Interpolation and extrapolation of fuzzy quantities revisited – an axiomatic approach

Abstract

This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the ®eld of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. For this purpose the well-known fuzzy inference mechanisms have to be extended or replaced by more general ones. The methods proposed so far in the literature for rule interpolation are mainly conceived for the application to fuzzy control and miss certain logical characteristics of an inference. First, a set of axioms is proposed in this paper. With this, a de®nition is given for the notion of interpolation, extrapolation, linear interpolation and linear extrapolation of fuzzy rules. The axioms include all the conditions that have been of interest in the previous attempts and others which either have logical characteristics or try to capture the linearity of the interpolation. A new method for linear interpolation and extrapolation of compact fuzzy quantities of the real line is suggested and analyzed in the spirit of the given de®nition. The method is extended to non-linear interpolation and extrapolation as well.

Fuzzy Rule Interpolation and Extrapolation Techniques: Criteria and Evaluation Guidelines

“…These limitations are waived using the Hausdorff-distance of α-cuts in defining the distance of fuzzy sets [15]:…”

“…The second approach, e.g., [15,20,21,32,33], uses the reference point concept [14,20,34], rp, to determine the position of fuzzy sets. By definition, A ≺ B if and only if rp (A) < rp (B), which is a total ordering, in which case the distance of fuzzy sets is determined as:…”

“…FRE is usually an extension of FRI. In the literature, FRI attracted more interest and FRE was mostly neglected [13][14][15][16][17], even though the latter significantly broadens FRI applicability. When we refer here to FRITs, we mean an approximative technique dealing with sparse rule bases including both FRI and FRE; when approximation is concerned, we explicitly specify either interpolation or extrapolation.…”

“…The several initiatives in the literature to set up FRIT criteria are divided into two groups -approach-oriented and summarization-oriented. The approach-oriented papers focus on a new FRI motivated by critiques of other methods or new analytical aspects, e.g., [15,20,21], using conditions in FRIT analysis that are, in our opinion, motivated to justify the proposed approach or aspect, i.e., are not comprehensive enough. Summarization-oriented papers include brief summaries mostly focusing on selected FRI aspects, e.g., [22][23][24][25], criticizable on the same ground.…”

“…[21], for example, analyzes and compares proposed MACI's general applicability, complexity, approximative power, and fuzziness of conclusion. Jenei [15] axiomatically treated FRITs by setting eight conditions on rule interpolation/extrapolation, focusing on applicability, consistency, functionality, and logic. Baranyi et al [20] investigated the general methodology for applicability, consistency, and shape-preservation -all of which appear in the second, which are also summarizations.…”