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

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)

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

binary-valued 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 user-specified parameter that measures the relative importance of recognition and error rates and is problem-dependent (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.