×
Klemelä, Jussi

Level set tree methods. (English) Zbl 07910826

Wiley Interdiscip. Rev., WIREs Comput. Stat. 10, No. 5, Article ID e1436, 14 p. (2018).

Cited in 1 Document

MSC:

62-08 Computational methods for problems pertaining to statistics

Keywords:

cluster analysis; mode detection; nonparametric function estimation; optimization of functions; visualization of functions
Cite Review PDF
Full Text: DOI

References:

[1] Aaron, C., & Bodart, O. (2016). Local convex hull support and boundary estimation. Journal of Multivariate Analysis, 147, 82-101. · Zbl 1334.62032
[2] Azzallini, A., & Torelli, N. (2007). Clustering via nonparametric density estimation. Statistics and Computing, 17, 71-80.
[3] Baillo, A. (2003). Total error in a plug‐in estimator of level sets. Statistics & Probability Letters, 65, 411-417. · Zbl 1116.62338
[4] Baíllo, A., Cuesta‐Albertos, J. A., & Cuevas, A. (2001). Convergence rates in nonparametric estimation of level sets. Statistics & Probability Letters, 53, 27-35. · Zbl 0980.62022
[5] Baíllo, A., Cuevas, A., & Justel, A. (2000). Set estimation and nonparametric detection. Canadian Journal of Statistics, 28, 765-782. · Zbl 1057.62026
[6] Biau, G., Cadre, B., & Pelletier, B. (2007). A graph‐based estimator of the number of clusters. ESAIM: Probability and Statistics, 11, 272-280. · Zbl 1187.62114
[7] Bugni, F. (2010). Bootstrap inference in partially identified models defined by moment inequalities: Coverage of the identified set. Econometrica, 76, 735-753. · Zbl 1188.62356
[8] Burman, P., & Polonik, W. (2009). Multivariate mode hunting: Data analytic tools with measures of significance. Journal of Multivariate Analysis, 100, 1198-1218. · Zbl 1159.62032
[9] Cadre, B. (2006). Kernel estimation of density level sets. Journal of Multivariate Analysis, 97(4), 999-1023. · Zbl 1085.62039
[10] Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society, 46(2), 255-308. · Zbl 1172.62002
[11] Carr, H., Snoeyink, J., & Axen, U. (2003). Computing contour trees in any dimension. Computational Geometry: Theory and Applications, 24(2), 75-94. · Zbl 1052.68098
[12] Chacón, J. E. (2015). A population background for nonparametric density‐based clustering. Statistical Science, 30(4), 518-532. · Zbl 1426.62181
[13] Chaudhuri, K., & Dasgupta, S. (2010). Rates of convergence for the cluster tree. In J. D.Lafferty (ed.), C. K. I.Williams (ed.), J.Shawe‐Taylor (ed.), R. S.Zemel (ed.), & A.Culotta (ed.) (Eds.), Advances in neural information processing systems 23 (pp. 343-351). Vancouver, BC: Curran Associates.
[14] Chernozhukov, V., Hong, H., & Tamer, E. (2007). Estimation and confidence regions for parameter sets in econometric models. Econometrica, 75, 1243-1284. · Zbl 1133.91032
[15] Cuevas, A., Febreiro, M., & Fraiman, R. (2000). Estimating the number of clusters. Canadian Journal of Statistics, 28, 367-382. · Zbl 0981.62054
[16] Cuevas, A., Febreiro, M., & Fraiman, R. (2001). Cluster analysis: A further approach based on density estimation. Computational Statistics and Data Analysis, 36, 441-459. · Zbl 1053.62537
[17] Cuevas, A., González‐Manteiga, W., & Rodriguez‐Casal, A. (2006). Plug‐in estimation of general level sets. Australian & New Zealand Journal of Statistics, 48, 7-19. · Zbl 1108.62036
[18] Cuevas, A., & Rodriguez‐Casal, A. (2004). On boundary estimation. Advances in Applied Probability, 36(2), 340-354. · Zbl 1045.62019
[19] Desforges, M. J., Jacob, P. J., & Cooper, J. E. (1998). Application of probability density estimation to the detection of abnormal conditions in engineering. Proceedings of the Institute Mechanical Engineers, 212, 687-703.
[20] Devroye, L., & Wise, G. L. (1980). Detection of abnormal behavior via nonparametric estimation of the support. SIAM Journal on Applied Mathematics, 38, 480-488. · Zbl 0479.62028
[21] Duong, T., Cowling, A., Koch, I., & Wand, M. P. (2008). Feature significance for multivariate kernel density estimation. Computational Statistics and Data Analysis, 52(9), 4225-4242. · Zbl 1452.62265
[22] Duong, T., Koch, I., & Wand, M. P. (2009). Highest density difference region estimation with application to flow cytometry data. Biometrical Journal, 51, 504-521. · Zbl 1442.62340
[23] Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2000). Topological persistence and simplification. In Proceedings of 41st annual IEEE symposium on foundations of computer science (pp. 454-463). Washington, DC: IEEE Computer Society.
[24] Fomenko, A. T. (ed.), & Kunii, T. L. (ed.) (Eds.). (1997). Topological modeling for visualization. Berlin: Springer. · Zbl 1120.37300
[25] Fraley, C., & Raftery, A. E. (2002). Model‐based clustering, discriminant analysis and density estimation. Journal of the American Statistical Association, 97, 611-631. · Zbl 1073.62545
[26] Gorban, A. N. (2013). Thermodynamic tree: The space of admissible paths. SIAM Journal on Applied Dynamical Systems, 12(1), 246-278. · Zbl 1283.37009
[27] Guillemin, V., & Pollack, A. (1974). Differential topology. Englewood Cliffs, NJ: Prentice‐Hall. · Zbl 0361.57001
[28] Hartigan, J. A. (1975). Clustering algorithms. New York: Wiley. · Zbl 0372.62040
[29] Hartigan, J. A. (1987). Estimation of a convex density cluster in two dimensions. Journal of the American Statistical Association, 82, 267-270. · Zbl 0607.62045
[30] Holmström, L., Karttunen, K., & Klemelä, J. (2017). Estimation of level set trees using adaptive partitions. Computational Statistics, 32, 1139-1163. · Zbl 1417.62063
[31] Indyk, P. (2004). Nearest neighbors in high‐dimensional spaces. In J. E.Goodman (ed.) & J.O’Rourke (ed.) (Eds.), Handbook of discrete and computational geometry (pp. 877-892). Boca Raton, FL: Chapman & Hall/CRC.
[32] Jang, W. (2006). Nonparametric density estimation and clustering in astronomical sky survey. Computational Statistics and Data Analysis, 50, 760-774. · Zbl 1432.62377
[33] Karttunen, K., Holmström, L., & Klemelä, J. (2014). Level set trees with enhanced marginal density visualization: Application to flow cytometry. Proceedings 5th international conference on information visualization theory and applications. Lisbon, Portugal, 210-217.
[34] Kent, B. P., Rinaldo, A., & Timothy, V. (2013). DeBaCl: A python package for interactive DEnsity‐BAsed CLustering. Technical report. arXiv:1307.8136.
[35] Klemelä, J. (2004a). Complexity penalized support estimation. Journal of Multivariate Analysis, 88, 274-297. · Zbl 1035.62027
[36] Klemelä, J. (2004b). Visualization of multivariate density estimates with level set trees. Journal of Computational and Graphical Statistics, 13(3), 599-620.
[37] Klemelä, J. (2005). Algorithms for the manipulation of level sets of nonparametric density estimates. Computational Statistics, 20, 349-368. · Zbl 1089.62037
[38] Klemelä, J. (2006). Visualization of multivariate density estimates with shape trees. Journal of Computational and Graphical Statistics, 15(2), 372-397.
[39] Klemelä, J. (2009). Smoothing of multivariate data: Density estimation and visualization. New York: Wiley. · Zbl 1218.62027
[40] Korostelev, A. P., & Tsybakov, A. B. (1993). Minimax theory of image reconstruction. In Lecture notes in statistics (Vol. 82). Berlin: Springer. · Zbl 0833.62039
[41] Kpotufe, S., & vonLuxburg, U. (2011). Pruning nearest neighbor cluster trees. In Proceedings of the 28th international conference on machine learning (Vol. 105, pp. 225-232). Madison, WI: International Machine Learning Society.
[42] Kriegel, H.‐P., Kröger, P., Sander, J., & Zimek, A. (2011). Density‐based clustering. WIREs Data Mining and Knowledge Discovery, 1, 231-240.
[43] Kunii, T. L. (ed.), & Shinagawa, Y. (ed.) (Eds.). (1992). Modern geometric computing for visualization. Berlin: Springer. · Zbl 0822.68108
[44] Maier, M., Hein, M., & vonLuxburg, U. (2009). Optimal construction of k‐nearest‐neighbor graphs for identifying noisy clusters. Theoretical Computer Science, 410(19), 1749-1764. · Zbl 1167.68045
[45] Mammen, E., & Polonik, W. (2013). Confidence regions for level sets. Journal of Multivariate Analysis, 122, 202-214. · Zbl 1280.62056
[46] Mammen, E., & Tsybakov, A. B. (1995). Asymptotical minimax recovery of sets with smooth boundaries. Annals of Statistics, 23, 502-524. · Zbl 0834.62038
[47] Mason, D., & Polonik, W. (2009). Asymptotic normality of plug‐in level set estimates. Annals of Applied Probability, 19, 1108-1142. · Zbl 1180.62048
[48] Matsumoto, Y. (2000). An introduction to Morse theory. In Translations of mathematical monographs (Vol. 208). Providence, RI: AMS Originally published 1997 (in Japanese).
[49] McMullen, P. (1970). The maximum numbers of faces of a convex polytope. Mathematika, 17, 179-184. · Zbl 0217.46703
[50] Menardi, G., & Azzalini, A. (2014). An advacement in clustering via nonparametric density estimation. Statistics and Computing, 24, 753-767. · Zbl 1322.62175
[51] Milnor, J. (1963). Morse theory. Princeton, NJ: Princeton University Press. · Zbl 0108.10401
[52] Müller, D. W., & Sawitzki, G. (1991). Excess mass estimates and tests of multimodality. Journal of the American Statistical Association, 86, 738-746. · Zbl 0733.62040
[53] Naumann, U., Luta, G., & Wand, M. P. (2010). The curvHDR method for gating flow cytometry samples. BMC Bioinformatics, 11, 44.
[54] Nocedal, J., & Wright, S. J. (1999). Numerical optimization. Berlin: Springer. · Zbl 0930.65067
[55] Nolan, D. (1991). The excess‐mass ellipsoid. Journal of Multivariate Analysis, 39, 348-371. · Zbl 0739.62042
[56] Ooi, H. (2002). Density visualization and mode hunting using trees. Journal of Computational and Graphical Statistics, 11, 328-347.
[57] Osher, S. J., & Fedkiw, R. P. (2002). Level set methods and dynamic implicit surfaces. Berlin: Springer.
[58] Polonik, W. (1995). Measuring mass concentration and estimating density contour clusters—An excess mass approach. Annals of Statistics, 23, 855-881. · Zbl 0841.62045
[59] Reeb, G. (1946). Sur les points singuliers d’une forme de pfaff completement integrable ou d’une fonction numerique. Comptes rendus de l’Académie des Sciences, 222, 847-849. · Zbl 0063.06453
[60] Rigollet, P., & Vert, R. (2009). Optimal rates for plug‐in estimators of density level sets. Bernoulli, 15, 1154-1178. · Zbl 1200.62034
[61] Rinaldo, A., & Wasserman, L. (2010). Generalized density clustering. Annals of Statistics, 38, 2678-2722. · Zbl 1200.62066
[62] Sethian, J. A. (1996). Level set methods: Evolving interfaces in geometry, fluid mechanics, computer vision, and materials science. Cambridge: Cambridge University Press. · Zbl 0859.76004
[63] Singh, G., Memoli, F., & Carlsson, G. (2007). Topological methods for the analysis of high dimensional data sets and 3D object recognition. In M.Botsch (ed.), R.Pajarola (ed.), B.Chen (ed.), & M.Zwicker (ed.) (Eds.), Symposium on point based graphics (pp. 91-100). Prague: Eurographics Association.
[64] Singh, A., Scott, C., & Nowak, R. (2009). Adaptive Hausdorff estimation of density level sets. Annals of Statistics, 37, 2760-2782. · Zbl 1173.62019
[65] Steinwart, I. (2015). Fully adaptive density‐based clustering. Annals of Statistics, 43, 2132-2167. · Zbl 1327.62382
[66] Stuetzle, W. (2003). Estimating the cluster tree of a density by analyzing the minimal spanning tree of a sample. Journal of Classification, 20(5), 25-47. · Zbl 1055.62075
[67] Stuetzle, W., & Nugent, R. (2010). A generalized single linkage method for estimating the cluster tree of a density. Journal of Computational and Graphical Statistics, 19, 397-418.
[68] Tarjan, R. E. (1972). Depth‐first search and linear graph algorithms. SIAM Journal on Computing, 1(2), 146-160. · Zbl 0251.05107
[69] Tarjan, R. E. (1976). Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22, 215-225. · Zbl 0307.68029
[70] Tsybakov, A. B. (1997). On nonparametric estimation of density level sets. Annals of Statistics, 25, 948-969. · Zbl 0881.62039
[71] Walther, G. (1997). Granulometric smoothing. Annals of Statistics, 25, 2273-2299. · Zbl 0919.62026
[72] Walther, G., Zimmerman, N., Moore, W., Parks, D., Meehan, S., Belitskaya, I., … Herzenberg, L. (2009). Automatic clustering of flow cytometry data with density‐based merging. Advances in Bioinformatics, 2009, 686-759.
[73] Weber, G., Bremer, P.‐T., & Pascucci, V. (2007). Topological landscapes: A terrain metaphor for scientific data. IEEE Transactions on Visualization Computer Graphics, 13(6), 1416-1423.
[74] Willett, R. M., & Nowak, R. D. (2005). Level set estimation in medical imaging. Proceedings of IEEE Statistical Signal Processing, 5, 1089-1092.
[75] Wishart, D. (1969). Mode analysis: A generalization of nearest neighbor which reduces chaining effects. In A. J.Cole (ed.) (Ed.), Numerical taxonomy (pp. 282-311). New York: Academic Press.
[76] Zhou, Q., & Wong, W. H. (2008). Bayesian inference of DNA sequence segmentation. Annals of Applied Statistics, 2(4), 1307-1331. · Zbl 1169.62008
[77] Zomorodian, A. (2012). Topological data analysis. In A.Zomorodian (ed.) (Ed.), Advances in applied and computational topology (Vol. 70, pp. 1-40). Providence: American Mathematical Society. · Zbl 06082690
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.