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An Improved Variable Kernel Density Estimator Based on L 2 Regularization

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  • Yi Jin

    (Department of Trace Inspection Technology, Criminal Investigation Police University of China, Shenyang 110854, China
    Key Laboratory of Impression Evidence Examination and Identification Technology, The Ministry of Public Security of the People’s Republic of China, Shenyang 110854, China)

  • Yulin He

    (Big Data Institute, College of Computer Science & Software Engineering, Shenzhen University, Shenzhen 518060, China
    National Engineering Laboratory for Big Data System Computing Technology, Shenzhen University, Shenzhen 518060, China)

  • Defa Huang

    (Big Data Institute, College of Computer Science & Software Engineering, Shenzhen University, Shenzhen 518060, China)

Abstract

The nature of the kernel density estimator (KDE) is to find the underlying probability density function ( p.d.f ) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share a fixed bandwidth (scalar for univariate KDE and vector for multivariate KDE) in the fixed KDE (FKDE). In this paper, we propose an improved variable KDE (IVKDE) which determines the optimal bandwidth for each data point in the given dataset based on the integrated squared error (ISE) criterion with the L 2 regularization term. An effective optimization algorithm is developed to solve the improved objective function. We compare the estimation performance of IVKDE with FKDE and VKDE based on ISE criterion without L 2 regularization on four univariate and four multivariate probability distributions. The experimental results show that IVKDE obtains lower estimation errors and thus demonstrate the effectiveness of IVKDE.

Suggested Citation

  • Yi Jin & Yulin He & Defa Huang, 2021. "An Improved Variable Kernel Density Estimator Based on L 2 Regularization," Mathematics, MDPI, vol. 9(16), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:2004-:d:618876
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    References listed on IDEAS

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    1. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2001. "Cluster analysis: a further approach based on density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 36(4), pages 441-459, June.
    2. Azzalini, Adelchi & Menardi, Giovanna, 2014. "Clustering via Nonparametric Density Estimation: The R Package pdfCluster," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 57(i11).
    3. Wu, Tiee-Jian & Chen, Ching-Fu & Chen, Huang-Yu, 2007. "A variable bandwidth selector in multivariate kernel density estimation," Statistics & Probability Letters, Elsevier, vol. 77(4), pages 462-467, February.
    4. Su Chen, 2015. "Optimal Bandwidth Selection for Kernel Density Functionals Estimation," Journal of Probability and Statistics, Hindawi, vol. 2015, pages 1-21, August.
    5. Jones, M. C., 1991. "The roles of ISE and MISE in density estimation," Statistics & Probability Letters, Elsevier, vol. 12(1), pages 51-56, July.
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    Cited by:

    1. Jenny Farmer & Eve Allen & Donald J. Jacobs, 2022. "Quasar Identification Using Multivariate Probability Density Estimated from Nonparametric Conditional Probabilities," Mathematics, MDPI, vol. 11(1), pages 1-19, December.

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