Abstract
We present a method to estimate the probability density function of multivariate distributions. Standard Parzen window approaches use the sample mean and the sample covariance matrix around every input vector. This choice yields poor robustness for real input datasets. We propose to use the L1-median to estimate the local mean and covariance matrix with a low sensitivity to outliers. In addition to this, a smoothing phase is considered, which improves the estimation by integrating the information from several local clusters. Hence, a specific mixture component is learned for each local cluster. This leads to outperform other proposals where the local kernel is not as robust and/or there are no smoothing strategies, like the manifold Parzen windows.
This is a preview of subscription content, log in via an institution to check access.
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
Bezdek, J.C.: Numerical taxonomy with fuzzysets. J. Math. Biol. 1, 57–71 (1974)
Bishop, C.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)
Dang, X., Serfling, R., Zhou, W.: Influence functions of some depth functions and application to depth-weighted L-statistics. Journal of Nonparametric Statistics 21(1), 49–66 (2009)
Gervini, D.: Robust functional estimation using the median and spherical principal components. Biometrika 95(3), 587–600 (2008)
Hobza, T., Pardo, L., Vajda, I.: Robust median estimator in logistic regression. Journal of Statistical Planning and Inference 138(12), 3822–3840 (2008)
Hjort, N.L., Jones, M.C.: Locally Parametric Nonparametric Density Estimation. Annals of Statistics 24(4), 1619–1647 (1996)
Hössjer, O., Croux, C.: Generalizing Univariate Signed Rank Statistics for Testing and Estimating a Multivariate Location Parameter. Non-parametric Statistics 4, 293–308 (1995)
Izenman, A.J.: Recent developments in nonparametric density estimation. Journal of the American Statistical Association 86(413), 205–224 (1991)
Lejeune, M., Sarda, P.: Smooth estimators of distribution and density functions. Computational Statistics & Data Analysis 14, 457–471 (1992)
López-Rubio, E., Ortiz-de-Lazcano-Lobato, J.M.: Soft clustering for nonparametric probability density function estimation. Pattern Recognition Letters 29, 2085–2091 (2008)
Newman, D.J., Hettich, S., Blake, C.L., Merz, C.J.: UCI Repository of machine learning databases. University of California, Department of Information and Computer Science, Irvine, CA (1998), http://www.ics.uci.edu/~mlearn/MLRepository.html
Parzen, E.: On the Estimation of a Probability Density Function and Mode. Annals of Mathematical Statistics 33, 1065–1076 (1962)
Tipping, M.E., Bishop, C.M.: Mixtures of Probabilistic Principal Components Analyzers. Neural Computation 11, 443–482 (1999)
Vapnik, V.N.: Statistical Learning Theory. John Wiley & Sons, New York (1998)
Vincent, P., Bengio, Y.: Manifold Parzen Windows. Advances in Neural Information Processing Systems 15, 825–832 (2003)
Zaman, A., Rousseeuw, P.J., Orhan, M.: Econometric applications of high-breakdown robust regression techniques. Economics Letters 71, 1–8 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
López-Rubio, E., Ortiz-de-Lazcano-Lobato, J.M., Vargas-González, M.C. (2009). Nonparametric Location Estimation for Probability Density Function Learning. In: Cabestany, J., Sandoval, F., Prieto, A., Corchado, J.M. (eds) Bio-Inspired Systems: Computational and Ambient Intelligence. IWANN 2009. Lecture Notes in Computer Science, vol 5517. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02478-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-02478-8_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02477-1
Online ISBN: 978-3-642-02478-8
eBook Packages: Computer ScienceComputer Science (R0)