Q2 Z2,(Q2) Z2(Q2) - Institute for Water Resources - U.S. Army

ilkii -
. (17) 2. 2 -1 -1(Z ij )2
(2 iij w) .0
( 19) P(J,k/i)
0111
while the probability of misclassifying an observation from w j as from
w i is
(18)
NUJ) '
Jr (51
ij
-
• 2
(2 i
a
ij
)4
i( TC - U )/a A R
i ij i ij 1 i ik
•
10
-1(w )2 '
.dW
ij
Furthermore, it is possible to compute the overall chances
of misclassifying an observation from w i as from w j or wk . With
U >U and
i
U
iJ ii s
k U ik'
where f(Z ij' Z ik ) is the standard form of the bivariate normal
f(Z ik )dZ ij dZ ik
distribution. The respective probabilities of correct classification
are equal to 1 - PQM, 1 - P(i/j), and 1 - P(j,k/i).
An alternative and equivalent, though more complicated,
, method of classification would be to compute the P i (X) 1 21, the
conditional probabilities that given X, the ith mode will be chosen,
and assign X to the mode with the greatest P1 (X). (1) Following (1)
above,
(20)
qi
N1 (X)
P '
i (X) Is
s
3
E 414 Nj (X) .
j=1 J
where multivariate normal distributions are assuTed& gut, as
(1) See S.L. Warner 'Multivariate Regression of Dummy Variates under
Normality Assumption' Journal o Amer can Statistic Association,
December, 1963, pp. 1054-1063.
•