Fuzzy fractal dimensions and fuzzy modeling

Authors:
Witold Pedrycz;Andrzej Bargiela
Affiliations:
Department of Electrical and Computer Engineering, University of Alberta, Alberta, Edmonton, Canada T6G 2G7;Department of Computing, The Nottingham Trent University, Nottingham NG1 4BU, UK
Venue:
Information Sciences: an International Journal
Year:
2003

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Abstract

Fractal dimensions describe self-similarity (structural complexity) of various phenomena (such as e.g., temporal signals, images). Their determination (through box counting or a correlation method) is inherently associated with the use of information granules--sets. The intent of this study is to generalize the idea to the domain of fuzzy sets and reveal associations between the mechanisms of fractal analysis and granular computing (including fuzzy modeling). First, we introduce the concept itself and discuss the role of fuzzy sets as a vehicle for constructing fractal dimensions. Second, we propose an algorithmic framework necessary to carry out all computing, and experimentally quantify a performance of regression models used to determine fractal dimensions and contrast it with the performance of the fractal models existing in the set-based environment. It is shown that fuzzy set approach produces more consistent models (in terms of their performance). We also postulate a power law of granularity and discuss its direct implications in the form of a variable granularity in fuzzy modeling. In particular, we show how the power law of granularity helps construct mappings between system's variables in rule-based models. Experimental studies involving several frequently used categories of fuzzy sets illustrate the main features of the approach.