Does 1/f noise really indicate self-organised criticality?
In [BTW], the authors used a 'sandpile' model to argue that
1/*f* noise can be explained in terms of self-organised
criticality.
However, Wentian Li,
creator of the
"1/f Noise Bibliography" explained to me in a series of personal
communications that it is not quite so straightforward:
"*Although Bak, Tang and Wiesenfeld claimed their model is able to
generate *1/f* noise, actually they made a mistake in forgetting to
square the Fourier transforms to get the spectrum. So their model as
it was would generate a *1/f^{2}* noise.
*1/f^{2}* noise is much less
interesting because it can be easily generated by a random walk
signal.*" [22/09/00]
"*The second author *[Chao Tang]* told me this fact in
person. They didn't write a retraction because another paper *[CFJ]*
mentioned the failure to generate *1/f* noise from the model. It
is unfortunate because (1) that second paper is not widely read and (2)
from that paper you cannot tell there is a simple error. You might
have thought that there was an uncertainty of the result.*"
[26/09/00]
Here is an excerpt from Bak's book on 1/f noise called
*How Nature Works: The Science of Self-Organized Criticality*
(page 95):
"*In an earlier work *([CFJ])*, performed while an undergraduate student in
Aarhus, Denmark, *[Kim Christensen]* showed that our analysis of
*1/f* noise in the original sandpile article was not fully
correct. Fortunately, we have since been able to recover from that
fiasco in a joint project by showing that for a large class of models,
*1/f* noise does indeed emerge in the SOC state.*"
W. Li comments:
"*...but then the whole premise that *1/f* noise is robust
breaks down - the model can generate anything depending on the
parameter setting...In my opinion, it was not just
"not fully correct", it was simply incorrect. *1/f^{2}*
is not the same as *1/f* noise. *1/f^{2}* lacks
long-range correlation, but *1/f* has *[it]*. Of course, in
real data, we may have *1/f^{1.8}*, *1/f^{0.5},
etc. *but if the exponent is very close to 2, it is just a minor
correction to the random walk.*" [28/09/00]
"[JCF] *tried to repeat the same spectral analysis of *[BTW]*,
but ended up with *1/f^{2}* noise. I was told by the
authors of *[BTW]* that because of this paper there is no need
for them to have a retraction.*"
"*Amazingly, the authors of *[BTW]* are not the only people
who made this mistake.
I list an erratum by Feder and Feder (1991) in which they admitted
that they forgot to square the Fourier amplitude.*" [01/11/00]
[BTW] P. Bak, C. Tang, and K. Wiesenfeld, "Self-organized
criticality", *Physical Review* B **38** (1988) 364.
[JCF] H. Jensen, K. Christensen, H. Fogedby, "1/*f* noise,
distribution of lifetimes, and a pile of sand",
*Physical Review* B **40** (10) (1989) 7425-7427.
K. Christensen's homepage
*Sadly, Per Bak died in October 2002 at the age of 54 (1948-2002).*
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