Fuzzy Multi-dimensional Analysis
Alexandr A. Savinov
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7. Finding Projections on Variables

The problem of finding projections on variables is one of the most important in multi-dimensional analysis. With the help of this operation we obtain a local distribution over the values of one variable from a multi-dimensional distribution. Let us suppose that it is required to find a projection of the distribution represented by the matrix of CNF D onto the variable xk. Generally, projection can be defined in different ways but we will suppose that it is defined by means of the operation of maximum. Namely, j-th value of the projection on the variable xk, i.e., its value in the point akj is equal to
max( D(x1,..., xk=akj,..., xn) )
on all values of all variables except for xk=akj. In other words, we take maximum in all points which have k-th component (k-th dimension) equal to akj. Thus to calculate the whole projection in nk points of k-th domain  we have to take maximum in all point of the universe of discourse. Obviously, the maximum of the projection on any variable is equal to the maximum of the whole multi-dimensional distribution and therefore we can solve the satisfiability problem (finding the consistency) by finding any projection.

We can also redefine the projection by means of the logical consequence relation. Let us suppose that n projections on all variables are represented by the conjunction c, i.e., each section ci of this conjunction represents a projection on i-th variable. Then all its components cij must be minimal provided that it is still a consequence of D. In other words, to find the conjunction of projections c we have to take the trivial conjunction consisting of all 1 and gradually decrease its components untill we reach the border where it ceases to be a consequence of D.

One procedure for finding projections is based on a theorem [S93a] which affirms that disjunction u is a consequence of the matrix of CNF D iff there exists such a prime disjunction p of this matrix that the disjunction d is its consequence:

D |= u <=> there exists prime p: p |= u

With the help of this criterion we can check if some disjunction is a consequence of the matrix, but we can also find minimal disjunctions which satisfy this condition. It can shown that the projection on k-th variable of the distribution represented by the matrix of CNF D is equal to the minimum of projections of all prime disjunctions on this variable. Note that we have to have all prime disjunctions to carry out this procedure.

Projection on k-th variable of one disjunction is equal to the k-th section of this disjunction plus constant M which is equal to maximal value in all the rest of the sections (i.e., M is equal to the maximum of all non-k-th section constants). For example, projection of the disjunction

{0.3, 0} {0, 0.4, 0} {0, 0.2, 0.7, 1}
on 3rd variable is equal to {0, 0.2, 0.7, 1}+0.4={0.4, 0.4, 0.7, 1} while its projections on the variables x1 and x2 is equal to the constant 1 (we say that the projection is absent).

Let us consider an example of the matrix consisting of all prime disjunctions from the previous section

D3 =  {0.3, 0} {1, 0.5, 0} {0, 0, 0, 0} 1
{0, 0} {0, 0.4, 1} {0, 0.2, 0.7, 1} 2
{0, 1} {0, 0, 0} {1, 0.3, 0, 0} 3
{0.3, 0} {0, 0.4, 0} {0, 0.2, 0.7, 1} 4
{0, 0} {1, 0.5, 0} {1, 0.3, 0, 0} 5
{0, 1} {0, 0.4, 0.3} {0, 0.2, 0, 0} 8
The projection of this matrix on the variable x1 is equal to {0.4, 1} (disjunction 8), on variable x2 -- {1, 0.5, 0.3} (disjunction 1), and on variable x3 -- {0.4, 0.4, 0.7, 1} (disjunction 4).
 
Fuzzy Multi-dimensional Analysis
Alexandr A. Savinov
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© Copyright 1997, Alexandr A. Savinov