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

The basis for this analysis is T.W. Anderson's
Introduction to Multivariate Statistical Analysis. 3 Let A
and S be two populations (air and sea) with density functions
z A (X) and z (X) respectively. The goal is to divide the
universe of observations into two mutually exclusive and
exhaustive regions RA and Rs . If an observation falls ,
into region RA , it is said that it comes from A.
The, design is to detetmine the probability that a given
observation comes from A based on discriminant analysis. If
the probability is of a certain level, that observation will
be assigned to population A. The goal is to chOose the R A
and R so that the costs of misclassification (i.e. the cost
associated with classifying an actual air shipment as a sea
shipment - placing an actual resident of R A into R s - and
vice versa) are minimized.
If the a priori, probabilities are known (q h is the a
priori probability for the h th population) for the popula-
tions A and S, the conditional probability of an observation
coming from a population given the values of the components
of the vector of explanatory variables can be defined. The
a priori probabilities may be based on previous studies done
or may be based on the assumption of equal ignorance, i.e.
q A =q S =.5.
The conditional probability of the observation (with
explanatory vector X) coming from A is:
( 1.) •
3 Ibid
h=A,S
Call this
q A
z A (X)
q z (X)
h h
, ( K).
'r
A
(Anderson, p. 143) .
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