Re: [Fis] definition(s) of disorder/chaos: an example

Following Karl and Michel:

On Jun 3, 2004, at 1:43 AM, Michel Petitjean wrote:

> Here is the excellent suggestion of Karl:
>> take a set of n objects. Investigate the collection of symbols on a
>> most
>> usual fragmentational state.
>> Linearise the objects.
>> Count, how many have a place with tension in the extent of: a) no, b)
>> unit,
>> c) multiples of unit.
>> This measure has a distribution.
>> Call the range where measure minimal (where the objects fit to their
>> places
>> nicely / most) as "ordered". Call other ranges "less ordered".
>
> This is indeed suitable in many situations.
> Now let us look to the case of N undistinguishable points on the
> real line: e.g. having (x1,x2,x3) and (y1,y2,y3), how could we
> produce a criteria to know which of (x1,x2,x3) and (y1,y2,y3) is
> the more disordered ? It is one of the siplest situations (having
> only 1 or 2 points would probably be too simple here). If it cannot
> be solved, why ?

I am not sure where this thread is leading, but I am intrigued by the
novelty of this approach.

The answer to Michel's question must be contingent on the choice of the
dimension(s) being measured. For example, the series (x1,x2,x3) may be
more ordered than the series (y1,y2,y3) along the dimension of size
(e.g., a monotonic increase compared with a non-monotonic series), but
the reverse may be true along the dimension of shape (e.g., a
non-monotonic series compared with a monotonic change toward simpler
shape). The orderliness of short series like these can be estimated
using test statistics from methods like trend analysis. I suppose that
a statistical metric could be devised that integrated the degree of
order estimated from many dimensions, but I cannot imagine a metric
that would take into account all possible dimensions simultaneously,
including those dimensions for which we have no conception.

It also seems to me that limiting this thought experiment to series of
only 3 points can easily lead to false notions, like the general
importance of monotonic vs. non-monotonic trends. If you allow for
longer strings of points in one dimension, I think that a better
indicator of orderliness would be the predictability (in principle) of
the value of any one point from the overall pattern of the series. A
powerful statistical approach to estimating the orderliness of strings
would be to resample (jackknife) the data to create variably sized
subsets and track the rate of growth in the predictability of point
values. The rate of predictability increase with sample size should be
strongly indicative of the orderliness of the string.

There's my 2-cents.

Cheers,

Guy Hoelzer

Department of Biology
University of Nevada Reno
Reno, NV 89557

Phone: 775-784-4860
Fax: 775-784-1302

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Received on Fri Jun 4 20:05:28 2004