Probabilities in the Problem of Classification - Probabilistic Composition of Preferences, Theory and Applications - page 76

To identify the class Ci, i , for each i from 1 to r are employed n(i) representative
pro les, each of them formed by a vector of evaluations by the m criteria.

The hth representative pro

le of the ith class is given by a vector of evaluations

by the m criteria (c ih1, … ,c ihm ).

The representative pro les are built in such a way that if i 1 <i 2 , then, for every j,

c i1hj ≤ c i2hj and, for at least one j o ,c i1hjo ≤ c i2hjo .

So, according to no criterion a pro le of a higher class may present an evaluation

lower than that of a pro le of a lower class by that same criterion and by at least one

criterion the pro le of the higher class presents evaluation higher than that of the

pro le of the lower class.

To allow for the probabilistic comparison, the evaluations a j of the alternative
being classi ed are replaced by probability distributions centered at a j . To classify

the alternative what is going to be compared are the probabilities of the alternative
presenting evaluations according to such distributions above or below the pro

les

of each class. The c ihj may be also replaced by random variables X ihj .
This characterization of the problem of classi cation follows that of Roy ( 1968 )

and Yu ( 1992 ). The representative pro les approach follows Almeida Dias et al.

( 2010 , 2012 ). More details about such characterization are available in Sant

' Anna

et al. ( 2012 ).

8.2 Computation of Probabilities of Preference

Following the principles of classical statistical modeling, it is reasonable to assume
here, as in the preceding chapters, not only a normal form, but also identical
distributions and independence between the disturbances causing the imprecision in
the evaluations according to any criterion. Alternatively, instead of normal, trian-
gular distributions may be a simpler starting point, like in the Theory of Fuzzy Sets
(Zadeh 1965 ). In advancing the analysis, the kind of evaluation involved may also
suggest a different distribution.
Since the mean of the normal distribution or the mode of the triangular distri-
bution are determined by the observed values, to complete modeling the distribu-
tions in these two cases it is enough to determine the values, respectively, of the
variance and of the extreme points. The available data are used to estimate these
parameters or other parameters that determine other distributions. If other infor-
mation on the parameters is available in advance, it can be used and will simplify
computations.
Assuming normality, for each j from 1 to m, the coordinates a j of the alternative
evaluated, for i from 1 to r, are considered as means of independent normal dis-
tributions with the same variance.
If the variance is small, is small the probability that an alternative with evalu-
ation a j according to the jth criterion may belong to the classes with pro les