A new method for fuzzy estimation of the fractal dimension and its applications to time series analysis and pattern recognition

Abstract

We describe a new method for the estimation of the fractal dimension of a geometrical object using fuzzy logic techniques. The fractal dimension is a mathematical concept, which measures the geometrical complexity of an object. The algorithms for estimating the fractal dimension calculate a numerical value using as data a time series for the specific problem. This numerical (crisp) value gives an idea of the complexity of the geometrical object (or time series). However, there is an underlying uncertainty in the estimation of the fractal dimension because we use only a sample of points of the object, and also because the numerical algorithms for the fractal dimension are not completely accurate. For this reason, we propose a new definition of the fractal dimension that incorporates the concept of a fuzzy set. This new definition can be considered a weaker definition (but more realistic) of the fractal dimension, and we have named this the fuzzy fractal dimension.