Probabilities in the Problem of Classification - Probabilistic Composition of Preferences, Theory and Applications - page 91

!
=

X n i X

r r

c j ¼ c iaj n i

i ¼ 1 a ¼ 1 i ¼ 1

The PseudoF measure of quality of the classi cation is then given by the ratio

between B/(r − 1) and W/(n − r) for B denoting the variance of the conditional

expectation given the classi cation, W the expectation of the conditional variance, r

the number of classes and n the number of alternatives classi ed.

Thus,

PseudoF ¼ ð B = ð r 1 Þ Þ= ð W = ð n r Þ Þ

for

X

n i X
j

r

2

B ¼ c ij c j ;

i ¼ 1

X X X

r

2

W ¼ c iaj c ij ;

i ¼ 1 j a

and

X

r

n ¼ n i :

i ¼ 1

Thus,

! !

r

! !

X

n i X
j

= X

X X

a ð c iaj c ij Þ

Þ= X

r r r

2 2

PseudoF ¼ ð c ij c j Þ = ð r 1 n i :

i ¼ 1 i ¼ 1 j i ¼ 1

Given the decomposition of the variance,

! ! !

X X

X
i

X X

= X
i

X X

Þ= X
i

2 2

2

PseudoF ¼ ð c iaj c ij Þ c iaj c ij c iaj c ij = ð r 1 n i r

j j a j a

i

This construction does not take into account the possibility of different scales of
measurement of the different criteria. An alternative measure for the case of mea-
sures of importance for the criteria being available would compute the ratios sep-
arately for each criterion and weigh these ratios with weights that take into account
such scales: