Fuzzy modelling using a simplified rule base

Abstract

dbsfruct-Transparency and complexity are two major concerns of fumy rule-based systems. To improve accuracy and precision of the outputs, we need to increase the partitioning of the input space. However, this increases the number of rules exponentially, thereby increasing the complexity of the system and decreasing its transparency. The main factor behind these t w o issues is the conjunctive canonical form of the fuzzy rules. We prment a novel method for replacing these rules with their singleton forms, and using aggregation operators to provide the mechanism for combining the crisp outputs.Keywords-Fuuy comiraint aggregation uperatur, fuziy ino&liing, singleton fuzzy rule 1NTRODUCTtONUntil recently, it has generally been perceived that systems implementing fuzzy modclling [ 13 provide better transparency than the black box models such as artificial neural networks. By transparency, we mean interpretability of rules and outputs. The main reason for this perception is that most conventional rule-based fuzzy systems are abstracted from human experts or heuristics, and they are usually easy to comprehend. This providcs the transparency enabling one to gain insights into the system and acquire important knowledge. However, as more and more fuzzy rules are automatically generated using training or experimenta1 data, fuzzy modelling becomes less easily understood by humans because of the increase in complexity of these rules. Another issue is that the number of inputs has to be kept low because the dimension of the input space and complexity grows exponentially in terms of the number of input variables [2, 3j. So we have a dilemma:' On one hand, the requirement of accuracy calls for the use of dense ruIe bases with Iarge numbers of antecedent variables and linguistic terms, on the other hand, exponential growth in the size of rule base creates problems with computational time and storage space requirements. Reduction of rules is dcsirable but limiting the number of rules may destroy the property of the model as a universal approximator [4].Many techniques have been proposed for rule reduction. We discuss briefly the problems associated with using these in Section 11. In Section 111 wc propose a new approach that aims to improve the interpretation of the much simpler fuzzy rules and reduce the complexity of the problem domain, using some examples to illustrate our approach. The paper concludes in Section IV. 0-where x,, i=I ,..., k is the ith input to the fuzzy system, which is dcfined on the universe of discourse U,; A, is a fizzy sct on U,; y is the system output defined on a universe of discourse V, and B is a fuzzy set on V. For simpIicity we are assuming B in this case to be on a one-dimensional output space. We discuss a multi-dimensional output space in the next section.To derive the fuzzy if-then rules, several approaches have been proposed. Some of the common approaches are outlined in the following subscctions: A . Grid PartitioningThe most common method to construct fuzzy rules is to partition the input sp...

A hybrid fuzzy cognitive model based on weighted OWA operators and single-antecedent rules

“…Furthermore, the single-antecedent fuzzy rule bases have higher transparency, lower complexity, and therefore better maintainability than the corresponding multipleantecedent rule base. 22 …”