this was supposed to go to the list.
---------- Forwarded message ---------- Date: Mon, 21 Aug 2006 01:10:13 +0200 From: jose manuel berenguer jmbeal@telefonica.net To: Mathieu Bouchard matju@artengine.ca Subject: Re: [PD] Colored (fractal, 1/f^a) noise generator for PD
El 20/08/2006, a las 20:58, Mathieu Bouchard escribió:
On Sun, 20 Aug 2006, Charles Henry wrote:
1/f^a noise is a fractal because it is self similar under different size windows. 1/(kf)^a = 1/k^a * 1/f^a, so the magnitude of spectrum viewed over a different window is just scaled by a certain amplitude. The same applies to the time domain as well (but it has to be interpreted as a probability function).
Then by this standard, the 1/x function is self-similar, and so are all hyperbolas. That is, as long as similarity is defined as modulo the group of diagonal matrices conjugated by rotation matrices.
this is true. but fractals are compact (closed and bounded) that shows self-similarity and fractional dimension. f(x) = 1/x is self- similar, but its dimension is 1 and it is not a compact set.
Isn't the definition of fractal requiring some kind of noninteger Hausdorff dimension?
yes, indeed. and this is the case of that noises. don't look at theoretical spectra in frequency domain but signals in time domain ... their dimension is fractional and they show self- similarity. I haven't now time to show they are compact sets, but, for shure, a proof could be found in Mandelbrot, Barnsley, Peitgen, etc ...
jmb
jose manuel berenguer jmbeal@telefonica.net +34932857046 +34696538403. http:// www.sonoscop.net/jmb/ jmberenguer@sonoscop.net +34933064128. http://www.sonoscop.net/ caos->sonoscop. cccb. montalegre, 5. 08001 barcelona spain
On Sun, 20 Aug 2006, jose manuel berenguer wrote:
and this is the case of that noises. don't look at theoretical spectra in frequency domain but signals in time domain ...
If a signal is random then how can I expect any part of it to be similar to any other part of it?
And then, if I look at more general tendencies using bigger signal blocks or averaging a lot of signal blocks together, how do I NOT approach a theoretical model from probability theory?
And then, how can a mapping from two incompatible dimensions (watts vs seconds) be analysed with methods that were made for two dimensions of the same kind? I mean, you could measure the Hausdorff dimension of a signal by setting some kind of equivalence rule between watts and seconds (huh?) and then seeing the signal as a planar curve; but, by itself, the signal is not a planar curve.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju | Freelance Digital Arts Engineer, Montréal QC Canada
On Mon, Aug 21, 2006 at 12:12:58AM -0400, Mathieu Bouchard wrote:
On Sun, 20 Aug 2006, jose manuel berenguer wrote:
and this is the case of that noises. don't look at theoretical spectra in frequency domain but signals in time domain ...
If a signal is random then how can I expect any part of it to be similar to any other part of it?
I'm speaking from mathematical ignorance, but isn't Perlin noise random but self similar at different scales? I guess the key is that it's periodical at some time scale.
Best,
Chris.
chris@mccormick.cx http://mccormick.cx
On 8/20/06, Mathieu Bouchard matju@artengine.ca wrote:
If a signal is random then how can I expect any part of it to be similar to any other part of it?
And then, if I look at more general tendencies using bigger signal blocks or averaging a lot of signal blocks together, how do I NOT approach a theoretical model from probability theory?
If you average signal blocks together, they should approach zero. How not to approach it from probability? I have no idea.
The only reason I know jack about this problem is that I just had a course on fractional Brownian motion and stochastic calculus this year. And still.... it's a friggin devil to wrap your mind around :)
The way these noises are self similar is in terms of their auto-covariance. The different values of the Hurst exponent, h, determine the exponent, a, in 1/f^a noise. A single stochastic process unfolds in a single way (it's a single deterministic function) out of a large number of possiblities (one big function space).
A single fractional Brownian motion is a probabilistic function of time that has 0 mean (relative to the point we start at) at all points in time, and a variance of t^(2h).
The auto-covariance is a measure of how the function correlates with itself with different amounts of lag. For different values of time, s and t, the auto-covariance of this type of "random" signal is |t-s|^(2h). So, in the time domain, it is correlated with itself by using a probabilistic interpretation. I may be wrong....I had substantial difficulty with the subject, and I'm certainly interested in finding out how this stuff really works.
Chuck
Charles Henry wrote:
On 8/20/06, Mathieu Bouchard matju@artengine.ca wrote:
If a signal is random then how can I expect any part of it to be similar to any other part of it?
And then, if I look at more general tendencies using bigger signal blocks or averaging a lot of signal blocks together, how do I NOT approach a theoretical model from probability theory?
... The auto-covariance is a measure of how the function correlates with itself with different amounts of lag. For different values of time, s and t, the auto-covariance of this type of "random" signal is |t-s|^(2h). So, in the time domain, it is correlated with itself by using a probabilistic interpretation. I may be wrong....I had substantial difficulty with the subject, and I'm certainly interested in finding out how this stuff really works.
Yes, it's not instantaneously self-similar, it's statistically self-similar and the autocorrelation and autocovariance both increase as a increases from zero in 1/(f^a). When a = 0 you have white noise and the signal is randomly correlated with itself, but as you add low-frequency power to the signal, the correlation at any given lag will have to increase because of the low frequencies present. Beyond red noise there is also 'long-tailed' noise (for example: http://www.pnas.org/cgi/content/full/102/13/4771 or the stock market) where there is more correlation than expected at greater lag, or clumping or non-gaussian distribution of values. This doesn't seem to be covered by the 1/(f^a) concept.
Martin
hello. sorry for the delayed answer. i was away.
El 21/08/2006, a las 6:12, Mathieu Bouchard escribió:
On Sun, 20 Aug 2006, jose manuel berenguer wrote:
and this is the case of that noises. don't look at theoretical
spectra in frequency domain but signals in time domain ...If a signal is random then how can I expect any part of it to be
similar to any other part of it?
similar doesn't mean equal... a function is similar to any other
funtion when their points look similar. thy would be equal if their
points where equal.
a function is self-similar because the distribution of points of f(x)
at any scale remembers distribution of points of f(x) at any other
scale. don't see why a random signal can't be self-similar. could
you guess the interval of a part of a white noise signal only by
knowing its points? at, say, [0, 1], it looks similar to what it
would be obtained at [0.5, 0.6]
And then, if I look at more general tendencies using bigger signal
blocks or averaging a lot of signal blocks together, how do I NOT
approach a theoretical model from probability theory?
i'm very sorry and apologize for this statement : i wasn't correct
here... for theoretical i wanted mean you say "expected" in another
email. we must, indeed, approachbig and averaging theoretical models
with theoretical tools, probability theory or any other else.
actually, this misunderdstanding is due to my so bad knowledge of
English.
And then, how can a mapping from two incompatible dimensions (watts
vs seconds) be analysed with methods that were made for two
dimensions of the same kind?
this is not a problem of fractals. it's moreover a problem arising
sometimes when some entity uses some fractal to model some phsysical
reality. fractals are mathematical objects, not physical ones.
mathematical approaches support infinity, physical ones don't. a
coast approaches some fractal at a rasonable set of scales, of
course, but there is a level of scales, little enough, in which the
coast doesen't approaches the fractal anymore.
this is the reason I do not use to enjoy "fractal music".
I mean, you could measure the Hausdorff dimension of a signal by
setting some kind of equivalence rule between watts and seconds
(huh?) and then seeing the signal as a planar curve; but, by
itself, the signal is not a planar curve._ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju | Freelance Digital Arts Engineer, Montréal QC
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jose manuel berenguer jmbeal@telefonica.net +34932857046 +34696538403. http:// www.sonoscop.net/jmb/ jmberenguer@sonoscop.net +34933064128. http://www.sonoscop.net/ caos->sonoscop. cccb. montalegre, 5. 08001 barcelona spain
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Charles Henry -
Chris McCormick -
jose manuel berenguer -
Martin Peach -
Mathieu Bouchard