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

p 1 (X) is the probability that X will be observed among all the X's
belonging to sub-population w i . To find pi (X), the probability
that a random observation X of w 1 be drawn from the total population
of all origins, pi (X) must be weighed by qi , the probability that
an observation (any observation) will come from it. In general,
the product q i p i (X) is the probability that X of w i will be drawn
and the sum of these products over the three sub-populations gives
the probability of X. It follows that the ratio.(1) is the
conditional probability that, given observation X, X comes from
wi . The expected loss of classifying that observation as from w 2 is:
qi p i (X)
(2) E 3 C(2/1),
is1,3 E q p (X)
i=1
where C(2/i) is the cost of classifying an observation from w l as
from w 2 . Then, one chooses the population ir k , or the mode of
transportation k, which minimizes the expected loss for X. Assuming
here that all the C(i/j) 's are equal, this is equivalent to minimizing
3
(3) E qi p i(X),
i=1
i0k
Proceeding in this fashion for successive observations, it is possible
to define decision or classification regions R 1 , R2 , R3 . The rule is:
assign X to Rk if
(4) E qi pi (x) c
i0k
E (1 4 p i (X), for all j 0 k.
•
This procedure minimizes the expected loss and is unique, if the
probability of equality between right-hand and left-hand sides of the
equation is zero . Subtracting from each side of (4) E qi p i (X),
(1) T.W. Anderson, 'Introduction to Multivariate Statistical Analysis'
1958, pp. 143-144. .
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