Probabilities in the Problem of Classification - Probabilistic Composition of Preferences, Theory and Applications - page 87

probabilities of outranking the pro les in at least one criterion, large intervals are

obtained for three more countries, Liechtenstein, Qatar and Kuwait.
All these results are explained by the construction of the sample. The four
countries with large intervals are those with strongly divergent classi

cations by

different criteria. The ten countries classi ed in the highest class without any doubt

are exactly those ten countries with the highest HDI. On the other end, the eight last
lines in the table, with coincident benevolent and hostile classi cation in the lowest

class, correspond to the countries with the eight lowest HDI values.
The results of the application of more complex forms of composition are pre-
sented in Table 8.6 . The

first applies the additive composition with constant
weights. In the second, is employed again a capacity built by assuming additivity
and equal importance, but treating as mutually interchangeable the two educational
indicators. In the third, this second capacity is modi

ed, becoming interchangeable
not only the two educational components but also the two indicators of longevity
and income. Assuming equal importance for each of these two sets of two elements,
numerically the capacity is formed by each set with one element of each pair of
interchangeable components having a value of 1, while the unitary sets as well as
the sets of two elements of the same interchangeable pair have the value of 1/2.
For these three capacities, the balanced classi cation is punctual. The benevolent

and hostile classi cations by reductions of 50 % are shown in Table 8.6 flanking the

balanced classi cations, a

divergence of more than one level never occurring between classifications in this
table or between a classi

cation. There is again high agreement between classi

cation of this table and a composition by joint probability

in Table 8.5 .

8.7 Evaluation of the Classification

To evaluate the homogeneity of the classes obtained after classifying a set of
alternatives, the measure proposed by Calinski and Harabasz ( 1974 ) may be
employed. It is based on the ratio between the sum of the variances within the
clusters, of the vectors of evaluations around a center of the cluster, and the vari-
ance of these centers around the mean of all evaluations. Due to the conceptual
similarity to the F distribution employed to evaluate normal regression, this mea-
sure is called PseudoF.
There are many different forms of determining the centers of the r classes and of
combining distances on the m criteria.
Let u s consid e r the centers of the classes, for i varying from 1 to r, given by the
vectors of averages of the evaluations of the alternatives in the classes

and the general center given by the vector

ð c i1 ; ... ; c im Þ

c ¼ c 1 ; ... ; c m ð Þ

of the averages of all the evaluations.