# Fuzzy Systems Are Universal Approximators for Random Dependencies: A Simplified Proof

• Mahdokht Afravi
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 276)

## Abstract

In many real-life situations, we do not know the actual dependence $$y=f(x_1,\ldots ,x_n)$$ between the physical quantities $$x_i$$ and y, we only know expert rules describing this dependence. These rules are often described by using imprecise (“fuzzy”) words from natural language. Fuzzy techniques have been invented with the purpose to translate these rules into a precise dependence $$y=\widetilde{f}(x_1,\ldots ,x_n)$$. For deterministic dependencies $$y=f(x_1,\ldots ,x_n)$$, there are universal approximation results according to which for each continuous function on a bounded domain and for every $$\varepsilon >0$$, there exist fuzzy rules for which the resulting approximate dependence $$\widetilde{f}(x_1,\ldots ,x_n)$$ is $$\varepsilon$$-close to the original function $$f(x_1,\ldots ,x_n)$$. In practice, many dependencies are random, in the sense that for each combination of the values $$x_1,\ldots ,x_n$$, we may get different values y with different probabilities. It has been proven that fuzzy systems are universal approximators for such random dependencies as well. However, the existing proofs are very complicated and not intuitive. In this paper, we provide a simplified proof of this universal approximation property.

## Notes

### Acknowledgements

This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, and by an award “UTEP and Prudential Actuarial Science Academy and Pipeline Initiative” from Prudential Foundation.

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