1 Introduction 91(x)=0 Class 1 9×)=0 Class 4 Fig. 1. 4 Decision-tree-based decision functions 1.1.2.4 Pairwise Formulation The third approach is to determine the decision functions by pairwise formu- lation 11. For classes i and j we determine the decision function gii(x)(i+ 1,2,3 1, 7), so that 91(x)>0 when x belongs to class i and 9i(x)<0 when x belongs to class j In this formulation, gij (x)=-gji(x), and we need to determine n(n-1)/2 decision functions. Classification is done by voting, namely, we calculate sign(gii(x)). (1.19) where sign(r) x≥0, -1x<0. (1.20) and we classify x into the class with the maximum gi(x). By this formu- lation also, unclassifiable regions exist if gi(x) take the maximum value for I We may define the sign function by x>0, sgn(x)={0x=0,6 1 Introduction Class 1 x1 x2 0 Class 2 g1 (x) = 0 g2 (x) = 0 g3 (x) = 0 Class 4 Class 3 Fig. 1.4 Decision-tree-based decision functions 1.1.2.4 Pairwise Formulation The third approach is to determine the decision functions by pairwise formu￾lation [11]. For classes i and j we determine the decision function gij (x) (i = j, i, j = 1,...,n), so that gij (x) > 0 (1.17) when x belongs to class i and gij (x) < 0 (1.18) when x belongs to class j. In this formulation, gij (x) = −gji(x), and we need to determine n(n−1)/2 decision functions. Classification is done by voting, namely, we calculate gi(x) = n j=i,j=1 sign(gij (x)), (1.19) where sign(x) = 1 x ≥ 0, −1 x < 0, (1.20) and we classify x into the class with the maximum gi(x). 1 By this formu￾lation also, unclassifiable regions exist if gi(x) take the maximum value for 1 We may define the sign function by sign(x) =  1 x > 0, 0 x = 0, −1 x < 0.