Page 341 - Computational Statistics Handbook with MATLAB

330 Computational Statistics Handbook with MATLAB

• We are building a classifier for a military command and control
system that will take features from images of objects and classify
them as targets or non-targets. If an object is classified as a target,
then we will destroy it. Target objects might be tanks or military
trucks. Non-target objects are such things as school buses or auto-
mobiles. We would want to make sure that when we build a clas-
sifier we do not classify an object as a tank when it is really a school
bus. So, we will control the amount of acceptable error in wrongly
saying it (a school bus or automobile) is in the target class. This is
the same as our Type I error, if we write our hypotheses as

H Object is a school bus, automobile, etc.
0
Object is a tank, military vehicle, etc.
H 1
• Another example, where this situation arises is in medical diagno-
sis. Say that the doctor needs to determine whether a patient has
cancer by looking at radiographic images. The doctor does not want
to classify a region in the image as cancer when it is not. So, we
might want to control the probability of wrongly deciding that
there is cancer when there is none. However, failing to identify a
cancer when it is really there is more important to control. There-
fore, in this situation, the hypotheses are

X-ray shows cancerous tissue
H 0
X-ray shows only healthy tissue
H 1
The terminology that is sometimes used for the Type I error in pattern recog-
nition is false alarms or false positives. A false alarm is wrongly classifying
something as a target ω 1 ) , when it should be classified as non-target ω 2 ) .
(
(
The probability of making a false alarm (or the probability of making a Type I
error) is denoted as

(
PFA) = α .
This probability is represented as the shaded area in Figure 9.7.
Recall that Bayes Decision Rule gives a rule that yields the minimum prob-
ability of incorrectly classifying observed patterns. We can change this
α
boundary to obtain the desired probability of false alarm . Of course, if we
do this, then we must accept a higher probability of misclassification as
shown in Example 9.4.
In the two class case, we can put our Bayes Decision Rule in a different
form. Starting from Equation 9.7, we have our decision as

(
(
P x ω 1 )P ω 1 ) > P x ω 2 )P ω 2 ) ⇒ x is in ω 1 , (9.9)
(
(
© 2002 by Chapman & Hall/CRC