Page 373 - Introduction to Statistical Pattern Recognition

7 Nonparametric Classification and Error Estimation 355

expected to be much smaller than the L-curve of Fig. 7-14(c) (5.5% at k = 5),
but larger than the R-curve (1.3% at k = 5). Note again that the curves of Fig.
7-14(c) were obtained as the 10 trial average by using the threshold of Option
4, while the dotted line of Fig. 7-15 was determined by human judgement for
one trial.
In order to apply the Neyman-Pearson test with = 2% for example, we
maintain the 45 .,slope of the line, and shift it until we find 2 misclassified 02-
samples out of 100.

Constant risk contours: It is frequently desirable to have risk informa-
tion about a point in a data display. For the two-class case, the risk at X, r (X),
is given by

I
Substituting pj(X) = (k,-l)/Njco I Cj li2dy(X), and taking logarithms,
1n'0
(k I -1)Nz I z2 I + ln~}
+
nlnd2(X) = n lnd I (X) - In
I (kZ--l)NI 1x1 I P2 1 -r (X)
(7.80)
Thus, for a given r-(X), a contour can be drawn on the display. The resulting
contour is a 45' line shifted by lnr(X)/[l-r(X)] [16]. Similarly, for
PIpI(X) > P2p2(X), the numerator of (7.79) is replaced by P2p2(X), and
(7.80) is modified to
(k I -1)lVz I x*
I
nlnd2(X) = n lnd I (X) - In + 1 - In-
I (k2-1)N1 1x1 I p2 1 -r- (X)
(7.8 1)

Comparison of (7.80) and (7.81) indicates that the constant risk lines are sym-
metric about the Bayes decision line where r-(X) = 0.5. As r(X) is decreased,
the constant risk lines move farther from the Bayes decision line. This result
indicates that points mapped near the Bayes decision line on the display do in
fact have a high degree of risk associated with them, while points mapped far
from the Bayes decision line have a low degree of risk. This desirable pro-