spectra to exhibit power-law behavior.

Marked departure from nominal 1−2H behavior of dFGN spectra occurs under several conditions. At high frequencies near f=0.5 (the Nyquist frequency for the data set sampled from cFGN), the dFGN spectra flatten out and the local spectral exponent is zero. When H is near zero, a significant portion of the frequency band beginning at f=0.5 and extending through mid-range frequencies also deviates from nominal 1−2H power-law behavior. The only spectrum of dFGN that obeys the nominal power law everywhere is for H=0.5, i.e., the ‘white’ noise spectrum with zero exponent everywhere. For H>0.5, the spectra of correlated or persistent dFGN exhibit a significant discrepancy from nominal power-law behavior only near the highest frequencies. The flatness of these spectra near f=0.5 is less remarkable than the overall impression of power-law behavior [9, p. 54]. For H<0.5, the spectra of anti-correlated or anti-persistent dFGN have more extensive regions of non-power-law behavior than what is observed for positively correlated dFGN spectra. More attention has been paid to positively correlated than to anti-correlated series, and consequently the impression of power-law behavior for all spectra of dFGN has been inadvertently reinforced.

The next three sections of this paper are devoted to (1) calculating the local Hurst coefficients, H(f,δ) from Eq. (2) for the spectra of dFGN; (2) comparing the local H(f,δ) from spectra and periodograms and showing the effects of series length, N; and (3) evaluating the utility of the local spectral exponents, e(f,δ), obtained at the lowest available frequencies of the periodograms. The results emphasize that the 1−2H power-law behavior of the spectra applies only to a very small range of frequencies when the Hurst coefficient is small; the expected value of the periodogram exhibits power-law behavior only over a small frequency band, if at all; the expected values of the periodograms at the lowest available frequencies for anti-correlated dFGN do not converge to the actual spectra as N increases; and for very small H, the local spectral exponent of the smallest available frequencies from the periodogram of dFGN is zero.