Probabilities in the Problem of Classification - Probabilistic Composition of Preferences, Theory and Applications - page 78

If there is no way to differentiate the relative importance of the criteria, simpler
compositions, like those by joint probabilities, may be employed In that case the
extremes of independence and of maximal dependence result in simple formulae.
The assumption of maximal dependence leads to simpler calculations than that
of independence. The foundation for this assumption may be strengthened in the
problem of classi cation by the fact that the criteria are made to agree with the

ordering of the classes. However, it should be taken into account that correlation
between the vectors of evaluations of the different criteria does not imply depen-
dence between the disturbances

that assign the random character to the

measurements.
On another direction, the agreement of the evolution of the values of the criteria
in the pro rm each other, is a

reason to take a pessimistic approach. In that case, the combination of the criteria by
joint probability will use, for the probability of the alternative being above the ith
class the formulae:

les along the classes, implying that the criteria con

A i ¼ min j A ij ;

for the hypothesis of maximal dependence

or
A i ¼ p j A ij ; for the hypothesis of independence.

And for the probability of the alternative being below the ith class

A i ¼ min j A ij

or
A i ¼ p j A ij :

The composition formulae for the Choquet integral will be:

X

m
j ¼ 1 ð

A i ¼ i þ ð i þ ð

A isi þ ðÞþ A isi þ j 1

Þþ Þ l ðf s

j Þ; ... ; s m ÞgÞ

ð

and

X

m
j ¼ 1 ð

A i ¼ i ð i ð A isi ðÞ A isi j 1

Þ Þ l ðf s

j Þ; ... ; s m ÞgÞ

ð

where is the capacity,

A i τ i+(0)

l

+ −

=A i τ i − (0) = 0, for all i,

i + and i − are the permutations of the set of indices {1,

and τ τ … , m} such that

A isi þð 1 Þ A isi þð 2 Þ A isi þð n 1 Þ A isi þð n Þ

and

A isi ðÞ A isi ðÞ A isi n 1

ð

A isi ðÞ :

Þ