Abstract
In this paper we show that the correlation integral can be decomposed into functions each related to a particular point of data space. For these functions, one can use similar polynomial approximations as used in the correlation integral. The essential difference is that the value of the exponent, which would correspond to the correlation dimension, differs in accordance to the position of the point in question. Moreover, we show that the multiplicative constant represents the probability density estimation at that point. This finding is used for the construction of a classifier. Tests with some data sets from the Machine Learning Repository show that this classifier can be very effective.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Camastra, F.: Data dimensionality estimation methods: a survey. Pattern Recognition 6, 2945–2954 (2003)
Camastra, F., Vinciarelli, A.: Intrinsic Dimension Estimation of Data: An Approach based on Grassberger-Procaccia’s Algorithm. Neural Processing Letters 14(1), 27–34 (2001)
Cover, T.M., Hart, P.E.: Nearest Neighbor Pattern Classification. IEEE Transactions on Information Theory IT-13(1), 21–27 (1967)
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern classification, 2nd edn. John Wiley and Sons, Inc., New York (2000)
Dvorak, I., Klaschka, J.: Modification of the Grassberger-Procaccia algorithm for estimating the correlation exponent of chaotic systems with high embedding dimension. Physics Letters A 145(5), 225–231 (1990)
Gama, J.: Iterative Bayes. Theoretical Computer Science 292, 417–430 (2003)
Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica 9D, 189–208 (1983)
Guerrero, A., Smith, L.A.: Towards coherent estimation of correlation dimension. Physics letters A 318, 373–379 (2003)
Lev, N.: Hausdorff dimension. Student Seminar, Tel-Aviv University (2006), www.math.tau.ac.il/~levnir/files/hausdorff.pdf
Merz, C. J., Murphy, P. M., Aha, D. W.: UCI Repository of Machine Learning Databases. Dept. of Information and Computer Science, Univ. of California, Irvine (1997), http://www.ics.uci.edu/~mlearn/MLSummary.html
Osborne, A.R., Provenzale, A.: Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35, 357–381 (1989)
Paredes, R., Vidal, E.: Learning Weighted Metrics to Minimize NearestNeighbor Classification Error. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(7), 1100–1110 (2006)
Takens, F.: On the Numerical Determination of the Dimension of the Attractor. In: Dynamical Systems and Bifurcations. Lecture Notes in Mathematics, vol. 1125, pp. 99–106. Springer, Berlin (1985)
Weisstein, E. W.: Information Dimension. From MathWorld–A Wolfram Web Resource (2007), http://mathworld.wolfram.com/InformationDimension.html
Friedmann, J.H.: Flexible Metric Nearest Neighbor Classification. Technical Report, Dept. of Statistics, Stanford University, p. 32 (1994)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jiřina, M., Jiřina, M. (2008). Correlation Integral Decomposition for Classification. In: Kůrková, V., Neruda, R., Koutník, J. (eds) Artificial Neural Networks - ICANN 2008. ICANN 2008. Lecture Notes in Computer Science, vol 5164. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87559-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-87559-8_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87558-1
Online ISBN: 978-3-540-87559-8
eBook Packages: Computer ScienceComputer Science (R0)