Correlation Dimension-Based Classifier

Correlation dimension (CD), singularity exponents, also called scaling exponents, are widely used in multifractal chaotic series analysis. CD and other measures of effective dimensionality are used for characterization of data in applications. A direct use of CD to multidimensional data classification has not been hitherto presented. There are observations that the correlation integral is a distribution function of distances between all pairs of data points, and that by using polynomial expansion of distance with exponent equal to the CD this distribution is transformed into locally uniform. The classifier is based on consideration that the influence of neighbor points of some class on the probability that the query point belongs to this class is inversely proportional to its distance to the CD, power. New classification approach is based on summing up all these influences for each class. We prove that a resulting formula gives an estimate of probability of the class, not a measure of membership to a class only, to which the query point belongs. For this assertion to be valid, it is necessary that exponent of the polynomial transformation must be the CD. We also propose an averaging approach that speeds up computation of the CD especially for large data sets. It is demonstrated that the CD-based classifier can outperform more sophisticated classifiers.