68
R. Bruggemann et al.
if ties are accepted. Whereas a complete (i.e., total or linear) order is a set of objects,
in which all elements x, y ∈ X are mutually comparable with x = y, a weak order
does not require the condition x = y, i.e., it accepts equivalent elements (or in terms
of statistics: it accepts ties).
A Hasse diagram allows identifying rankings for subsets of X, without any
subjectivity beyond the data matrix. It is clear that the task to get a weak order
should be parameter free too. Hence, the typical procedure to aggregate the values
of the m indicators into a composite indicator by a numerical procedure, where
weights for each indicator and other parameters are required, is to be avoided. An
important device, how to get a weak order without the need of finding additional
parameters, such as weights for the indicators, out of a partially ordered set is found
in the paper of Winkler (1982). The crucial term is the average height, denoted as
Hav.
2.5 Average Height
Any poset can be represented by a set of linear order, whose elements are called
linear extensions (Davey and Priestley 1990; Trotter 1992). A linear extension is a
linear order, respecting all order relations within a poset. For example the set X = {a,
b, c, d} may have the following order relations:
a < b, a < c, a < d.
(4)
Obviously, b c and c d, i.e. U = 2. Then the set of linear extensions is:
{(a, b, c, d) , (a, c, b, d) , (a, c, d, b)} .
Within the above set {(a, b, c, d), (a, c, b, d), (a, c, d, b)} a linear extension is for
example (a, b, c, d), others are (a, c, b, d) and (a, c, d, b). Each single linear extension
indicates a complete ordered set, for example (a, b, c, d) denotes: a < b, a < c, a <
d, b < c, b < d, c < d). All order relations of Eq. 4 are reproduced. The fact that the
poset in Eq. 4 includes some incomparabilities leads to the necessity to consider the
3 linear extensions simultaneously. Within each linear extension any object x has a
height that is the number of objects ≤ x. For example, in the linear extension (a, b,
c, d) object a has the height 1, b the height 2, whereas in the linear extension (a, c,
d, b) object b has the height 4.
The idea of Winkler (1982) is to calculate the average of all heights of all objects,
denoted as Hav(x). Let L(k) be the kth linear extension and h(L(k),x) the height of
x in L(k), then, after Winkler (1982)
R. Bruggemann et al.
if ties are accepted. Whereas a complete (i.e., total or linear) order is a set of objects,
in which all elements x, y ∈ X are mutually comparable with x = y, a weak order
does not require the condition x = y, i.e., it accepts equivalent elements (or in terms
of statistics: it accepts ties).
A Hasse diagram allows identifying rankings for subsets of X, without any
subjectivity beyond the data matrix. It is clear that the task to get a weak order
should be parameter free too. Hence, the typical procedure to aggregate the values
of the m indicators into a composite indicator by a numerical procedure, where
weights for each indicator and other parameters are required, is to be avoided. An
important device, how to get a weak order without the need of finding additional
parameters, such as weights for the indicators, out of a partially ordered set is found
in the paper of Winkler (1982). The crucial term is the average height, denoted as
Hav.
2.5 Average Height
Any poset can be represented by a set of linear order, whose elements are called
linear extensions (Davey and Priestley 1990; Trotter 1992). A linear extension is a
linear order, respecting all order relations within a poset. For example the set X = {a,
b, c, d} may have the following order relations:
a < b, a < c, a < d.
(4)
Obviously, b c and c d, i.e. U = 2. Then the set of linear extensions is:
{(a, b, c, d) , (a, c, b, d) , (a, c, d, b)} .
Within the above set {(a, b, c, d), (a, c, b, d), (a, c, d, b)} a linear extension is for
example (a, b, c, d), others are (a, c, b, d) and (a, c, d, b). Each single linear extension
indicates a complete ordered set, for example (a, b, c, d) denotes: a < b, a < c, a <
d, b < c, b < d, c < d). All order relations of Eq. 4 are reproduced. The fact that the
poset in Eq. 4 includes some incomparabilities leads to the necessity to consider the
3 linear extensions simultaneously. Within each linear extension any object x has a
height that is the number of objects ≤ x. For example, in the linear extension (a, b,
c, d) object a has the height 1, b the height 2, whereas in the linear extension (a, c,
d, b) object b has the height 4.
The idea of Winkler (1982) is to calculate the average of all heights of all objects,
denoted as Hav(x). Let L(k) be the kth linear extension and h(L(k),x) the height of
x in L(k), then, after Winkler (1982)