Abstract
In this paper we propose to consider Probability Measures (PM) as generalized functions belonging to some functional space endowed with an inner product. This approach allows to introduce a new family of distances for PMs. We propose a particular (non parametric) metric for PMs belonging to this class, based on the estimation of density level sets. Some real and simulated data sets are used for a first exploration of its performance.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., Rao, C.R.: Differential Geometry in Statistical Inference. Lecture Notes-Monograph Series, vol. 10 (1987)
Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society (2007)
Atkinson, C., Mitchell, A.F.S.: Rao’s Distance Measure. The Indian Journal of Statistics, Series A 43, 345–365 (1981)
Müller, A.: Integral Probability Metrics and Their Generating Classes of Functions. Advances in Applied Probability 29(2), 429–443 (1997)
Banerjee, A., Merugu, S., Dhillon, I., Ghosh, J.: Clustering whit Bregman Divergences. Journal of Machine Learning Research, 1705–1749 (2005)
Burbea, J., Rao, C.R.: Entropy differential metric, distance and divergence measures in probability spaces: A unified approach. Journal of Multivariate Analysis 12, 575–596 (1982)
Cohen, W.W., Ravikumar, P., Fienberg, S.E.: A Comparison of String Distance Metrics for Name-matching Tasks. In: Proceedings of IJCAI 2003, pp. 73–78 (2003)
Devroye, L., Wise, G.L.: Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 38, 480–488 (1980)
Dryden, I.L., Koloydenko, A., Zhou, D.: Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. The Annals of Applied Statistics 3, 1102–1123
Dryden, I.L., Koloydenko, A., Zhou, D.: The Earth Mover’s Distance as a Metric for Image Retrieval. International Journal of Computer Vision 40, 99–121 (2000)
Gretton, A., Borgwardt, K., Rasch, M., Schlkopf, B., Smola, A.: A kernel method for the two sample problem. In: Advances in Neural Information Processing Systems, pp. 513–520 (2007)
Hastie, T., Tibshirani, R., Friedman, J.: The elements of statistical learning, 2nd edn. Springer (2009)
Hayashi, A., Mizuhara, Y., Suematsu, N.: Embedding Time Series Data for Classification. In: Perner, P., Imiya, A. (eds.) MLDM 2005. LNCS (LNAI), vol. 3587, pp. 356–365. Springer, Heidelberg (2005)
Kylberg, G.: The Kylberg Texture Dataset v. 1.0. Centre for Image Analysis, Swedish University of Agricultural Sciences and Uppsala University, Uppsala, Sweden, http://www.cb.uu.se/gustaf/texture/
Lebanon, G.: Metric Learnong for Text Documents. IEEE Trans. on Pattern Analysis and Machine Intelligence 28(4), 497–508 (2006)
Mallat, S.: A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Trans. on Pattern Analysis and Machine Intelligence 11(7), 674–693
Marriot, P., Salmon, M.: Aplication of Differential Geometry to Econometrics. Cambridge University Press (2000)
Moon, Y.I., Rajagopalan, B., Lall, U.: Estimation of mutual information using kernel density estimators. Physical Review E 52(3), 2318–2321
Muñoz, A., Moguerza, J.M.: Estimation of High-Density Regions using One-Class Neighbor Machines. IEEE Trans. on Pattern Analysis and Machine Intelligence 28(3), 476–480
Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis. Springer, New York (2005)
Sriperumbudur, B.K., Fukumizu, K., Gretton, A., Scholkopf, B., Lanckriet, G.R.G.: Non-parametric estimation of integral probability metrics. In: International Symposium on Information Theory (2010)
Strichartz, R.S.: A Guide to Distribution Theory and Fourier Transforms. World Scientific (1994)
Székely, G.J., Rizzo, M.L.: Testing for Equal Distributions in High Dimension. InterStat (2004)
Ullah, A.: Entropy, divergence and distance measures with econometric applications. Journal of Statistical Planning and Inference 49, 137–162 (1996)
Xing, E.P., Ng, A.Y., Jordan, M.I., Russell, S.: Distance Metric Learning, with Application to Clustering with Side-information. In: Advances in Neural Information Processing Systems, pp. 505–512 (2002)
Zolotarev, V.M.: Probability metrics. Teor. Veroyatnost. i Primenen 28(2), 264–287 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Muñoz, A., Martos, G., Arriero, J., Gonzalez, J. (2012). A New Distance for Probability Measures Based on the Estimation of Level Sets. In: Villa, A.E.P., Duch, W., Érdi, P., Masulli, F., Palm, G. (eds) Artificial Neural Networks and Machine Learning – ICANN 2012. ICANN 2012. Lecture Notes in Computer Science, vol 7553. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33266-1_34
Download citation
DOI: https://doi.org/10.1007/978-3-642-33266-1_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33265-4
Online ISBN: 978-3-642-33266-1
eBook Packages: Computer ScienceComputer Science (R0)