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Flexible integrated functional depths. (English) Zbl 1512.62103

Summary: This paper develops a new class of functional depths. A generic member of this class is coined \(J\)th order \(k\)th moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show that they are uniformly consistent and satisfy an inequality related to the law of the iterated logarithm. Moreover, limiting distributions are derived under mild regularity assumptions. The versatility displayed by the new class of depths makes them particularly amenable for capturing important features of functional distributions. This is illustrated in supervised learning, where we show that the corresponding maximum depth classifiers outperform classical competitors.

MSC:

62R10 Functional data analysis
62H30 Classification and discrimination; cluster analysis (statistical aspects)
68T05 Learning and adaptive systems in artificial intelligence

Software:

ddalpha; fda (R)
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Full Text: DOI Euclid

References:

[1] Biau, G., Bunea, F. and Wegkamp, M.H. (2005). Functional classification in Hilbert spaces. IEEE Trans. Inf. Theory 51 2163-2172. Zentralblatt MATH: 1285.94015
Digital Object Identifier: doi:10.1109/TIT.2005.847705
· Zbl 1285.94015
[2] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. Zentralblatt MATH: 0822.60002
· Zbl 0822.60002
[3] Borggaard, C. and Thodberg, H.H. (1992). Optimal minimal neural interpretation of spectra. Anal. Chem. 64 545-551.
[4] Cérou, F. and Guyader, A. (2006). Nearest neighbor classification in infinite dimension. ESAIM Probab. Stat. 10 340-355. Zentralblatt MATH: 1187.62115
Digital Object Identifier: doi:10.1051/ps:2006014
· Zbl 1187.62115
[5] Chakraborty, A. and Chaudhuri, P. (2014). The spatial distribution in infinite dimensional spaces and related quantiles and depths. Ann. Statist. 42 1203-1231. Zentralblatt MATH: 1305.62141
Digital Object Identifier: doi:10.1214/14-AOS1226
Project Euclid: euclid.aos/1403276912
· Zbl 1305.62141
[6] Chernozhukov, V., Galichon, A., Hallin, M. and Henry, M. (2017). Monge-Kantorovich depth, quantiles, ranks and signs. Ann. Statist. 45 223-256. Zentralblatt MATH: 1426.62163
Digital Object Identifier: doi:10.1214/16-AOS1450
Project Euclid: euclid.aos/1487667622
· Zbl 1426.62163
[7] Claeskens, G., Hubert, M., Slaets, L. and Vakili, K. (2014). Multivariate functional halfspace depth. J. Amer. Statist. Assoc. 109 411-423. Zentralblatt MATH: 1367.62162
Digital Object Identifier: doi:10.1080/01621459.2013.856795
· Zbl 1367.62162
[8] Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Comput. Statist. 22 481-496. Zentralblatt MATH: 1195.62032
Digital Object Identifier: doi:10.1007/s00180-007-0053-0
· Zbl 1195.62032
[9] Cuevas, A. and Fraiman, R. (2009). On depth measures and dual statistics. A methodology for dealing with general data. J. Multivariate Anal. 100 753-766. Zentralblatt MATH: 1163.62039
Digital Object Identifier: doi:10.1016/j.jmva.2008.08.002
· Zbl 1163.62039
[10] Delaigle, A. and Hall, P. (2012). Achieving near perfect classification for functional data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 74 267-286. Zentralblatt MATH: 1411.62164
Digital Object Identifier: doi:10.1111/j.1467-9868.2011.01003.x
· Zbl 1411.62164
[11] Donoho, D.L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803-1827. Zentralblatt MATH: 0776.62031
Digital Object Identifier: doi:10.1214/aos/1176348890
Project Euclid: euclid.aos/1176348890
· Zbl 0776.62031
[12] Dudley, R.M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge: Cambridge Univ. Press. Zentralblatt MATH: 0951.60033
· Zbl 0951.60033
[13] Dudley, R.M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge: Cambridge Univ. Press.
[14] Ferraty, F. and Vieu, P. (2003). Curves discrimination: A nonparametric functional approach 44 161-173. Zentralblatt MATH: 1429.62241
Digital Object Identifier: doi:10.1016/S0167-9473(03)00032-X
· Zbl 1429.62241
[15] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. New York: Springer. Zentralblatt MATH: 1119.62046
· Zbl 1119.62046
[16] Fraiman, R. and Muniz, G. (2001). Trimmed means for functional data. TEST 10 419-440. Zentralblatt MATH: 1016.62026
Digital Object Identifier: doi:10.1007/BF02595706
· Zbl 1016.62026
[17] Ghosh, A.K. and Chaudhuri, P. (2005). On maximum depth and related classifiers. Scand. J. Stat. 32 327-350. Zentralblatt MATH: 1089.62075
Digital Object Identifier: doi:10.1111/j.1467-9469.2005.00423.x
· Zbl 1089.62075
[18] Gijbels, I. and Nagy, S. (2015). Consistency of non-integrated depths for functional data. J. Multivariate Anal. 140 259-282. Zentralblatt MATH: 1327.62305
Digital Object Identifier: doi:10.1016/j.jmva.2015.05.012
· Zbl 1327.62305
[19] Grinblat, L.Š. (1976). A limit theorem for measurable random processes and its applications. Proc. Amer. Math. Soc. 61 371-376. Zentralblatt MATH: 0379.60009
Digital Object Identifier: doi:10.1090/S0002-9939-1976-0423450-2
· Zbl 0379.60009
[20] Hall, P., Poskitt, D.S. and Presnell, B. (2001). A functional data-analytic approach to signal discrimination. Technometrics 43 1-9. Zentralblatt MATH: 1072.62686
Digital Object Identifier: doi:10.1198/00401700152404273
· Zbl 1072.62686
[21] Hallin, M., Paindaveine, D. and Šiman, M. (2010). Multivariate quantiles and multiple-output regression quantiles: From \(L_1\) optimization to halfspace depth. Ann. Statist. 38 635-669. Zentralblatt MATH: 1183.62088
Digital Object Identifier: doi:10.1214/09-AOS723
Project Euclid: euclid.aos/1266586607
· Zbl 1183.62088
[22] Kuelbs, J. and Dudley, R.M. (1980). Log log laws for empirical measures. Ann. Probab. 8 405-418. Zentralblatt MATH: 0442.60031
Digital Object Identifier: doi:10.1214/aop/1176994716
Project Euclid: euclid.aop/1176994716
· Zbl 0442.60031
[23] Kuelbs, J. and Zinn, J. (2013). Concerns with functional depth. ALEA Lat. Am. J. Probab. Math. Stat. 10 831-855. Zentralblatt MATH: 1277.60049
· Zbl 1277.60049
[24] Li, B., Van Bever, G., Oja, H., Sabolova, R. and Critchley, F. (2019). Functional independent component analysis: An extension of fourth order blind identification. Technical Report, Univ. Namur.
[25] Li, J., Cuesta-Albertos, J.A. and Liu, R.Y. (2012). \(DD\)-classifier: Nonparametric classification procedure based on \(DD\)-plot. J. Amer. Statist. Assoc. 107 737-753. Zentralblatt MATH: 1261.62058
Digital Object Identifier: doi:10.1080/01621459.2012.688462
· Zbl 1261.62058
[26] Liu, R.Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414. Zentralblatt MATH: 0701.62063
Digital Object Identifier: doi:10.1214/aos/1176347507
Project Euclid: euclid.aos/1176347507
· Zbl 0701.62063
[27] Liu, R.Y., Parelius, J.M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference. Ann. Statist. 27 783-858. Zentralblatt MATH: 0984.62037
Project Euclid: euclid.aos/1018031260
· Zbl 0984.62037
[28] Liu, X., Mosler, K. and Mozharovskyi, P. (2019). Fast computation of Tukey trimmed regions and median in dimension \(p>2\). J. Comput. Graph. Statist. 28 682-697.
[29] López-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. J. Amer. Statist. Assoc. 104 718-734. Zentralblatt MATH: 1388.62139
Digital Object Identifier: doi:10.1198/jasa.2009.0108
· Zbl 1388.62139
[30] López-Pintado, S. and Romo, J. (2011). A half-region depth for functional data. Comput. Statist. Data Anal. 55 1679-1695. Zentralblatt MATH: 1328.62029
Digital Object Identifier: doi:10.1016/j.csda.2010.10.024
· Zbl 1328.62029
[31] Magnano, L., Boland, J.W. and Hyndman, R.J. (2008). Generation of synthetic sequences of half-hourly temperature. Environmetrics 19 818-835.
[32] Massart, P. (1986). Rates of convergence in the central limit theorem for empirical processes. Ann. Inst. Henri Poincaré Probab. Stat. 22 381-423. Zentralblatt MATH: 0615.60032
· Zbl 0615.60032
[33] Massé, J.-C. (2004). Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean. Bernoulli 10 397-419. · Zbl 1053.62021
[34] Mosler, K. (2013). Depth statistics. In Robustness and Complex Data Structures (C. Becker, R. Fried and S. Kuhnt, eds.) 17-34. Heidelberg: Springer.
[35] Mosler, K. and Polyakova, Y. (2016). General notions of depth for functional data. Preprint. Available at arXiv:1208.1981. arXiv: 1208.1981
[36] Nagy, S. (2017). Integrated depth for measurable functions and sets. Statist. Probab. Lett. 123 165-170. Zentralblatt MATH: 1360.62236
Digital Object Identifier: doi:10.1016/j.spl.2016.12.012
· Zbl 1360.62236
[37] Nagy, S. and Ferraty, F. (2019). Data depth for measurable noisy random functions. J. Multivariate Anal. 170 95-114. Zentralblatt MATH: 1415.62028
Digital Object Identifier: doi:10.1016/j.jmva.2018.11.003
· Zbl 1415.62028
[38] Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. J. Comput. Graph. Statist. 26 883-893.
[39] Nagy, S., Gijbels, I., Omelka, M. and Hlubinka, D. (2016). Integrated depth for functional data: Statistical properties and consistency. ESAIM Probab. Stat. 20 95-130. Zentralblatt MATH: 1357.62201
Digital Object Identifier: doi:10.1051/ps/2016005
· Zbl 1357.62201
[40] Nagy, S., Helander, S., Van Bever, G., Viitasaari, L. and Ilmonen, P. (2020). Supplement to “Flexible integrated functional depths.” https://doi.org/10.3150/20-BEJ1254SUPP
[41] Narisetty, N.N. and Nair, V.N. (2016). Extremal depth for functional data and applications. J. Amer. Statist. Assoc. 111 1705-1714.
[42] Pokotylo, O., Mozharovskyi, P. and Dyckerhoff, R. (2019). Depth and depth-based classification with R package ddalpha. J. Stat. Softw. 91 1-46.
[43] Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. New York: Springer. Zentralblatt MATH: 1079.62006
· Zbl 1079.62006
[44] Sguera, C., Galeano, P. and Lillo, R. (2014). Spatial depth-based classification for functional data. TEST 23 725-750. Zentralblatt MATH: 1312.62083
Digital Object Identifier: doi:10.1007/s11749-014-0379-1
· Zbl 1312.62083
[45] Shao, W. and Zuo, Y. (2020). Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm. Comput. Statist. 35 203-226. Zentralblatt MATH: 07206119
Digital Object Identifier: doi:10.1007/s00180-019-00906-x
· Zbl 1505.62372
[46] Tukey, J.W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2 523-531.
[47] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. New York: Springer. Zentralblatt MATH: 0862.60002
· Zbl 0862.60002
[48] Zuo, Y. · Zbl 1106.62334
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