Correlation Dimension-Based Classifier

Authors: Marcel Jiřina, M. Jiřina jr.
Publication year: 2014
Publication type: Článek v odborném periodiku
Periodical: IEEE Transactions on Cybernetics
Projects: Advanced statistical analysis and non-statistical separation techniques for physical processing detection in data sets sampled by means of elementary particle accelerators.
Bibliographic citation: JIŘINA, Marcel, JIŘINA JR., M. Correlation Dimension-Based Classifier. IEEE Transactions on Cybernetics. 2014, 44(12), 2253-2263. ISSN 2168-2267. E-ISSN 2168-2275
Dostupné z: doi:10.1109/TCYB.2014.2305697
DOI: 10.1109/TCYB.2014.2305697
ID: 421968

Abstract

Correlation dimension, singularity exponents, also scaling exponents are widely used in multifractal chaotic series analysis. Correlation dimension and other measures of effective dimensionality are used for characterization of data in applications. A direct use of correlation dimension 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 correlation dimension 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 correlation dimension - 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 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 correlation dimension. We also propose an "averaging approach" that speeds up computation of the correlation dimension especially for large data sets. It is demonstrated that the correlation dimension based classifier can outperform more sophisticated classifiers.