Appendix G. Is There An Alternative to 1/f Noise?
Can we be certain that our spectra are indicative of 1/f noise? Indeed, spectra with a shape roughly similar to that of 1/f noise can result from other models. In particular, it could easily be conceived that the ecological variables we have measured act as first-order autoregressive (AR1) processes. In this case, the value of a variable at one point in space or time depends in part on the value at an adjacent point, as well as on chance. For example, if we follow an autoregressive variable q through time:
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(G.1)
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where q(t) is the value of the variable at time t, and q(t-1) is the value at time t-1. a is the regression coefficient that determines how alike two adjacent points are (on average), and e is a random variable that can be either positive or negative (Priestley 1981). Autoregressive processes can act in space as well as in time.
The spectra of first-order autoregressive
process are in some respects similar to those of 1/f noise (Fig.
G1). For a portion of the spectrum, the log of the frequency-specific variance
decreases approximately linearly with the log of frequency. The most important
difference between the spectra of 1/f noise and autoregressive processes
lies in the low-frequency tail. Whereas a 1/f-noise spectrum continues
to increase without limit at low frequencies (for any
> 0), a first-order autoregressive spectrum "flattens out" and
both the spectrum and the total variance approach a limit. The definable limit
associated with an AR1 process is an appealing alternative to those 1/f-noise
processes for which b
1.
The applicability of autoregressive processes is most apparent for temporal data. If the current state of a variable depends at least in part on its state a moment ago, the process is intrinsically autoregressive (although not necessarily to first order). Furthermore, the directional nature of the autoregressive process described by Eq. G.1 is in accord with the nature of time itself. Time's "arrow" ensures that the state of a variable can depend on history, but cannot depend on the future. The application of Eq. G.1 to spatial variables is less intuitive, and perhaps less appropriate. The state of a variable at location x might easily depend as much on what is happening at location x + 1 as it does on location x - 1, but this type of bi-directional dependence is not incorporated in the simple AR1 process described by Eq. G.1.
The distinction between 1/f-noise
and first-order autoregressive processes disappears under certain circumstances.
When a is 1, Eq. G.1 describes a random walk, which has a spectrum identical
to that of a 1/f-noise process with
= 2. When a is 0, the result is equivalent to white noise, a 1/f-noise
process with
= 0.
We compared our data (those for which there was no dominant spectral peak) to these two null models (1/f noise, AR1) by calculating on a log-log plot the mean squared error between our spectral estimates and either the best-fit line of a 1/f-noise process or the best-fit curve calculated for a first-order autoregressive process (Priestley 1981). In 18 of the 24 cases, the linear (that is, 1/f) fit yielded a smaller mean-squared error. Three of the six exceptions occurred on the short transect (maximum temperature, species diversity, and mussel disturbance). If these short-transect exceptions are indeed examples of an autoregressive process, the AR1 fit should be even more evident on the medium transect, which, with its greater extent, should reveal more of the flat, low-frequency tail of the spectrum. In each case, however, on the medium transect a linear fit has a lower mean-squared error than does the AR1 fit. In only two cases (species diversity and grazer abundance) does the AR1 process provide the best fit on the long transect.
Given the relatively small number
of points in each of our spectra, and the relatively broad confidence limits
on each spectral estimate (see Appendix E), it
would be difficult to differentiate definitively between these two null models,
but based on this preliminary evidence we proceed on the assumption that the
overall spatial variation we have measured is better modeled by a 1/f-noise
process rather than an autoregressive one. The average
among our variables is 0.69 (SD = 0.36). Thus, taken
as a whole, our data are "pinkish" noise, in concert with the concepts
discussed by Halley (1996), and in general they have a
< 1, which avoids the problems mentioned above regarding unlimited variance.
Halley, J. M. 1996. Ecology, evolution and 1/f noise. Trends in Ecology and Evolution 11:3337.
Priestley, M. B. 1981. Spectral analysis and time series. Academic Press. New York, New York, USA.
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FIG. G1. A comparison of the spectra for a 1/f-noise process and a first order autoregressive process. The spectra diverge at low frequencies. |