一类二叉树型分层模糊系统的等效性条件

为了进一步证明分层模糊系统在很好地解决了多变量模糊系统的规则数随输入变量个数指数增长的问题的同时,是否能够解决系统的参数个数随输入变量指数式的增长,推导一类二叉树型分层模糊系统和一类典型多输入单输出(MISO)模糊系统的输入输出表达式,证明了两者的等效性条件。得出结论:此类二叉树型分层模糊系统要达到典型MISO模糊系统同样的逼近精度,其参数个数至少要与典型模糊系统一样多。这一特例对分层模糊系统是否能从根本上解决MISO模糊系统的"维数灾"问题提出了质疑。

Equivalence conditions for binary-tree-type hierarchical fuzzy systems

To prove whether hierarchical fuzzy systems can be solved as the number of parameters exponentially increases when the number of input variables increases.This paper analyzes the problem of the number of fuzzy rules exponentially increasing as the number of input variables increases.This analysis gives equivalence conditions for a binary-tree-type hierarchical fuzzy system and a typical multi-input-single-output(MISO) fuzzy system by deriving their input-output expressions.The analysis shows that to reach the same approximation accuracy,the number of parameters used by the binary-tree-type hierarchical fuzzy system must not be less than of the typical fuzzy system.This example casts doubt on whether hierarchical fuzzy systems can ultimately solve the "curse of dimensionality" problem of general MISO fuzzy systems.