Re: sampling from a circular distribution

From: Einar Andreas Rødland (e.a.rodland_at_labmed.uio.no)
Date: 08/13/04 

Date: Fri, 13 Aug 2004 10:58:35 +0200
To: David Alexander <scalesymmetry@optusnet.com.au>

David Alexander wrote:
> I have an image made up of grey-scale pixel values--1 to 180 (pi/2) in
> integer steps. These values are circular i.e. 180+1=1.

To consider grey-scale values to be circular immediately seems odd to
me: that amounts to saying that black=white. Or are the grey-scale
just an imperfect representation of values on a circular scale?

> What I want to show is that the sample is not a random sampling of the
> population i.e. that it is systematically biased.

Since the data points are circular, a natural way to model the data is
in terms of sine and cosine: i.e., for X(i)=number of pixels of color
i, use a model on the form

X(i) ~ a0 + a1*cos(i/180*2*pi) + b1*sin(i/180*2*pi) + ...
          ... + bn*cos(n*i/180*2*pi) + bn*sin(n*i/180*2*pi)

or, more generally,

F(X(i)) ~ a0 + ....

for some transformation F() where n is and integer indicating the
order of your model. The assumption behind this model is that if X(i)
is biased in any direction, X(j) is likely to have a similar bias for
any j close to i: i.e. the X(i) are auto-correlated. If no such
assumption is made, you have 180 totally independent data points, and
the suggestion of Ross Clement is more appropriate.

This can then be analysed using either GLM (or regression) if you're
willing to assume that X(i) is roughly normally distributed (probably
not a very good assumption) or generalised linear models assuming
f.ex. a Poisson distribution (i.e. Poisson regression).

You can also use rk = sqrt(ak^2+bk^2) as a measure of the amount of
variation with frequency k and plot r1,r2,r3,... to give you an
indication of the range over which the X(i) auto-correlate.

However, note that the null hypothesis of this model is that the image
is complete noise, which is a very strict assumption: e.g. if you take
an image which is complete noise and scale it up by a factor 2, each
pixel will be repeated 4 times and the null hypothesis violated as the
pixels are no longer independent.

Einar

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