random systems

[tester]

From time to time I wonder about so called "random systems" and their level of non/randomness. We call "random" what we can't predict or narrow (sometimes it's due to complexity, sometimes due to lack of it). But can we narrow the non-randomness in so called "random systems"?

Here is a thought. Let say that you have a soup. The soup is made of flavored water and tiny tiny pieces of macaroni, so tiny, that the soup looks like a uniform mass. But "macaroni" = granulation and repeatable patterns in that mass. So now the questions. Let say that we have specific type of noise; matrixes that are made of segments - segments made of smaller sets - sets made of unique samples - samples that have values in certain range.

Can we determine whether in that noise - are there so called "short patterns"? A tiny tiny macaroni pieces? These short patterns would be made of few samples per set or samples that are connected between/through few segments.

Are there "shapes" of these patterns to seek for (like in wavelet comparisons?), or can be they generated somehow from the noise? What are the shapes of these tiny tiny macaroni pieces?

How to determine how big (how many samples) these patters would be? How long are these tiny tiny macaroni pieces

Can we determine the density of short patterns in the noise? How many tiny tiny macaroni pieces are in the soup per plate?

And final question - can we determine which samples in current sampleset are involved in these short patterns, and can we predict what next samples (values) might refer to these samples (or surrounding holes)?

These questions are related to measurement of local non-randomnes in random pattern. They should help to determine whether the noise is random or pseudorandom, or to be more precise - whether in noise there are patterns/attractors/regularities or local fluctuations, that are not shown by classic gaussian probability.

pufff... any examples in FS?

p.s.: just a short note - ruby random generator seems to be less random than FS/SM based one. It produces more numbers of certain kind. I suspect this is because there are less real-time-dependent triggers within the process of generation. But I don't want to measure ruby here; ruby generator just made me to ask such questions.

[tester]

Here is an example on audio noises.

Question. Should a white noise "sound" like anything?
If it "sounds", and it sounds in a uniform way, they its a mixture of predictable patterns.
More uniform than white noise is a DC (yep, pure offset level), which is transparent.
Lossless compression of audio files also proves that audio noise is patterned.

[tester]

Interesting? Not interesting?
http://www.wallenberg.com/kaw/en/resear ... m-patterns

[trogluddite]

Not so fast - you're making some big assumptions here!

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If it "sounds", and it sounds in a uniform way, they its a mixture of predictable patterns.

There's no need for a pattern - human hearing does not work by following the contour of the waveform.
Consider this - toss a coin many times, and tally how many heads and how many tails. Assuming that neither the coin, nor the method of tossing, bias the results, then after enough tosses of the coin, we discover that the probability of tossing heads approaches 0.5 more and more exactly. We have discovered a property of the system which we are able to classify - but this in no way enables us to predict what the result of the next coin toss will be.
Now consider the definition of white noise...

white noise is a random signal with a flat (constant) power spectral density. In other words, a signal that contains equal power within any frequency band with a fixed width.

And human hearing works in a similar way - even at the physical level of the cochlea of the inner ear. Effectively, this organ works something like an FFT, measuring the amount of power contained within critical bands of frequencies.
Thus we can tell that "white"/"pink"/"brown" noise are subjectively different sounds - because the "analysis" can tell that certain frequency bands contain more/less power than others.
As with the coin tossing, there is no need for there to be any kind of pattern - only a reliable and predictable statistical outcome over some time period. We can no more predict the instantaneous sound pressure level at a given moment than we can predict the outcome of a single coin toss.
This of course does not prove that there is NOT a pattern - but, as no pattern is required, neither can we conclude that there IS a pattern.

Lossless compression of audio files also proves that audio noise is patterned.

No proof there either, I'm afraid.
Principle of information theory tell us that IF every single bit of every single number was random, the signal would not compress.
However, some compression is nearly always possible because that condition is very rarely met.

Consider - the quiet passage between two songs. There is inevitably some noise here, even if it is only dithering noise - however, it is at a very low level - so only a few of the lowest value bits in the number are changing. Thus, during this passage, the audio can be encoded as, say, only eight bit numbers without any data loss, because the upper bits of the number are predictably zero for some length of time. This does not require the lowest bits to be in any way predictable - the "redundancy" is an inherent part of the medium, not a quality of the signal.

Another method used is to encode not the actual sample values, but rather the difference between adjacent samples. For most real audio signals, the amount of random noise is small, and the regular fluctuations of "music" waveforms are relatively slow. So, much of the time, the difference signal can be encoded using far fewer bits than those needed to record the full sample value.

Unlike "lossy" compression, these methods do not rely on predictive or perceptual methods to identify redundant data - but can still yield a useful amount of data compression on 'real world' signals. But this also explains why lossless compression very rarely achieves the same degree of compaction as lossy compression.

Again, this does not prove the absence of patterns, but nor does it prove that there are.


Interesting link, BTW.
But look closely at the language - the use of the word "pattern" in particular. It is not describing patterns in the sense of the microscopic level of individual data point - the patterns they refer to are macroscopic patterns contained in the statistical distributions - "meta-patterns", if you like.

[tester]

Generally I'm trying to figure out something else than audio, but using signal processing similar to audio approaches. It's like the difference between ocean surf and blowing wind, or ocean surf in various places / at varous weather, or different rivers/streams (see the freesound.org).

I "assume", that although various generative systems (like lottery systems) produce some sort of unpredictable stuff, these "noise generators" - let say "sound different" from each other (sonification as an alternate way of visualization) due to different constructions. I'd like to find a way of determining these differences in practical way. Lottery systems use machines, and these machines have specific "room response" (but not necessarily "spectral") so to speak (fractals, attractors, call it whatever you wish). I'm not breaking the probability nor cracking the game, I'm just defining in which way their randomness is non-random (local/temporary whirls so to speak), because nature combines coherent systems (otherwise humans would not be possible designs?). Seeing various facts around me and in (not only my) past - I'm not satisfied with current explanations provided by mainstream gurus.

Statistics is not my domain. But I use word "pattern" in both ways (meta-pattern is a good word). Distribution seems go into direction of narrowing soup to water and maccaroni pieces of probability and possibility at certain degree of confidence (whatever).