50
R. Bruggemann and L. Carlsen
6. Us and other quantities derived from U:
The poset, based on the data matrix alone, with n objects and m indicators will
have a number of incomparabilities which is the “total incomparability” and which
is called U0. It is then clear that
U (g, 0) = 0
(8a)
and
U (g, 1) = U0
(8b)
In former publications (Bruggemann and Carlsen 2017) it is shown that the Eq.
(9) describes sufficiently the general situation for a certain g and s varying from 0
to 1:
Us = s ∗ U0
(9)
In the following the incomparabilities are called
• Udirectly, if the determination of U is done, by checking each poset in env(g,s)
• Uan if U can be determined by means of other quantities, such as the crucial
weights (see below)
• Us from Eq. (9)
• U0 the number of incomparabilities of the poset under m indicators, i.e. without
any additional information beyond the data matrix.
In Fig. 1 the evolution of U on the basis of Eq. (9) and Udirectly, i.e. directly
determined from all the posets resulting from different values of s is shown. To
be clear: each environment env(g,s) allows a set of weights, which in turn allows
different CI, and these CI, evaluated similar to Eqs. 1, 2 and 3 (replace x(I,j) by
CI(x(i),k), k being a label for the resulting CI and check x CI y).
Hereto, a fictitious data matrix with 14 objects and m = 3 indicators was applied.
Fig. 1 Evolution of U, based
on Eq. (9), and of the number
of incomparabilities directly
determined from all posets,
resulting from env(g,s),
“Udirectly” is shown. Data
are shown in Table 1
R. Bruggemann and L. Carlsen
6. Us and other quantities derived from U:
The poset, based on the data matrix alone, with n objects and m indicators will
have a number of incomparabilities which is the “total incomparability” and which
is called U0. It is then clear that
U (g, 0) = 0
(8a)
and
U (g, 1) = U0
(8b)
In former publications (Bruggemann and Carlsen 2017) it is shown that the Eq.
(9) describes sufficiently the general situation for a certain g and s varying from 0
to 1:
Us = s ∗ U0
(9)
In the following the incomparabilities are called
• Udirectly, if the determination of U is done, by checking each poset in env(g,s)
• Uan if U can be determined by means of other quantities, such as the crucial
weights (see below)
• Us from Eq. (9)
• U0 the number of incomparabilities of the poset under m indicators, i.e. without
any additional information beyond the data matrix.
In Fig. 1 the evolution of U on the basis of Eq. (9) and Udirectly, i.e. directly
determined from all the posets resulting from different values of s is shown. To
be clear: each environment env(g,s) allows a set of weights, which in turn allows
different CI, and these CI, evaluated similar to Eqs. 1, 2 and 3 (replace x(I,j) by
CI(x(i),k), k being a label for the resulting CI and check x CI y).
Hereto, a fictitious data matrix with 14 objects and m = 3 indicators was applied.
Fig. 1 Evolution of U, based
on Eq. (9), and of the number
of incomparabilities directly
determined from all posets,
resulting from env(g,s),
“Udirectly” is shown. Data
are shown in Table 1