Convergence of Generalized Probability Mixtures That Describe Adaptive Fuzzy Rule-based Systems

Uniform Mixture Convergence of Continuously Transformed Fuzzy Systems

“…We now show first that a 2-bell-curve Gaussian mixture p(y|x) can exactly represent any continuously transformed bounded real function f for any continuous function φ. This result extends the recent result of representing just a bounded target function f [13]. We present this and other results only for the scalar-valued case even though they extend componentwise to vector-valued systems.…”

“…The new mixture convergence theorem exploits the fact that a generalized probability mixture p(y|x) of just two Gaussian bell curves can exactly represent any bounded real function f as the average of the mixture: [12,13]. The conditional expectation E[Y |X = x] integrates or sums with respect to the conditional Gaussian mixture p(y|x):…”

“…The mixing weights always depend on x. The 2-bell-curve Gaussian mixture p(y|x) in (1) gives an efficient way to represent a fuzzy or neural approximator and still have access to the mixture's XAI moment and Bayesian structure [13,20]. Figure 1 shows the mixture representation p φ (y)|x) of the square of the target function f (x) = sin x.…”