Analysis- — KlFFENBURGH and CLUNIES-ROSS 
255 
Fig, L Four regions in (B, C) corresponding to the 
properties : ( 1 ) no linear discriminator reasonable; 
(2) c{m) is a compromise between c(ms), c(mx); 
(3) c(ms) is better than c(m); (4) both c(mx), 
c(ms ) are better than c{m) . In general, the larger the 
C, the stronger will be the discriminator. 
The lower curve in Figure 1 separates the re - 
gions in (B, C) for which (5.4) is true, un- 
true. Thus in region 1 a quadratic discriminator 
is appropriate; elsewhere a linear discriminator 
is appropriate. 
Investigation of tv ben c(ms) is better than 
c(m). Let us denote the larger conditional 
probability of misclassification, P x , using c ( m ) , 
c ( ms ) by P x ( m ) , P x ( ms ) respectively. 
Now c.(mx) is the point on either side of 
which the probabilities of misclassification are 
equal, so that a c < c(mx) indicates Pj = P x 
and a c > c(mx) indicates Pjj = P x - Further, 
m 2 mi, o- 2 o”i imply that both c(m) and 
c(ms) are greater than c(mx) since: 
- 0 - ) 
(5. 5. a) c(m) -c(mx) = — — -- — - — 
(5.5. b) c(ms)-c(mx) 
Yz 
q +2(. 
(o’ 2 -< r 1 )(m 1 tr 2 + m 2 o- 1 ) 
Therefore, P x (m) = Pn(m) and P x (ms) = 
Pll(ms). 
It follows immediately that a necessary and 
sufficient condition for P x (m) > P x (ms) is 
c(m) > c ( ms ) , 
rrv 2 . ( Ba i + - rn 2 )(# tr i) - 
r ^ i 
in ^]i > ° 
(m 2 " 1 " n i )(<r z' <r l ) > Z<T i °2 jj 1 j%^ rt, :1 i 2+2 (' r 2 2 - a \). In ■- 
which may be rewritten as: 
(B + 1) 
The center curve in Figure 1 separates the 
regions of (B, C) for which P x using AX' = 
c(ms) is greater, less than those using AX' = 
c(m). Thus in regions 1 and 2, c(m) is better 
with respect to the minimax criterion; in re- 
gions 3 and 4, c(ms) is better with respect to 
the minimax criterion. 
Investigation of when c(mx) is better than 
c(m ). Let us denote" the sum of conditional 
probabilities of misclassification, P s , using c(m) , 
c(mx) by P s (m), P B (mx). 
On expressing Pj, Pjj in terms of c(m), 
c ( mx ) and hence in terms of B, C, it follows, 
after rearrangement, that 
' - p 
■ N(0 , l)dx - 
N(0 , l)dx - 
Jo ^ 
0 L 
C(Btl) 
2B 
N(0,l)d> 
P (m) -P (rnx) 
- g(B,C) 
say. From differential-geometrical considerations 
and the fact that both c(m),c(ms) are greater 
than c(mx), it follows that c(m) < c(rns) 
implies that P s (m) < P s (mx). The upper 
curve in Figure 1 is the curve g(B, C) = 0, 
which separates the regions of (B, C) for which 
the sum of conditional probabilities of mis- 
classification using AX' = c(mx) is greater, 
less than those using AX' = c(m). Thus in 
region 4, c(mx) is better with respect to the 
minisum criterion; elsewhere c(m) is better 
with respect to the minisum criterion. The 
asymptote as B tends to infinity is, approxi- 
mately, C 1.029. 
Figure 2 shows g(B, C) plotted against C 
for several values of B. 
