A new method of estimation of the box-counting dimension of multivariate objects using space-filling curves

Box-counting dimension is one of the most widely used fractal dimensions, because it is relatively easy for empirical estimation. Unfortunately, the task is simple only if one operates on one- or two-dimensional data. Otherwise, the amount of computation grows exponentially with the number of space coordinates. A new method of estimation of the box-counting dimension of multivariate objects using space-filling curves is presented. The method is based on a quasi-inverse to a space-filling curve which maps data to one dimension. Such a transformation retains box-counting dimension up to a known factor. This factor equals the embedding dimension. Thus, the box-counting dimension of multidimensional objects can be estimated by means of a one-dimensional procedure. (C) 2005 Elsevier Ltd. All rights reserved.