Evaluations as Sets over Lattices – Application Point of View
93
2.3 Role of Mapping A, Standard Norm and the Data Matrix
A central role in the data exploration applying data continuous in concept plays the
“has”-relation:
An object x ∈ X has an indicator q ∈ Q
( 1 )
Because the data are supposed to be continuous, the term “has” must be
interpreted as a degree, i.e. a truth value (tv) of the “has”-relation. If tv = 0 then
object x has not the indicator q, if tv = 1, then the object has certainly the indicator
q, hence in general tv ∈ [0,1]. Statement (1) in terms of A, Q and x(i,j) is given by
Eq. 2:
A
(x) = ∧ q(j )∈Q1⊂Q s ∗ (A (q(j )) , x (i, j ))
(2)
Therein is Q1 a crisp subset of Q. The meet-operation in Eq. 2 can in the case of
data continuous in concept replaced by the Min-operation. When Q1 is a singleton,
say Q1 = {q(1)} the evaluation of Eq. 2 is very easy:
A
(x) = Min {s ∗ (A (q (j)) , x (i, j))} .
As shown in BK the residuum of the standard norm is given by Eq. (3):
s ∗ (α, β) :=
1
if α ≤ β
0 otherwise
(3)
α, β being real numbers ≥ 0.
A is a tuple of length |Q|. A(q(j)) means, the value of the tuple A at position j.
Let us select a 1 at the jth position and 0 otherwise, in order to describe the crisp
subset {q(j)}. Then, the residuum of the standard norm delivers for x(i) everywhere,
where A(q(j)) = 0 the value 1, and only in the jth position the value x(i,j). Hence a
singleton Q1 = {q(j)} selects just the entry x(i,j) and if all x ∈ X are considered just
the jth column of the data matrix.
Consequently, the subset Q1, with several indicators, say {q(j1), q(j2), q(j3)}
delivers for the object x(i) first the entries x(i,j1), x(i,j2) and x(i,j3) and A’(x) is
0 besides at the positions j = j1, j = j2, j = j3) where the actual values of x(i,j) are
to be inserted (which nevertheless can also be 0) and after this selection the minimal
value among the set of entries {x(i,j1), x(i,j2), x(i,j3)} is to be found.
In Fig. 1 the situation, due to Eq. (2) is schematically shown, assuming that
|Q| = 7 and A = (0,1,0,0,1,0,0) describing the crisp subset {q(2), q(5)}.
93
2.3 Role of Mapping A, Standard Norm and the Data Matrix
A central role in the data exploration applying data continuous in concept plays the
“has”-relation:
An object x ∈ X has an indicator q ∈ Q
( 1 )
Because the data are supposed to be continuous, the term “has” must be
interpreted as a degree, i.e. a truth value (tv) of the “has”-relation. If tv = 0 then
object x has not the indicator q, if tv = 1, then the object has certainly the indicator
q, hence in general tv ∈ [0,1]. Statement (1) in terms of A, Q and x(i,j) is given by
Eq. 2:
A
(x) = ∧ q(j )∈Q1⊂Q s ∗ (A (q(j )) , x (i, j ))
(2)
Therein is Q1 a crisp subset of Q. The meet-operation in Eq. 2 can in the case of
data continuous in concept replaced by the Min-operation. When Q1 is a singleton,
say Q1 = {q(1)} the evaluation of Eq. 2 is very easy:
A
(x) = Min {s ∗ (A (q (j)) , x (i, j))} .
As shown in BK the residuum of the standard norm is given by Eq. (3):
s ∗ (α, β) :=
1
if α ≤ β
0 otherwise
(3)
α, β being real numbers ≥ 0.
A is a tuple of length |Q|. A(q(j)) means, the value of the tuple A at position j.
Let us select a 1 at the jth position and 0 otherwise, in order to describe the crisp
subset {q(j)}. Then, the residuum of the standard norm delivers for x(i) everywhere,
where A(q(j)) = 0 the value 1, and only in the jth position the value x(i,j). Hence a
singleton Q1 = {q(j)} selects just the entry x(i,j) and if all x ∈ X are considered just
the jth column of the data matrix.
Consequently, the subset Q1, with several indicators, say {q(j1), q(j2), q(j3)}
delivers for the object x(i) first the entries x(i,j1), x(i,j2) and x(i,j3) and A’(x) is
0 besides at the positions j = j1, j = j2, j = j3) where the actual values of x(i,j) are
to be inserted (which nevertheless can also be 0) and after this selection the minimal
value among the set of entries {x(i,j1), x(i,j2), x(i,j3)} is to be found.
In Fig. 1 the situation, due to Eq. (2) is schematically shown, assuming that
|Q| = 7 and A = (0,1,0,0,1,0,0) describing the crisp subset {q(2), q(5)}.