**Stephen P. King** (*stephenk1@home.com*)

*Mon, 26 Jul 1999 19:36:14 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 498] Re: [time 497] [Fwd: Fuzzy projections and cylindrical extensions]"**Previous message:**Stephen P. King: "[time 496] Re: [time 491]Forms, Entailment structures, intersections, etc."**Next in thread:**Stephen P. King: "[time 498] Re: [time 497] [Fwd: Fuzzy projections and cylindrical extensions]"

Hi All,

Toward the discussion of how to formulate interactions between Local

Systems and to address the question of distributivity raised by Ben, I

offer this attached message and this URL:

http://www.born-again.demon.nl/phd.html

Onward!

Stephen

**attached mail follows:**

*> I was trying to understand Fuzzy projections from the meterial that I
*

=

*> have - Not so descriptive. Can some body please explain the concept
*

more =

*> descriptively or give pointers towards some papers, tutorial =
*

*> books,applications etc.
*

Projection can be viewed as aggregation or consolidation of

information. Two notions are important for defining the

projection: dimension and measure. Dimensions are sets

(of values of variables) which define the space along which

we aggregate information (they are usual variables). Measure

is a special variable which represents the information we

aggregate. For example, if measure is SALES and dimension is

TIME then we can aggregate the sales along the time and thus

formally project the sales distribution over the time dimension.

The time dimension is then removed from the consideration.

Generally, projection can be viewed as an operation of

decreasing the number of dimensions over which the distribution

is defined. For example, finding sales over all dimensions

means that we find projection onto the 0-dimensional space

which does not include any dimensions and consists of one

point (empty set).

Cylindrical extension is an opposite operation which can be

viewed as deaggregation, deconsolidation of information along

new dimensions. In this sense good name for this operation is

deprojection. It always increases the distribution dimensionality.

For example, suppose we have 0-dimensional distribution over 1

point which is meant as the total sales. Then we can deproject

this measure represented by the one number, say, on the time

dimension and obtain 1-dimensional distribution which obviously

should be constant since no new information is added. It is also

natural, that the projection of this distribution along the time

should result in the same initial number.

It is frequently important what concrete operations are used

to define projection and deprojection. For many real world

applications usual arithmetic operations (sum and division,

respectively) are the most natural. The case of probabilistic

distributions also can be viewed as based on arithmetic

operations. For fuzzy case, however, it is supposed that the

measure takes values from [0,1] and the operations minimum or

maximum are used for projecting. Note that it depends on the

modality we assign to the fuzzy distribution what operation

(maximum or minimum) to choose. If we interpret the fuzzy

distribution as possibilistic then the maximum is used (it

can be considered definition of possibility distributions).

For example, if we have 1-dimensional distribution p(x)

over the variable x, then its projectin onto 0-dimensional

distribution is calculated as maximum of all numbers p(x)

for all x: p = max(p(x)), x \in X. Deprojection for fuzzy

case is defined equal to the initial distribution, e.g.,

when we add one new dimension x we obtain: p(x)=p. This is

due to the definition of fuzzy division: p/n=p, where

p \in [0,1], and n is natural number (the power of new

dimensions). Interestingly, the term "distribution" of some

value means that there is some quantity (say, initial mass equal

to 1) which is then *distributed* over some space, i.e., over

a set of points each of which obtains some part of

this value. In the case of no additional information all

points have equal rights and it is exactly deprojection

operation.

Deprojection can be thought of as intensionalisation since it

allows representing distributions (knowledge) over large space

with the help of a relatively small number of lower dimensional

distributions (usually 1-dimensional). This is why these

operations are important for knowledge representation techniques.

In fact, the problem of logical inference can be formulated

as finding projections of some intensionally represented

distribution. In other words, we have a distribution which

is represented by several lower dimensional distributions,

say, obtained from rules; it is necessary to find its projection

on the target variable making use of only this representation

(i.e., without calculation of its values in each point).

In general case, and particularly in probabilistic case this

is rather difficult (unsolved) problem. For fuzzy case, i.e.,

finding projectins of fuzzy multidimensional distributions it is

much easier since minimum and maximum operations are in some

sense degenerated (exterme) operations with rather well

properties.

Regards,

-- Alexandr A. Savinov, PhD GMD - German National Research Center for Information Technology AiS.KD - Autonomous Intelligent Systems Institute, Knowledge Discovery Team Schloss Birlinghoven, Sankt-Augustin, D-53754 Germany tel: +49-2241-142629, fax: +49-2241-142072 mailto:savinov@usa.net, http://ais.gmd.de/~savinov/Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't.

############################################################################ This message was posted through the fuzzy mailing list. (1) To subscribe to this mailing list, send a message body of "SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at (2) To unsubscribe from this mailing list, send a message body of "UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL yoursubscription@email.address.com" to listproc@dbai.tuwien.ac.at (3) To reach the human who maintains the list, send mail to fuzzy-owner@dbai.tuwien.ac.at (4) WWW access and other information on Fuzzy Sets and Logic see http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info (5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

**Next message:**Stephen P. King: "[time 498] Re: [time 497] [Fwd: Fuzzy projections and cylindrical extensions]"**Previous message:**Stephen P. King: "[time 496] Re: [time 491]Forms, Entailment structures, intersections, etc."**Next in thread:**Stephen P. King: "[time 498] Re: [time 497] [Fwd: Fuzzy projections and cylindrical extensions]"

*
This archive was generated by hypermail 2.0b3
on Sun Oct 17 1999 - 22:36:58 JST
*