The majority vote one the easiest implement
286 Performance
applications. However, it is a rule that is relatively insensitive to errors in the estimation of the joint densities and would be applied to classifiers used for a common input pattern (Figure 8.8).
Min, max and median combiners
The max combiner may be derived by approximating the posterior probabilities in (8.20) by an upper bound, Lmaxi p.!kjxi/, to give the decision rule: assign Z to class ! j if
p.! jjxi/ > .1 � L/p.!k/ C L max  p.!kjxi/  k D 1; : : : ; C;  

k 6D j 
give the decision rule: assign Z to class ! j if We can also approximate the product in (8.17) by an upper bound, mini p.!kjxi/, to
[p.! j/]�.L�1/min i p.! jjxi/ > [p.!k/]�.L�1/min p.!kjxi/ k D 1; : : : ; C; k 6D j (8.25)
p.! jjxi/ > min  p.!kjxi/  (8.26) 

Finally, the median combiner is derived by noting that the sum rule calculates the mean of the classifier outputs and that a robust estimate of the mean is the median. Thus, under equal priors, the median combiner is: assign Z to class ! j if

p.! jjxi/ > med  p.!kjxi/  (8.27) 

binaryvalued function, 1ki, where p.!kjxi/, have been ‘hardened’ (Kittler et al., 1998); that is, p.!kjxi/ is replaced by the
( 1 if p.!kjxi/ D max p.! jjxi/
1ki D 0 otherwise
objective function and an appropriate optimisation procedure. One
approach is to define the objective function
F D Re � þE
where Re is the recognition rate and E is the error rate of the combiner (they do not sum to unity as the individual classifiers may reject patterns – see Chapter 1); þ is a userspecified parameter that measures the relative importance of recognition and error rates and is problemdependent (Lam and Suen, 1995). Rejection may be treated as an extra class by the component classifiers and thus the combiner will reject a pattern if the weighted majority of the classifiers also predicts a rejection. In a study of combination schemes applied to a problem in optical character recognition, Lam and Suen (1995) used a genetic optimisation scheme (a scheme that adjusts the weights using a learning method loosely motivated by an analogy to biological evolution) to maximise F and concluded that simple majority voting (all weights equal) gave the easiest and most reliable classification.