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/f2 noise. 1/f2 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/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
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