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M. Fattore and A. Arcagni
be consistent with the input partial order relation, the score function s(·) is required
to be strictly order preserving, i.e. such that x y (which means x y and x = y)
in the input poset implies s(x) < s(y); thus, the ranking problem reduces to the
definition of “reasonable” strictly order-preserving maps on posets.
In the daily practice of socio-economic statistics, it is quite typical to compute
the score function, by coding ordinal scores as numbers and by applying tools from
classical data analysis, or even by computing simple averages. This approach is
definitely inconsistent, for two main reasons: first, since ordinal scores cannot be
treated as cardinals, unless forcing the nature of the data; second, since partially
ordered data need not be obtained from MISes, so that no attribute scores even
exist (for example, one could partially order products or services based on personal
taste, with no explicit reference to any underlying quality dimensions). In a posetic
setting, however, all the information useful for scoring and ranking is comprised in
the structure of the partial order relation adopted to describe the data and must be
extracted out of it. The issue thus becomes how score functions can be computed
directly over partial orders.
Currently there are two main algorithms, to score units and to extract rankings
out of a partially ordered set, namely the average height algorithm (Bruggemann
and Patil 2011) and the dominance eigenvector algorithm (Fattore et al. 2019); both
draw upon mutual ranking probabilities, which carry information on the relative
dominance of pairs of poset elements.
3.1.1 Average Height
Given a finite poset π , the average height avh(x i ) of an element x i ∈ π is defined
as the arithmetic mean of the heights of x i in the linear extensions of π ,
avh(x i ) =
1
|(π)|
λ∈
h λ (x i ),
(1)
where the height h λ (x i ) of x i in λ is given by 1 + the number of elements below
x i in λ. It can be shown (De Loof 2009) that the average height is linked to mutual
ranking probabilities by the following simple relation:
avh(x i ) =
n
j =1
p ji .
(2)
Once the average height is computed for each element of the poset, a ranking
is obtained by ordering elements in a decreasing way. By construction, the average
height is a strictly order preserving map.
M. Fattore and A. Arcagni
be consistent with the input partial order relation, the score function s(·) is required
to be strictly order preserving, i.e. such that x y (which means x y and x = y)
in the input poset implies s(x) < s(y); thus, the ranking problem reduces to the
definition of “reasonable” strictly order-preserving maps on posets.
In the daily practice of socio-economic statistics, it is quite typical to compute
the score function, by coding ordinal scores as numbers and by applying tools from
classical data analysis, or even by computing simple averages. This approach is
definitely inconsistent, for two main reasons: first, since ordinal scores cannot be
treated as cardinals, unless forcing the nature of the data; second, since partially
ordered data need not be obtained from MISes, so that no attribute scores even
exist (for example, one could partially order products or services based on personal
taste, with no explicit reference to any underlying quality dimensions). In a posetic
setting, however, all the information useful for scoring and ranking is comprised in
the structure of the partial order relation adopted to describe the data and must be
extracted out of it. The issue thus becomes how score functions can be computed
directly over partial orders.
Currently there are two main algorithms, to score units and to extract rankings
out of a partially ordered set, namely the average height algorithm (Bruggemann
and Patil 2011) and the dominance eigenvector algorithm (Fattore et al. 2019); both
draw upon mutual ranking probabilities, which carry information on the relative
dominance of pairs of poset elements.
3.1.1 Average Height
Given a finite poset π , the average height avh(x i ) of an element x i ∈ π is defined
as the arithmetic mean of the heights of x i in the linear extensions of π ,
avh(x i ) =
1
|(π)|
λ∈
h λ (x i ),
(1)
where the height h λ (x i ) of x i in λ is given by 1 + the number of elements below
x i in λ. It can be shown (De Loof 2009) that the average height is linked to mutual
ranking probabilities by the following simple relation:
avh(x i ) =
n
j =1
p ji .
(2)
Once the average height is computed for each element of the poset, a ranking
is obtained by ordering elements in a decreasing way. By construction, the average
height is a strictly order preserving map.