[om-list] Dimensions in knowledge space

[Thomas L. Packer at home]

Om

    Again, I like what Mark has said.

    (I also appreciate the footnote :-)

tomp

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----- Original Message -----
From: "Mark Butler" <butlerm at middle.net>
To: <om-list at onemodel.org>
Sent: Monday, 02 September, 2002 15:16
Subject: Re: [om-list] Dimensions in knowledge space


> MrM0j0r15n Rajeev the kanehbosm wrote:
>
> > I have a question regarding your discussion on dimensions in knowledge
> > space. My questions is, wouldnt, any "non-normative" KRS based on
dimensions
> > of quality, immediately degenrate into an infinite dimensional knowledge
> > space ? How does one utilise the useful concept of dimensions without
going
> > overboard ?
>
> Yes, the number of dimensions is arbitrary.  Most dimensions (like words)
are completely artificial, and once cannot pretend to represent everything
there is to know about an object in terms of a finite number of attributes
or dimensions.
>
> > For, eg, if OM were to be used in say political analysis, we might
arrive at
> > a point where we need to "ask" OM, of how one event is similiar to
another
> > event. In my opinion (uneducated), we would now need an additional
dimension
> > just to formulate such a question. Going in this direction, where can we
> > find the upper limit on the number of dimensions required ?
>
> There is no such limit, but no need to find it either.  One should strive
to have as few as possible, but many dimensions are irreducible.  For
example, friendship - friendship is a function of each pair of possible
friends and there is no necessary correlation between friend(A,B)
friend(B,C) and friend(A,C), which means that is least a three dimensional
problem, if not a six dimensional problem.
>
> The number of dimensions has no consequence - no fixed length vectors
here. In any given calculation, we might simply identify the relevant shared
dimensions and ignore all of the rest.
>
> Shared dimensions imply a very specific relationship between any pair of
objects.  More importantly, known relationships can be used to derive
unknown relationships, which is the whole point.  Given the position vectors
Rab and Rbc, Rac is easy to identify as the vector addition Rab + Rbc.  (*)
>
> Even when dimensions are strictly independent, as in the case of the
(N*(N+1))/2 relationships between N parties, there are very strong
statistical correlations between similar relationships.  If A is a  likes B
and B likes C, common experience surely indicatates that the probability
that A will like C is greater than the probability that A will like someone
chosen at random, notwithstanding extremely strong contrary experiences
where A can't stand the sight of C.
>
> Classical data modeling is all about determining a minimal set of
non-redundant single-valued attributes to describe each member of a set of
similar objects, which makes any data set from a well designed database well
suited to this type of dimensional analysis.  The whole field of data mining
is based on using techniques to inductively discover hidden relationships in
large bodies of such data.  This is only practical because data in a typical
database is highly normalized to begin with - if instead of having a nearly
complete data set about a large number of objects, you had a random
collection of uncorrelated propositions, one would have to normalize the
data somehow to some reasonable set of analyzable dimensions to begin with.
>
> I guess what I am saying is that the fact that objects share common
dimensions is the only thing that makes them relatable or comparable in the
first place, so viewing objects as points or extents in an arbitrarily
dimensioned space, at least as an abstraction, seems to me to be a both a
convenient perspective and one that appears to be complete enough to
represent any real world problem.
>
> - Mark
>
> (*) It was Tom's idea to treat relationships between objects as multi
dimensional vectors in a conceptual space, and more especially to apply
geometric analysis to the relationships between those vectors, particularly
in abstract dimensions where the existence of transitive geometric
relationships between objects is not at all obvious.
>
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