The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.
Statist. Sci.
31(1):
61-79
(February 2016).
DOI: 10.1214/15-STS532
Barnett, V. (1976). The ordering of multivariate data. J. Roy. Statist. Soc. Ser. A 139 318–355. MR445726 10.2307/2344839Barnett, V. (1976). The ordering of multivariate data. J. Roy. Statist. Soc. Ser. A 139 318–355. MR445726 10.2307/2344839
Beale, N., Rand, D., Battey, H., Croxson, K., Nowak, M. A. and May, R. M. (2011). Individual versus systemic risk and the Regulator’s Dilemma. Proc. Natl. Acad. Sci. 108 12647–12652.Beale, N., Rand, D., Battey, H., Croxson, K., Nowak, M. A. and May, R. M. (2011). Individual versus systemic risk and the Regulator’s Dilemma. Proc. Natl. Acad. Sci. 108 12647–12652.
Chakraborty, A. and Chaudhuri, P. (2014a). The spatial distribution in infinite dimensional spaces and related quantiles and depths. Ann. Statist. 42 1203–1231. MR3224286 10.1214/14-AOS1226 euclid.aos/1403276912
Chakraborty, A. and Chaudhuri, P. (2014a). The spatial distribution in infinite dimensional spaces and related quantiles and depths. Ann. Statist. 42 1203–1231. MR3224286 10.1214/14-AOS1226 euclid.aos/1403276912
Chakraborty, A. and Chaudhuri, P. (2014b). On data depth in infinite dimensional spaces. Ann. Inst. Statist. Math. 66 303–324. MR3171407 10.1007/s10463-013-0416-yChakraborty, A. and Chaudhuri, P. (2014b). On data depth in infinite dimensional spaces. Ann. Inst. Statist. Math. 66 303–324. MR3171407 10.1007/s10463-013-0416-y
Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862–872. MR1395753 10.1080/01621459.1996.10476954Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862–872. MR1395753 10.1080/01621459.1996.10476954
Chen, D. and Müller, H.-G. (2012). Nonlinear manifold representations for functional data. Ann. Statist. 40 1–29. MR3013177 10.1214/11-AOS936 euclid.aos/1331830772
Chen, D. and Müller, H.-G. (2012). Nonlinear manifold representations for functional data. Ann. Statist. 40 1–29. MR3013177 10.1214/11-AOS936 euclid.aos/1331830772
Chiou, J.-M. and Müller, H.-G. (2014). Linear manifold modelling of multivariate functional data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 605–626. MR3210730 10.1111/rssb.12038Chiou, J.-M. and Müller, H.-G. (2014). Linear manifold modelling of multivariate functional data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 605–626. MR3210730 10.1111/rssb.12038
Claeskens, G., Hubert, M., Slaets, L. and Vakili, K. (2014). Multivariate functional halfspace depth. J. Amer. Statist. Assoc. 109 411–423. MR3180573 10.1080/01621459.2013.856795Claeskens, G., Hubert, M., Slaets, L. and Vakili, K. (2014). Multivariate functional halfspace depth. J. Amer. Statist. Assoc. 109 411–423. MR3180573 10.1080/01621459.2013.856795
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. MR2336349 10.1007/s00180-007-0053-0Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Comput. Statist. 22 481–496. MR2336349 10.1007/s00180-007-0053-0
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. MR2478196 10.1016/j.jmva.2008.08.002Cuevas, A. and Fraiman, R. (2009). On depth measures and dual statistics. A methodology for dealing with general data. J. Multivariate Anal. 100 753–766. MR2478196 10.1016/j.jmva.2008.08.002
Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Applications of Mathematics (New York) 31. Springer, New York. MR1383093Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Applications of Mathematics (New York) 31. Springer, New York. MR1383093
Dutta, S., Ghosh, A. K. and Chaudhuri, P. (2011). Some intriguing properties of Tukey’s half-space depth. Bernoulli 17 1420–1434. MR2854779 10.3150/10-BEJ322 euclid.bj/1320417511
Dutta, S., Ghosh, A. K. and Chaudhuri, P. (2011). Some intriguing properties of Tukey’s half-space depth. Bernoulli 17 1420–1434. MR2854779 10.3150/10-BEJ322 euclid.bj/1320417511
Fraiman, R. and Muniz, G. (2001). Trimmed means for functional data. TEST 10 419–440. MR1881149 10.1007/BF02595706Fraiman, R. and Muniz, G. (2001). Trimmed means for functional data. TEST 10 419–440. MR1881149 10.1007/BF02595706
Genton, M. G. and Hall, P. (2015). A tilting approach to ranking influence. J. R. Stat. Soc. Ser. B. Stat. Methodol. DOI:10.1111/rssb.12102. To appear.Genton, M. G. and Hall, P. (2015). A tilting approach to ranking influence. J. R. Stat. Soc. Ser. B. Stat. Methodol. DOI:10.1111/rssb.12102. To appear.
Hampel, F. R. (1971). A general qualitative definition of robustness. Ann. Math. Statist. 42 1887–1896. MR301858 10.1214/aoms/1177693054 euclid.aoms/1177693054
Hampel, F. R. (1971). A general qualitative definition of robustness. Ann. Math. Statist. 42 1887–1896. MR301858 10.1214/aoms/1177693054 euclid.aoms/1177693054
Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York. MR829458Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York. MR829458
Hlubinka, D., Gijbels, I., Omelka, M. and Nagy, S. (2015). Integrated data depth for smooth functions and its application in supervised classification. Comput. Statist. DOI:10.1007/s00180-015-0566-x. To appear.Hlubinka, D., Gijbels, I., Omelka, M. and Nagy, S. (2015). Integrated data depth for smooth functions and its application in supervised classification. Comput. Statist. DOI:10.1007/s00180-015-0566-x. To appear.
Huber, P. J. (1972). The 1972 Wald lecture. Robust statistics: A review. Ann. Math. Statist. 43 1041–1067. MR314180 10.1214/aoms/1177692459 euclid.aoms/1177692459
Huber, P. J. (1972). The 1972 Wald lecture. Robust statistics: A review. Ann. Math. Statist. 43 1041–1067. MR314180 10.1214/aoms/1177692459 euclid.aoms/1177692459
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. MR2980081 10.1080/01621459.2012.688462Li, 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. MR2980081 10.1080/01621459.2012.688462
Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405–414. MR1041400 10.1214/aos/1176347507 euclid.aos/1176347507
Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405–414. MR1041400 10.1214/aos/1176347507 euclid.aos/1176347507
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. MR1724033 euclid.aos/1018031260
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. MR1724033 euclid.aos/1018031260
López-Pintado, S. and Jornsten, R. (2007). Functional analysis via extensions of the band depth. In Complex Datasets and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 54 103–120. IMS, Beachwood, OH. MR2459182 10.1214/074921707000000085López-Pintado, S. and Jornsten, R. (2007). Functional analysis via extensions of the band depth. In Complex Datasets and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 54 103–120. IMS, Beachwood, OH. MR2459182 10.1214/074921707000000085
López-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. J. Amer. Statist. Assoc. 104 718–734. MR2541590 10.1198/jasa.2009.0108López-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. J. Amer. Statist. Assoc. 104 718–734. MR2541590 10.1198/jasa.2009.0108
Mosler, K. (2013). Depth statistics. In Robustness and Complex Data Structures 17–34. Springer, Heidelberg. MR3135871 10.1007/978-3-642-35494-6_2Mosler, K. (2013). Depth statistics. In Robustness and Complex Data Structures 17–34. Springer, Heidelberg. MR3135871 10.1007/978-3-642-35494-6_2
Nieto-Reyes, A. (2011). On the properties of functional depth. In Recent Advances in Functional Data Analysis and Related Topics (F. Ferraty, ed.). Contrib. Statist. 239–244. Physica-Verlag/Springer, Heidelberg. MR2815588 10.1007/978-3-7908-2736-1_37Nieto-Reyes, A. (2011). On the properties of functional depth. In Recent Advances in Functional Data Analysis and Related Topics (F. Ferraty, ed.). Contrib. Statist. 239–244. Physica-Verlag/Springer, Heidelberg. MR2815588 10.1007/978-3-7908-2736-1_37
Paindaveine, D. and Van Bever, G. (2013). From depth to local depth: A focus on centrality. J. Amer. Statist. Assoc. 108 1105–1119. MR3174687 10.1080/01621459.2013.813390Paindaveine, D. and Van Bever, G. (2013). From depth to local depth: A focus on centrality. J. Amer. Statist. Assoc. 108 1105–1119. MR3174687 10.1080/01621459.2013.813390
Paindaveine, D. and Van Bever, G. (2015). Nonparametrically consistent depth-based classifiers. Bernoulli 21 62–82. MR3322313 10.3150/13-BEJ561 euclid.bj/1426597064
Paindaveine, D. and Van Bever, G. (2015). Nonparametrically consistent depth-based classifiers. Bernoulli 21 62–82. MR3322313 10.3150/13-BEJ561 euclid.bj/1426597064
Rice, S. O. (1945). Mathematical analysis of random noise. Bell Syst. Tech. J. 24 46–156. MR11918 10.1002/j.1538-7305.1945.tb00453.xRice, S. O. (1945). Mathematical analysis of random noise. Bell Syst. Tech. J. 24 46–156. MR11918 10.1002/j.1538-7305.1945.tb00453.x
Serfling, R. (2002). A depth function and a scale curve based on spatial quantiles. In Statistical Data Analysis Based on the $L_{1}$-Norm and Related Methods (Neuchâtel, 2002). Stat. Ind. Technol. 25–38. Birkhäuser, Basel. MR2001302 10.1007/978-3-0348-8201-9_3Serfling, R. (2002). A depth function and a scale curve based on spatial quantiles. In Statistical Data Analysis Based on the $L_{1}$-Norm and Related Methods (Neuchâtel, 2002). Stat. Ind. Technol. 25–38. Birkhäuser, Basel. MR2001302 10.1007/978-3-0348-8201-9_3
Serfling, R. (2006). Depth functions in nonparametric multivariate inference. In Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 72 1–16. Amer. Math. Soc., Providence, RI. MR2343109Serfling, R. (2006). Depth functions in nonparametric multivariate inference. In Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 72 1–16. Amer. Math. Soc., Providence, RI. MR2343109
Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. MPS/SIAM Series on Optimization 9. SIAM, Philadelphia, PA. MR2562798Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. MPS/SIAM Series on Optimization 9. SIAM, Philadelphia, PA. MR2562798
Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1–24. MR1389877 10.1214/aos/1033066196 euclid.aos/1033066196
Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1–24. MR1389877 10.1214/aos/1033066196 euclid.aos/1033066196
Swierenga, H., de Weijer, A. P., van Wijk, R. J. and Buydens, L. M. C. (1999). Strategy for constructing robust multivariate calibration models. Chemom. Intell. Lab. Syst. 49 1–17.Swierenga, H., de Weijer, A. P., van Wijk, R. J. and Buydens, L. M. C. (1999). Strategy for constructing robust multivariate calibration models. Chemom. Intell. Lab. Syst. 49 1–17.
Tukey, J. W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 2 523–531. Canad. Math. Congress, Montreal, QC. MR426989Tukey, J. W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 2 523–531. Canad. Math. Congress, Montreal, QC. MR426989
Vardi, Y. and Zhang, C.-H. (2000). The multivariate $L_{1}$-median and associated data depth. Proc. Natl. Acad. Sci. USA 97 1423–1426 (electronic). MR1740461 10.1073/pnas.97.4.1423Vardi, Y. and Zhang, C.-H. (2000). The multivariate $L_{1}$-median and associated data depth. Proc. Natl. Acad. Sci. USA 97 1423–1426 (electronic). MR1740461 10.1073/pnas.97.4.1423
Zuo, Y. and Serfling, R. (2000a). On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. J. Statist. Plann. Inference 84 55–79. MR1747497 10.1016/S0378-3758(99)00142-1Zuo, Y. and Serfling, R. (2000a). On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. J. Statist. Plann. Inference 84 55–79. MR1747497 10.1016/S0378-3758(99)00142-1
Zuo, Y. and Serfling, R. (2000b). General notions of statistical depth function. Ann. Statist. 28 461–482. MR1790005 10.1214/aos/1016218226 euclid.aos/1016218226
Zuo, Y. and Serfling, R. (2000b). General notions of statistical depth function. Ann. Statist. 28 461–482. MR1790005 10.1214/aos/1016218226 euclid.aos/1016218226