From: bhv@areaplg2.corp.mot.com (Bronis Vidugiris)
Subject: Re: a test for white noise?
Organization: Motorola, CCR&D, CORP, Schaumburg, IL
Distribution: na
Date: Wed, 7 Oct 1992 22:53:56 GMT
Keywords: white noise

In article <gilligan.718429928@camelot> gilligan@camelot.bradley.edu (Edward Henigin) writes:
)Hi y'all . . . . 
)If you had a set of data points (a 2d array: position versus time for ex.)
)what set of criteria does it have to meet to be classified as
)'white noise' or 'brownian motion' or whatnot?  How could you tell
)if it was purely random? Is there a degree of randomness, ie, a
)scale of 0 to 100 where 0 is totally un-random and 100 is totally
)random?  And, is my head up my wazoo for thinking that brownian motion,
)white noise, and pure randomness are essentially the same thing?
)
)Either direct info or references to books or articles would be 
)greatly appreciated.

Absolutely 'pure' white noise requires that there statistically be no
correlation between samples at different times - each data point is
independent of every other.

Almost any physical process can't change its value instantaneously, so it
won't really be 'white'.

The 'whiteness' of noise can be addressed mathematically by looking at the
auto-correlation function - if you've got x = f[t], you look at the
fuction r(t1,t2) = f[t1]*f[t2] (actually the expected value of f[t1]*f[t2]]. 

If the 'noise' is not time varying, and generally otherwise well-behaved,
you'll find that r(t1,t2) is really a function r1(t2-t1) of the time
differeince - i.e. it's independent of the actual time.

r1 is the 'autocorrelation' fuction.  r1 will be zero for all non-zero
arguments for truly 'white' noise - i.e. there will be no correlation between
any two data points regardless of how close they are together as long as
there is some nonzero time difference between them.

Probably any book about 'random processes' or 'noise' will go into a lot more
depth on this aspect of things.

Most real continuous processes have some characteristic auto-correlation
function which is not a pure 'impulse' function (i.e r1(t) is unbounded when
t = 0 and zero otherwise)  - this is equivalent to saying they are
bandlimited.  If the correlation dies out sufficiently fairly rapidly, they
can be considered to be 'white enough' (or broadband enough) for experimental
purposes, however.

Things get a lot tricker when one is talking about whether arbitrary
sequences generated by say a computer are random or not.  Any sequence of
numbers generated by an algorithm isn't 'really' random in some sense.
For one thing, it will eventually repeat.

For 'practical' (as opposed to highbrow algorithmical-theoretical)
statistical tests done on computer 'pseudorandom' number generators, Knuth
writes about this somewhat in his Art of Computer Programming series.

Terminology can get confusing here - you could write a computer program
that generated numbers that appeared to have the property of 'white' random
noise I mentioned (being non-correlated with each other), but they wouldn't
really be 'really' random in some senses of the word.  And really 'random'
things, like zener diode noise, might not (and would not) in practice be
totally 'white'. Hope this helps.