Approximation capabilities of hierarchical fuzzy systems

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    Abstract

    Derived from practical application in location analysis and pricing, and based on the approach of hierarchical structure analysis of continuous functions, this paper investigates the approximation capabilities of hierarchical fuzzy systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters. Then, based on Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property. Further, the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed. © 2005 IEEE.
    Original languageEnglish
    Pages (from-to)659-672
    Number of pages13
    JournalIEEE Transactions on Fuzzy Systems
    Volume13
    Issue number5
    DOIs
    Publication statusPublished - Oct 2005

    Keywords

    • Approximation accuracy
    • Hierarchical fuzzy systems
    • Hierarchical structure
    • Kolmogorov's theorem
    • Universal approximation

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