52
R. Bruggemann and L. Carlsen
the curve for Udirectly. So far, the number of “jumps” in Fig. 1 is obvious. However,
where will the jumps appear?
2.3 Understanding the Fine-Structure
It is completely clear that the ordinal structure of orderings within partial order
theory react on the evolution due to s only in discrete steps. The reason is that the
partial order is changing, when a transition from x ⊥ y to xy appears. Nevertheless
it is of interest at which values of weights, within a series of env(g,s) such a
transition and hence an enhancement of U is to be expected. In order to analyze
the fine-structure, the simplest system is assumed, where still incomparabilities
can appear, i.e., the m = 2 system. Starting from low s-values there are only so
few CI’s possible that almost for all object pairs (x, y) it will be found x ⊥ y.
Increasing s, some CI’s may exist such that a comparability relation is transferred to
an incomparability relation. This means that the extension of the range for possible
weight values can be formulated as follows:
s small : CI1 (x) ≤ CI1 (y) , CI2 (x) ≤ CI2 (y) → s enlarged : CI1 (x) ≤ CI1 (y) but CI2 (x) ≤ CI2 (y)
⇓
⇓
s small g ∈ env (g, s) , such that x≤ {CI} y
s enlarged : there are weights so that
CI (g, x) ≤ CI (g, y) and
CI
g
, x
≤ CI
g
, y
Hence, within an m = 2-system, there must be a value gc1 (gc1, (gc2 = 1- gc1)
such that CI((gc1, 1-gc1),x) = CI((gc1,1-gc1),y) and if g1 < gc1, then x ≤ CI y
whereas for g1 ≥ gc1, then x CI y. These decisive weights (within the m = 2-system)
are called crucial weights. Crucial weights can be calculated in closed form, see Eqs.
10a and 10b and for more details Bruggemann et al. 2008b.
gc1 = (x (2, 2) − x (1, 2)) / [(x (2, 2) − x (1, 2)) − (x (2, 1) − x (1, 1))]
(10a)
gc2 = (x (2, 1) − x (1, 1)) / [(x (2, 1) − x (1, 1)) − (x (2, 2) − x (1, 2))]
(10b)
If an environment env(g, s) encompasses one of the possible crucial weights, then
there is a chance that composite indicators are generated which are countercurrent
(i.e. not co-monotonic) to others and which therefore generate incomparabilities.
Therefore the crucial weights in the g-space are the location, where the number of
incomparabilities will increase.
As an object set may have more than two elements, many more possible crucial
weights can be calculated, according to different object pairs. As the weights have
to follow Eqs. 5 and 6, the relevant gc-values must also be in the range [0,1].
With the focus on gc1 (the other value in m-2-systems is just 1-gc1) a distribution
of gc-values is possible and instead of a statistical oriented notation we write simply
0 ≤ gc1(1) ≤ gc1(2) ≤ . . . . ≤ 1.
R. Bruggemann and L. Carlsen
the curve for Udirectly. So far, the number of “jumps” in Fig. 1 is obvious. However,
where will the jumps appear?
2.3 Understanding the Fine-Structure
It is completely clear that the ordinal structure of orderings within partial order
theory react on the evolution due to s only in discrete steps. The reason is that the
partial order is changing, when a transition from x ⊥ y to xy appears. Nevertheless
it is of interest at which values of weights, within a series of env(g,s) such a
transition and hence an enhancement of U is to be expected. In order to analyze
the fine-structure, the simplest system is assumed, where still incomparabilities
can appear, i.e., the m = 2 system. Starting from low s-values there are only so
few CI’s possible that almost for all object pairs (x, y) it will be found x ⊥ y.
Increasing s, some CI’s may exist such that a comparability relation is transferred to
an incomparability relation. This means that the extension of the range for possible
weight values can be formulated as follows:
s small : CI1 (x) ≤ CI1 (y) , CI2 (x) ≤ CI2 (y) → s enlarged : CI1 (x) ≤ CI1 (y) but CI2 (x) ≤ CI2 (y)
⇓
⇓
s small g ∈ env (g, s) , such that x≤ {CI} y
s enlarged : there are weights so that
CI (g, x) ≤ CI (g, y) and
CI
g
, x
≤ CI
g
, y
Hence, within an m = 2-system, there must be a value gc1 (gc1, (gc2 = 1- gc1)
such that CI((gc1, 1-gc1),x) = CI((gc1,1-gc1),y) and if g1 < gc1, then x ≤ CI y
whereas for g1 ≥ gc1, then x CI y. These decisive weights (within the m = 2-system)
are called crucial weights. Crucial weights can be calculated in closed form, see Eqs.
10a and 10b and for more details Bruggemann et al. 2008b.
gc1 = (x (2, 2) − x (1, 2)) / [(x (2, 2) − x (1, 2)) − (x (2, 1) − x (1, 1))]
(10a)
gc2 = (x (2, 1) − x (1, 1)) / [(x (2, 1) − x (1, 1)) − (x (2, 2) − x (1, 2))]
(10b)
If an environment env(g, s) encompasses one of the possible crucial weights, then
there is a chance that composite indicators are generated which are countercurrent
(i.e. not co-monotonic) to others and which therefore generate incomparabilities.
Therefore the crucial weights in the g-space are the location, where the number of
incomparabilities will increase.
As an object set may have more than two elements, many more possible crucial
weights can be calculated, according to different object pairs. As the weights have
to follow Eqs. 5 and 6, the relevant gc-values must also be in the range [0,1].
With the focus on gc1 (the other value in m-2-systems is just 1-gc1) a distribution
of gc-values is possible and instead of a statistical oriented notation we write simply
0 ≤ gc1(1) ≤ gc1(2) ≤ . . . . ≤ 1.