[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 |