Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

Example X.2. As an example, synthetic data was used to verify this approach. Parameters for the
multiplicative hybrid operators were randomly generated and assigned to each node. Then, a table of
1000 input values was randomly generated and corresponding outputs were calculated from successive
applications of Equation X.5. The training data consisted of 800 data points and the test data had 200 data
points. Table X.3 shows a sample of the training and test data from one experiment while Table X.4
shows the original and recovered hybrid parameters. With an easy and effective training mechanism,
such fuzzy aggregation networks are attractive tools for hierarchical confidence fusion. Besides the
ability to approximate input/output training data, an additional advantage of these networks is that after
training, each node can be associated with a linguistic connective (disjunction, conjunction, mean), based
on the corresponding value of γ, and the weights give an indication of the importance of the particular
criteria towards the fused result.
Table X.4 Actual and recovered parameters corresponding to Table X.3.
Parameter Actual Recovered
Node
1
δ 1 0.440 0.446
δ 2 1.559 1.553
γ 0.255 0.341
Node
2
Final
Node
δ 1 0.161 0.163
δ 2 1.838 1.836
γ 0.180 0.198
δ 1 0.786 0.816
δ 2 1.213 1.183
γ 0.0846 0.028
Exercises
x. A trapezoid membership function is defined by 4 parameters a < ≤ b c < d. (If b = c, you have a
triangular function). Define the equations for trapezoid and triangular fuzzy sets over a fixed interval of
reals, [r, s]. Note that these membership functions do not need to be symmetric about the “center”. What
happens if b < r? What about s < c?
Let X = [0, 10]. Define and draw the graphs of the Trap(x; 1,2,3,5), Trap(x; 3,5,7,9), Trap(x; 8,9,9,10),
Trap(x; -2,-1,3,5) and Trap(x; 8,9,11,12).