Hybrid intelligent systems for time series prediction using neural networks, fuzzy logic, and fractal theory

In this paper, 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 have proposed 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." We can apply this new definition of the fractal dimension in conjunction with soft computing techniques for the problem of time series prediction. We have developed hybrid intelligent systems combining neural networks, fuzzy logic, and the fractal dimension, for the problem of time series prediction, and we have achieved very good results.