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Bivariate density estimation using BV regularisation. (English) Zbl 1445.62073

Summary: The problem of bivariate density estimation is studied with the aim of finding the density function with the smallest number of local extreme values which is adequate with the given data. Adequacy is defined via Kuiper metrics. The concept of the taut-string algorithm which provides adequate approximations with a small number of local extrema is generalised for analysing two- and higher dimensional data, using Delaunay triangulation and diffusion filtering. Results are based on equivalence relations in one dimension between the taut-string algorithm and the method of solving the discrete total variation flow equation. The generalisation and some modifications are developed and the performance for density estimation is shown.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference

Software:

KernSmooth; ftnonpar
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Full Text: DOI

References:

[1] Abraham, C.; Biau, G.; Cadre, B., Simple estimation of the mode of a multivariate density, Canad. J. Statist., 31, 23-34 (2003) · Zbl 1035.62046
[2] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures (2005), Birkhäuser: Birkhäuser Basel · Zbl 1090.35002
[3] Aurenhammer, F., Klein, R., 2000. Handbook of Computational Geometry. Elsevier, Amsterdam, pp. 201-290 (Chapter 5).; Aurenhammer, F., Klein, R., 2000. Handbook of Computational Geometry. Elsevier, Amsterdam, pp. 201-290 (Chapter 5).
[4] Azzalini, A.; Bowman, A. W., A look at some data on the Old Faithful Geyser, Applied Statistics, 39, 357-365 (1990) · Zbl 0707.62186
[5] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Number 5 in North-Holland Mathematics Studies (1973), North-Holland: North-Holland Amsterdam, London
[6] Davies, P., Data features, Statistica Neerlandica, 49, 185-245 (1995) · Zbl 0831.62001
[7] Davies, P. L.; Kovac, A., Local extremes, runs, strings and multiresolution (with discussion), Ann. Statist., 29, 1-65 (2001) · Zbl 1029.62038
[8] Davies, P. L.; Kovac, A., Densities, spectral densities and modality, Ann. Statist., 32, 3, 1093-1136 (2004) · Zbl 1093.62042
[9] Davies, P.L., Gather, U., Weinert, H., 2006. Nonparametric regression as an example of model choice. Technical Report 24/06, Sonderforschungsbereich 475, Fachbereich Statistik, University of Dortmund, Germany.; Davies, P.L., Gather, U., Weinert, H., 2006. Nonparametric regression as an example of model choice. Technical Report 24/06, Sonderforschungsbereich 475, Fachbereich Statistik, University of Dortmund, Germany. · Zbl 1138.62021
[10] Devroye, L., Recursive estimation of the mode of a multivariate density, Canad. J. Statist., 7, 159-167 (1979) · Zbl 0444.60023
[11] Dudley, R., Real Analysis and Probability (1989), Wadsworth and Brooks/Cole: Wadsworth and Brooks/Cole California · Zbl 0686.60001
[12] Eddy, W., Optimum kernel estimators of the mode, Ann. Statist., 8, 870-882 (1980) · Zbl 0438.62027
[13] Edelsbrunner, H., Geometry and Topology for Mesh Generation (2001), Cambridge University Press: Cambridge University Press Cambridge, London · Zbl 0981.65028
[14] Eilers, P.; Marx, B., Flexible smoothing with b-splines and penalties, Statist. Sci., 11, 89-102 (1996) · Zbl 0955.62562
[15] Evans, L. C.; Gariepy, R. F., Measure theory and fine properties of functions. Studies in Advanced Mathematics (1992), CRC Press: CRC Press Boca Raton, FL
[16] Ferraty, F.; Vieu, P., Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics (2006), Springer: Springer New York
[17] Fisher, N. I.; Mammen, E.; Marron, J. S., Testing for multimodality, Comput. Statist. Data Anal., 18, 499-512 (1994) · Zbl 0825.62444
[18] Gasser, T.; Hall, P.; Presnell, B., Nonparametric estimation of the mode of a distribution of random curves, J. Roy. Statist. Soc., 60, 681-691 (1998) · Zbl 0909.62030
[19] Good, I.; Gaskins, R., Density estimating and bump-hunting by the penalizedlikelihood method exemplified by scattering and meteorite data, J. Amer. Statist. Assoc., 75, 42-73 (1980) · Zbl 0432.62024
[20] Grasmair, M., 2006. The equivalence of the taut string algorithm and bv-regularization. J. Math. Imaging Vision.; Grasmair, M., 2006. The equivalence of the taut string algorithm and bv-regularization. J. Math. Imaging Vision.
[21] Hartigan, J. A.; Hartigan, P. M., The dip test of unimodality, Ann. Statist., 13, 70-84 (1985) · Zbl 0575.62045
[22] Hengartner, N.; Stark, P., Finite-sample confidence envelopes for shape-restricted densities, Ann. Statist., 23, 525-550 (1995) · Zbl 0828.62043
[23] Herrmann, E.; Ziegler, K., Rates on consistency for nonparametric estimation of the mode in absence of smoothness assumptions, Statist. Probab. Lett., 68, 359-368 (2004) · Zbl 1086.62046
[24] Hinterberger, W., Tube methods for bv regularization, J. Math. Imaging Vision, 19, 219-235 (2003) · Zbl 1101.68927
[25] Jacob, P.; Oliveira, P., Kernel estimators of general radon-nikodym derivatives, Statistics, 30, 25-46 (1997) · Zbl 0906.62034
[26] Jang, W., Nonparametric density estimation and clustering in astronomical sky surveys, Comput. Statist. Data Anal., 50, 760-774 (2006) · Zbl 1432.62377
[27] Kovac, A., Smooth functions and local extreme values, Comput. Statist. Data Anal., 51, 5156-5171 (2007) · Zbl 1162.62358
[28] Mammen, E., On qualitative smoothness of kernel density estimators, Statistics, 26, 253-267 (1995) · Zbl 0811.62046
[29] Mammen, E.; van de Geer, S., Locally adaptive regression splines, Ann. Statist., 25, 387-413 (1997) · Zbl 0871.62040
[30] Minotte, M. C.; Scott, D. W., The mode tree: a tool for visualization of nonparametric density features, J. Comput. Graphical Statist., 2, 51-68 (1993)
[31] Müller, D. W.; Sawitzki, G., Excess mass estimates and tests for multimodality, J. Amer. Statist. Assoc., 86, 415, 738-746 (1991) · Zbl 0733.62040
[32] Nadaraya, E. A., On estimating regression, Theory Probab. Appl., 10, 186-190 (1964) · Zbl 0134.36302
[33] Park, B.-U.; Turlach, B., Practical performance of several data-driven bandwidth selectors (with discussion), Comput. Statist., 7, 251-285 (1992) · Zbl 0775.62100
[34] Parzen, E., On estimation of a probability density function and mode, Ann. Math. Statist., 33, 1065-1076 (1962) · Zbl 0116.11302
[35] Polonik, W., Density estimation under qualitative assumptions inhigher dimensions, J. Mulitivariate Anal., 55, 61-81 (1995) · Zbl 0847.62027
[36] Polonik, W., Measuring mass concentrations and estimating density contour clusters-an excess mass approach, Ann. Statist, 23, 855-881 (1995) · Zbl 0841.62045
[37] Polonik, W., Concentration and goodness of fit in higherdimensions: (aymptotically) distribution-free methods, Ann. Statist., 27, 1210-1229 (1999) · Zbl 0961.62041
[38] Polzehl, J.; Spokoiny, V., Adaptive weights smoothing with applications to image resoration, J. Roy. Statist. Soc. Ser. B., 62, 335-354 (2000)
[39] Pöschl, C., Scherzer, O., 2006. Characterization of minimizers of convex regularization functionals. Technical Report 42, Institute for Computer Science, University Innsbruck.; Pöschl, C., Scherzer, O., 2006. Characterization of minimizers of convex regularization functionals. Technical Report 42, Institute for Computer Science, University Innsbruck. · Zbl 1163.47004
[40] Richardson, S.; Green, P., On bayesian analysis of mixtures with an unknown number of components (with discussion), J. Roy. Statist. Soc., 59, 731-792 (1997) · Zbl 0891.62020
[41] Rinehart, J. S., Thermal and seismic indications of old faithful geyser’s inner workings, J. Geophys. Res., 74, 566 (1969)
[42] Roeder, K.; Wasserman, L., Practical Bayesian density estimation using mixtures of normals, J. Amer. Statist. Assoc., 92, 439, 894-902 (1997) · Zbl 0889.62021
[43] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268 (1992) · Zbl 0780.49028
[44] Sain, S.; Scott, D., On locally adaptive density estimation, J. Amer. Statist. Assoc., 91, 1525-1534 (1996) · Zbl 0882.62035
[45] Scherzer, O., Taut-string algorithm and regularization programs with g-norm data fit, J. Math. Imaging Vision, 23, 2, 135-143 (2005) · Zbl 1452.62481
[46] Scott, D., Multivariate Density Estimation. Theory, Practice and Visualization (1992), Wiley: Wiley New York · Zbl 0850.62006
[47] Scott, D., Mulitvariate density estimation. Handbook of Statistics (2004), Springer: Springer Berlin, pp. 517-538
[48] Silverman, B. W., Density Estimation for Statistics and Data Analysis (1986), Chapman & Hall: Chapman & Hall London · Zbl 0617.62042
[49] Steidl, G.; Weickert, J.; Brox, J.; Mrazek, P.; Welk, M., On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides, SIAM J. Numer. Anal., 42, 2, 686-713 (2004) · Zbl 1083.94001
[50] Vidakovic, B., Statistical modeling based on wavelets (1999), Wiley: Wiley New York
[51] Vieu, P., A note on density mode estimation, Statist. Probab. Lett., 26, 297-307 (1996) · Zbl 0847.62024
[52] Wand, M. P.; Jones, M. C., Kernel Smoothing (1995), Chapman & Hall: Chapman & Hall London · Zbl 0854.62043
[53] Watson, G. S., Smooth regression analysis, Sankhyā, 26, 101-116 (1964) · Zbl 0138.13906
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