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
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/f2 noise.
1/f2 noise is much less
interesting because it can be easily generated by a random walk
"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."
Here is an excerpt from Bak's book on 1/f noise called
How Nature Works: The Science of Self-Organized Criticality
"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/f2
is not the same as 1/f noise. 1/f2 lacks
long-range correlation, but 1/f has [it]. Of course, in
real data, we may have 1/f1.8, 1/f0.5,
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/f2 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).
1/f noise, signal processing and number theory