Page 48 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

DETECTION: THE TWO-CLASS CASE 37

For vectors drawn from normal distributions, the log-likelihood ratio is:

1 T 1
LðzÞ¼ ln jC 1 j ln jC 2 jþðz m Þ C ðz m Þ
1
1
1
2
ð2:40Þ

T 1
ðz m Þ C ðz m Þ
2
2
2
which is much easier than the likelihood ratio. When the covariance
matrices of both classes are equal (C 1 ¼ C 2 ¼ C) the log-likelihood ratio
simplifies to:
T

1 1
LðzÞ¼ z ð m þ m Þ C ð m m Þ ð2:41Þ
1
1
2
2
2
Two types of errors are involved in a detection system. Suppose that ^ !(z)
!
is the result of a decision based on the measurement z. The true (but
unknown) class ! of an object is either ! 1 or ! 2 . Then the following four
states may occur:
! ¼ ! 1 ! ¼ ! 2
correct decision I type II error
^ ! !(z) ¼ ! 1
type I error correct decision II
^ ! !(z) ¼ ! 2
Often, a detector is associated with a device that decides whether an
object is present (! ¼ ! 2 ) or not (! ¼ ! 1 ), or that an event occurs or not.
These types of problems arise, for instance, in radar systems, medical
diagnostic systems, burglar alarms, etc. Usually, the nomenclature for
the four states is as follows then:

! ¼ ! 1 ! ¼ ! 2
^ ! !(z) ¼ ! 1 true negative missed event
or false negative
^ ! !(z) ¼ ! 2 false alarm detection (or hit)
or false positive or true positive

Sometimes, the true negative is called ‘rejection’. However, we have
reserved this term for Section 2.2, where it has a different denotation.
The probabilities of the two types of errors, i.e. the false alarm and the
missed event, are performance measures of the detector. Usually these