Varying functional representations of a crisp discrete function under fuzzy environment to coin new siso algorithms with application to time series modeling

Abstract

One of the main categories of fuzzy reasoning mechanisms for a single-input single-output (SISO) fuzzy system based on linguistically described model is the category of composition based algorithms. The main ingredients of a composition based SISO algorithm consist of two parts: a system relation matrix (representing the structure information of a fuzzy system) and a compositional rule of inference (to obtain the corresponding output for the crisp or fuzzy value of an input variable). In literature, there are various composition based algorithms based on a fuzzy-logic-based approach via different versions of the theory of approximate reasoning (AR) since it is initiated by Zadeh in 1979. In this article, we provide an alternate approach to construct composition based SISO algorithms by viewing the collection of the (antecedent part, consequent part) of IF-THEN rules of a fuzzy system as a class of weighted information and using weighted t-norm and weighted t-conorm operations to aggregate such weighted information to build its system relation matrix instead of the usual AR approach. The novelty of our approach is that not only the construction of the system relation matrix but also the process of composition can be executed in a similar way. To justify our approach, we examine various functional representations of a crisp discrete function under fuzzy environment and using them as patterns to coin four classes of composition based SISO algorithms, in which, two classes are of new type while the other two classes rediscover the well-known Mandani-type and Zadeh's logic-type algorithms. Then, as an application, in the latter part of this article, we provide a generalization of the time series modeling proposed by Štĕpnička et al. in which, they suggested using the fuzzy transform method to estimate the trend-cyclic component of a time series. To accomplish this, we will show that, under the contexts of this article, we may view the (discrete) fuzzy transform (and the corresponding inverse fuzzy transform) of a time series as a composition of specific SISO algorithms and hence, the time series modeling proposed by Štĕpnička et al. may be extended via suitable techniques of SISO algorithms.