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/
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