Fuzzy fractal dimensions and fuzzy modeling. (English) Zbl 1046.93501

Summary: 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 the 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 to construct mappings between systems’ variables in rule-based models. Experimental studies involving several frequently used categories of fuzzy sets illustrate the main features of the approach.


93A30 Mathematical modelling of systems (MSC2010)
28A80 Fractals
Full Text: DOI


[1] Barnsley, M., Fractals Everywhere (1988), Academic Press: Academic Press Boston · Zbl 0691.58001
[2] Carlin, M., Measuring the complexity of non-fractal shapes by a fractal method, Pattern Recognition Letters, 21, 11, 1013-1017 (2000) · Zbl 0966.37005
[4] Crownover, R. M., Introduction to Fractals and Chaos (1995), Jones and Bartlett Publishers: Jones and Bartlett Publishers Boston
[5] Gleick, J., Chaos, Making a New Science (1987), Viking Penguin: Viking Penguin NY · Zbl 0706.58002
[6] Mandelbrot, B. B., The Fractal Geometry of Nature (1982), W.H. Freeman: W.H. Freeman San Francisco · Zbl 0504.28001
[7] (Manderbrot, B., The Fractal Geometry of Nature (1983), Freeman: Freeman NY), 14-57
[8] (Mullin, T., The Nature of Chaos (1993), Claredon Press: Claredon Press Oxford) · Zbl 0784.58001
[9] (Zadeh, L. A.; Kacprzyk, J., Computing with Words in Information/Intelligent Systems, vol. I and II (1999), Physical Verlag: Physical Verlag Heidelberg) · Zbl 0931.00022
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