Abstract
Kernel discriminants are greatly appreciated because 1) they permit to establish nonlinear boundaries between classes and 2) they offer the possibility of visualizing graphically the data vectors belonging to different classes. One such method, called Generalized Discriminant analysis (GDA) was proposed by Baudat and Anouar (2000). GDA operates on a kernel matrix of size N x N, (N denotes the sample size) and is for large N prohibitive. Our aim was to find out how this method works in a real situation, when dealing with relatively large data. We considered a set of predictors of erosion risk in the Kefallinia island categorized into five classes of erosion risk (together N=3422 data items). Direct evaluation of the discriminants, using entire data, was computationally demanding. Therefore, we sought for a representative sample. We found it by a kind of sieve algorithm. It appeared that using the representative sample, we could greatly speed up the evaluations and obtain discriminative functions with good generalization properties. We have worked with Gaussian kernels which need one declared parameter SIGMA called kernel width. We found that for a large range of parameters the GDA algorithm gave visualization with a good separation of the considered risk classes.
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References
Bartkowiak, A., Evelpidou, N.: Visualizing some multi-class erosion data using kernel methods. In: A. Rizzi, M. Vichi, Eds., Compstat 2006, Proceedings in Computational Statistics, Physica-Verlag, A Springer Company, pp. 805–812, 2006.
Bartkowiak, A., Evelpidou, N.: Visualization of multivariate data with additional class information. In: K. Saeed et al., Image Analysis, Computer Graphics, Security Systems and Artificial Intelligence Applications. WSFiZ Bialystok, 307–320, 2005.
Bartkowiak A., Vassilopoulos A., Evelpidou N.: Choosing data vectors representing a huge data set: Kohonen’s SOM applied to the Kefallinia erosion data. Proc. First Int. Conf. on Environmental Research and Assessment, Bucharest, March 23-27, 2003. Ars Docendi P. H., Bucharest, Romania, pp. 505–521, 2003.
Bartkowiak A., Szustalewicz A., Evelpidou N., Vassilopoulos A.: Choosing data vectors representing a huge data set: a comparison of Kohonen’s maps and the neural gas method. Proc. First Int. Conf. on Environmental Research and Assessment, Bucharest, March 23–27, 2003. Ars Docendi P. H., Bucharest, Romania, pp. 561–572, 2003.
Baudat, G., Anouar, F.: Generalized discriminant analysis using a kernel approach. Neural Computation 12 (2000), pp. 2385–2404.
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd Edition, Wiley. 2001.
Gournelos, Th., Vassilopoulos, A., Evelpidou, N., Giotitsas, Il.: Modelling slop erosional processes. COST Action 634 conference: ‘Soil conservation management, perception and policy’, Mont Saint Aignan, France, 5-7 June 2005, under edition.
Gournelos, Th., Vassilopoulos, A., Evelpidou, N.: Examples of erosion risk maps using Boolean and fuzzy logical rules in GIS web platforms, Proceedings of 20th International cartographic conference, Beijing China 2001,, Vol. 4. pp. 2472–2479.
Gournelos, Th., Vassilopoulos, A., Evelpidou, N.: An erosion risk map on Samos island, based on fuzzy models, taking into consideration landuse situation after the fire of July 2000, Proceedings of the 7th Conference of Environmental Science and Technology, Syros 2001, pp. 284–290.
Gournelos, Th., Vassilopoulos, A. Evelpidou, N.: Fire danger index using soft computing approach in GIS web platform – a case study from Greece. Proc. First Int. Conf. on Environmental Research and Assessment, Bucharest, March 23-27, 2003. Ars Docendi, Bucharest Romania 2003, pp. 586–595.
Hastie T., Tibshirani R., Friedman J.,: The Elements of Statistical Learning. Springer 2001.
Lachenbruch, P.: Discriminant Analysis. Hafner Press 1975.
Martinetz, M., Berkovich, S., Schulten, K.: ’Neural-gas’ network for vector quantization and its application to time series prediction. IEEE Trans. Neural Networks, 4 (1993), pp. 558–569.
Mika S., Rätsch G., Weston J., Schölkopf B., Muller K.R.: Fisher Discriminant Analysis with Kernels. Neural Networks for Signal Processing IX, IEEE 1999, pp. 41–48.
Muller, K-R., Mika, S., Rätsch, G., Tsuda, K., Schölkopf, B.: An introduction to kernel-based learning algorithms. IEEE Trans. on Neural Networks 12(2), 2001, pp. 181–202.
Roth V., Steinhage V.: Nonlinear discriminant analysis using kernel functions. NIPS, Advances in Neural Information Processing Systems 12, Proceedings of the 1999 Conference, MIT Press Cambridge, MA, London, England 2000, pp. 568–574.
Schölkopf, B., Mika S., Burges Ch.J.C, Knirsch, P.H., Muller, K.R., Rätsch G., Smola, A.: Input space versus feature space in kernel-based methods. IEEE Trans. on Neural Networks 10 (1999), No. 5, pp. 1000–1017.
Shawe-Taylor, J., Christianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press UK. 2004.
Szustalewicz, A.: A practice in applying radial basis functions (RBF) in real situation. J. Soldek, J. Pejas (Eds), ACS’2000, Advanced Computer Systems, Proceedings. Fac. of Computer Science, Technical University of Szczecin. Informa, 2000, pp. 294–299.
Vassilopoulos, A., Gournelos, Th., Evelpidou, N., Land use – Land cover classification for the island of Cephalonia, Proceedings of the VI International conference ‘Protection and Restoration of the Environment”, 1-5 July, Skiathos, 2002, pp. 1761–1768.
Vassilopoulos, A.: Coastal geomorphological classification in GIS environment of Kefallinia Island. Proc. 6th Pan-Hellenic Geographical Congress of the Hellenic Geographical Society. Thessaloniki 3-5.10.2002. V.1 388–394.
Vesanto, J., Himberg J., Alhoniemi, E., Parhankangas J.: SOM Toolbox for Matlab 5. Som Toolbox team, Helsinki University of Technology, Finland, Libella Oy, Espoo 2000, pp. 1–54. Version 0beta 2.0, November 2001, actualized 2005. http://www.cis.hut.fi/projects/somtoolbox/
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Bartkowiak, A., Evelpidou, N., Vasilopoulos, A. (2007). Visualization of Five Erosion Risk Classes using Kernel Discriminants. In: Pejaś, J., Saeed, K. (eds) Advances in Information Processing and Protection. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-73137-7_15
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DOI: https://doi.org/10.1007/978-0-387-73137-7_15
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