1 Introduction 1.1 Decision Functions 1.1.1 Decision Functions for Two-Class Problems Consider classifying an m-dimensional vector x=(a1,., Im) into one of wo classes. Suppose that we are given scalar functions gi(x) and g2(x) for Classes 1 and 2, respectively, and we classify x into Class l if 91(x)>0,g2(x)<0, (1.1) Class 2 if q1(x)<0.92(x)>0. We call these functions decision functions. By the preceding decision func tions. if fo 1(x)g2(x)>0 (1.3) is satisfied, x is not classifiable(see the hatched regions in Fig. 1. 1; the arrows show the positive sides of the functions) 91(x)=0 Fig. 1.1 Decision functions in a two-dimensional space To resolve unclassifiable regions, we may change(1.1)and(1. 2)as follows We classify x into Class 1 if2 1 Introduction 1.1 Decision Functions 1.1.1 Decision Functions for Two-Class Problems Consider classifying an m-dimensional vector x = (x1,...,xm) into one of two classes. Suppose that we are given scalar functions g1(x) and g2(x) for Classes 1 and 2, respectively, and we classify x into Class 1 if g1(x) > 0, g2(x) < 0, (1.1) and into Class 2 if g1(x) < 0, g2(x) > 0. (1.2) We call these functions decision functions. By the preceding decision func￾tions, if for x g1(x) g2(x) > 0 (1.3) is satisfied, x is not classifiable (see the hatched regions in Fig. 1.1; the arrows show the positive sides of the functions). Class 1 x1 x2 0 Class 2 g1 (x) = 0 g2 (x) = 0 Fig. 1.1 Decision functions in a two-dimensional space To resolve unclassifiable regions, we may change (1.1) and (1.2) as follows. We classify x into Class 1 if g1(x) > g2(x) (1.4)