Page 44 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

REJECTION 33

!
þ
function itself is a mapping ^ !(z): R N ! O . In order to develop a Bayes
classifier, the definition of the cost function must also be extended:
C(^ !j!): O O ! R. The definition is extended so as to express the
þ
!
cost of rejection. C(! 0 j! k ) is the cost of rejection while the true class of
the object is ! k .
With these extensions, the decision function of a Bayes classifier
becomes (2.8):

K
( )
X
^ ! ! BAYES ðzÞ¼ argmin Cð!j! k ÞPð! k jzÞ ð2:30Þ
!2O þ k¼1
The further development of the classifier follows the same course as in (2.8).

2.2.1 Minimum error rate classification with reject option

The minimum error rate classifier can also be extended with a reject
option. Suppose that the cost of a rejection is C rej regardless of the true
class of the object. All other costs are uniform and defined by (2.9).
We first note that if the reject option is chosen, the risk is C rej .Ifit is
not, the minimal conditional risk is the e min (z) given by (2.15). Minimiza-
tion of C rej and e min (z) yields the following optimal decision function:

(
! 0 if C rej < e min ðzÞ
^ ! !ðzÞ¼ ð2:31Þ
^ ! !ðzÞ¼ argmaxfPð!jzÞg otherwise
!2O
The maximum posterior probability maxfP(!jz)g is always greater than
or equal to 1/K. Therefore, the minimal conditional error probability is
bounded by (1 1/K). Consequently, in (2.31) the reject option never
wins if C rej 1 1/K.
The overall probability of having a rejection is called the reject rate.
It is found by calculating the fraction of measurements that fall inside
the reject region:

Z
Rej-Rate ¼ pðzÞdz ð2:32Þ
fzjC rej <e min ðzÞg

The integral extends over those regions in the measurement space for
which C rej < e(z). The error rate is found by averaging the conditional