×

zbMATH — the first resource for mathematics

FFT-based fast bandwidth selector for multivariate kernel density estimation. (English) Zbl 06917858
Summary: The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One of such methods is based on the Fast Fourier Transform. However, the currently available implementation works very well only for the univariate KDE and its multivariate extension suffers from a very serious limitation as it can accurately operate only with diagonal bandwidth matrices. A more general solution is presented where the above mentioned limitation is relaxed. Moreover, the presented solution can be easily adopted also for the task of efficient computation of integrated density derivative functionals involving an arbitrary derivative order. Consequently, bandwidth selection for kernel density derivative estimation is also supported. The practical usability of the new solution is demonstrated by comprehensive numerical simulations.

MSC:
62 Statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andrzejewski, W.; Gramacki, A.; Gramacki, J., Graphics processing units in acceleration of bandwidth selection for kernel density estimation, Int. J. Appl. Math. Comput. Sci., 23, 4, 869-885, (2013) · Zbl 1284.93221
[2] Azzalini, A.; Bowman, A. W., A look at some data on the old faithful geyser, Appl. Stat., 39, 357-365, (1990) · Zbl 0707.62186
[3] Bowman, A. W., An alternative method of cross-validation for the smoothing of density estimates, Biometrika, 71, 2, 353-360, (1984)
[4] Chacón, J. E., Data-driven choice of the smoothing parametrization for kernel density estimators, Canad. J. Statist., 37, 249-265, (2009) · Zbl 1176.62028
[5] Chacón, J. E.; Duong, T., Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices, TEST, 19, 375-398, (2010) · Zbl 1203.62054
[6] Chacón, J. E.; Duong, T., Unconstrained pilot selectors for smoothed cross validation, Aust. N. Z. J. Stat., 53, 331-351, (2011) · Zbl 1334.62049
[7] Chacón, J. E.; Duong, T., Data-driven density estimation, with applications to nonparametric clustering and bump hunting, Electron. J. Stat., 7, 499-532, (2013)
[8] Chacón, J. E.; Duong, T., Efficient recursive algorithms for functionals based on higher order derivatives of the multivariate Gaussian density, Stat. Comput., 25, 959-974, (2015) · Zbl 1332.62170
[9] Chiu, S.-T., The effect of discretization error on bandwidth selection for kernel density estimation, Biometrika, 78, 2, 436-441, (1991)
[10] Duong, T., Bandwidth selectors for multivariate kernel density estimation, (2004), University of Western Australia, School of Mathematics and Statistics, (Ph.D. thesis)
[11] Duong, T., 2015. Kernel Smoothing, R package version 1.10.3. URL http://CRAN.R-project.org/package=ks.
[12] Duong, T.; Hazelton, M. L., Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation, J. Multivariate Anal., 93, 417-433, (2005) · Zbl 1066.62059
[13] Duong, T.; Hazelton, M. L., Cross-validation bandwidth matrices for multivariate kernel density estimation, Scand. J. Statist., 32, 485-506, (2005)
[14] Fan, J.; Marron, J. S., Fast implementations of nonparametric curve estimators, J. Comput. Graph. Statist., 3, 35-56, (1994)
[15] González-Manteigaa, W.; Sánchez-Selleroa, C.; Wand, M., Accuracy of binned kernel functional approximations, Comput. Statist. Data Anal., 22, 1-16, (1996) · Zbl 0900.62195
[16] Gramacki, A.; Gramacki, J., FFT-based fast computation of multivariate kernel estimators with unconstrained bandwidth matrices, J. Comput. Graph. Statist., (2016), (in press)
[17] Hall, P.; Marron, J. S.; Park, B. U., Smoothed cross validation, Probab. Theory Related Fields, 92, 1-20, (1992) · Zbl 0742.62042
[18] Henderson, H. V.; Searle, S. R., Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics, Canad. J. Statist., 7, 1, 65-81, (1979) · Zbl 0435.15022
[19] Jones, M. C.; Marron, J. S.; Sheather, S. J., A brief survey of bandwidth selection for density estimation, J. Amer. Statist. Assoc., 91, 433, 401-407, (1996) · Zbl 0873.62040
[20] Jones, M. C.; Marron, J. S.; Sheather, S. J., Progress in data-based bandwidth selection for kernel density estimation, Comput. Statist., 11, 3, 337-381, (1996) · Zbl 0897.62037
[21] Łukasik, S., Parallel computing of kernel density estimates with mpi, Lecture Notes in Comput. Sci., 4489, 726-734, (2007)
[22] Park, B. U.; Marron, J. S., Comparison of data-driven bandwidth selectors, J. Amer. Statist. Assoc., 85, 409, 66-72, (1990)
[23] Raykar, V. C.; Duraiswami, R., Very fast optimal bandwidth selection for univariate kernel density estimation, tech. rep. CS-TR-4774/UMIACS-TR-2005-73, (2006), Dept. of Computer Science, University of Maryland College Park
[24] Raykar, V. C.; Duraiswami, R.; Zhao, L. H., Fast computation of kernel estimators, J. Comput. Graph. Statist., 19, 1, 205-220, (2010)
[25] Rudemo, M., Empirical choice of histograms and kernel density estimators, Scand. J. Statist., 9, 2, 65-78, (1982) · Zbl 0501.62028
[26] Sain, S. R.; Baggerly, K. A.; Scott, D. W., Cross-validation of multivariate densities, J. Amer. Statist. Assoc., 89, 427, 807-817, (1994) · Zbl 0805.62059
[27] Scott, D. W., Multivariate density estimation: theory, practice, and visualization, (1992), John Wiley & Sons, Inc. · Zbl 0850.62006
[28] Scott, D. W.; Terrell, G. R., Biased and unbiased cross-validation in density estimation, J. Amer. Statist. Assoc., 82, 1131-1146, (1987) · Zbl 0648.62037
[29] Sheather, S. J.; Jones, M. C., A reliable data-based bandwidth selection method for kernel density estimation, J. R. Stat. Soc. Ser. B Stat. Methodol., 53, 3, 683-690, (1991) · Zbl 0800.62219
[30] Silverman, B. W., Kernel density estimation using the fast Fourier transform. algorithm AS 176, Appl. Stat., 31, 93-99, (1982) · Zbl 0483.62032
[31] Silverman, B. W., Density estimation for statistics and data analysis, (1986), Chapman & Hall/CRC · Zbl 0617.62042
[32] Wand, M. P., Fast computation of multivariate kernel estimators, J. Comput. Graph. Statist., 3, 4, 433-445, (1994)
[33] Wand, M. P.; Jones, M. C., Multivariate plug-in bandwidth selection, Comput. Statist., 9, 2, 97-116, (1994) · Zbl 0937.62055
[34] Wand, M. P.; Jones, M. C., Kernel smoothing, (1995), Chapman & Hall · Zbl 0854.62043
[35] Żychaluk, K.; Patil, P., A cross-validation method for data with ties in kernel density estimation, Ann. Inst. Statist. Math., 60, 21-44, (2008) · Zbl 1184.62058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.