Classification Methods for Remotely Sensed Data, Second Edition

Page 139

nated by the subscripts or superscripts a and b. Figure 3.18b illustrates the linkages between F_ab and both F2_a (in the case of three neurones) and F2_b (in the case of two desired categories). The structure of the F2_a and F2_b layers is the same as shown in Figure 3.15b. However, for simplicity, both self-linkage and inter-neurone linkages are ignored. Each neurone in F2_a is linked to all F_ab neurones by ways of w_ki^a, while neurones in F2_b are connected to F_ab in terms of one-to-one pathways in both directions (F2_b → F_ab and F2_b → F_ab). The variables u_k^b and z_k^ab each denotes the output of F2_b and F_ab, respectively. The inputs for both ART_a and ART_b are normalised in terms of complement coding. An input vector X, becomes , where , for i =1 to n, while the rest of the elements remain the same. Complement coding provides a useful means of solving a potential fuzzy ART category proliferation problem (Carpenter et al., 1991a).

During training, the initialisation process for weights within the ART_a and ART_b is the same as described in Section 3.5.1. The weights (F2_a →F_ab) are initialised to the value of unity. Once a training pair (i.e. data features and desired category) is presented to the fuzzy ARTMAP, ART_a and ART_b are triggered to determine their own winning neurones on F2_a and F2_b using the procedures described in Figure 3.16, steps (1) and (2). Let I denote the winning neurone in F2_a. The variable in F_ab is then computed by

and subjected to the vigilance test:

where ρ^ab is the vigilance parameter for F_ab. The purpose of Equation (3.55) is in fact to test if ART_a favours the same category as shown in ART_b. If the test in Equation (3.55) does not succeed, the vigilance in ART_a, denoted by ρ^a, is increased by an amount which is sufficiently large so as to make the current ART_a winner neurone invalid, i.e.

The variables in Equation (3.56) are the same as those defined in Section