From: mailbox@dmitry-kazakov.de
On Sun, 20 Jul 2003 23:14:17 GMT, "project2501"
wrote:
>i'm looking into a clustering algorithm (modififcation of standard k-means
>to start with) for fuzzy data.
>
>a simplified 3-dimensional data instance may take the form:
> (
> length = short/0.2 + medium/0.8 + long/0.3, texture = smooth/0.3 +
> rough/0.7,
> colour = red/0.3 + blue/0.7 + green/0.1 )
>
>where short, medium, long are predefined (different) shapes, or fuzzy sets
>on length. and similar for texture and colour.
>
>my question is this: are there existing standard distance measures between
>data instance of this type?
No in general, yes if there is a prefered distance defined. For
example if length has a predefined Euclidean distance, then it would
induce some function which for L1 = short/0.2, medium/0.8, long/0.3
and L2 = short/0.4, medium/1, long/0 would yeld a fuzzy number
describing d(L1, L2).
1D vs. 3D or nD plays no role.
> i have spent some time looking around the web,
>citeseer, etc ... and have reach the opinion that there is no generally
>agreed upon general measure? (keeping the scaling ussue aside, assume
>normalised according to application).
>
>even if we use the standard euclidian distances, how does one combine the
>components:
> (length1^2 + length2^2) + (texture1^2 + texture2^2) + ... ?
>
>thanks for opinions, advice or pointers.
It is easy if you have some crisp distance (like Euclidean one). Then
you might simply ask the question: how it is possible that the
distance between A and B is x? So the answer is a fuzzy set which for
any given number x has the membership function fd(A,B) (x) =
Sup min {fA(w1), fB(w2)}
(w1, w2) | d(w1,w2) = x
N.B. There could well be (and are) alternative ways to define a
distance between fuzzy sets. So the answer: is there is no "standard"
way.
---
Regards,
Dmitry Kazakov
www.dmitry-kazakov.de
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