Chaotic Mining: Knowledge Discovery Using the Fractal Dimension

Nature is lled with examples of phenomena that exhibit seemingly chaotic behavior, such as air turbulence, forest res and the like. However, under this behavior it is almost always possible to nd self-similarity, i.e. an invariance with respect to the scale used. The structures that appear as a consequence of self-similarity are known as fractals [12]. Fractals have been used in numerous disciplines (for a good coverage of the topic of fractals and their applications see [14]). In the database arena, fractals have been sucessfully used to analyze R-trees [6], Quadtrees [5], model distributions of data [7] and selectivity estimation [3]. Fractal sets are characterized by their fractal dimension. In truth, there exists an in nite family of fractal dimensions. By embedding the dataset in an n-dimensional grid which cells have sides of size r, we can compute the frequency with which data points fall into the i-th cell, pi, and compute Dq, the generalized fractal dimension [8, 9], as shown in Equation 1.