Robust Product Functional Principal Component Analysis

Abstract

We introduce a novel approach for modeling two-way functional trajectories, relying on Kendall’s tau representation of marginal covariance functions and utilizing the concept of product functional principal component analysis. The developed estimation procedure is intuitive and straightforward to implement. Theoretical results supporting its validity are also established. Numerical simulation studies validate its superior performance compared to recently developed methods. Furthermore, the application of this approach to analyze two-way air pollution trajectories demonstrates its practical superiority.

Keywords

Functional data analysis / Kendall’s tau function / Product functional principal component analysis / Robustness / Two-way functional data / 62R10 / 62H25 / 62G35

References

[1] Aston JA, Pigoli D, Tavakoli S. Tests for separability in nonparametric covariance operators of random surfaces. Ann. Stat., 2017, 45(4): 1431-1461

[2] Benko M, Härdle W, Kneip A. Common functional principal components. Ann. Stat., 2009, 37(1): 1-34

[3] Boente G, Salibian-Barrera M. S-estimators for functional principal component analysis. J. Am. Stat. Assoc., 2015, 110(511): 1100-1111

[4] Chen KH, Delicado Useros PF, Müller H-G. Modelling function-valued stochastic processes, with applications to fertility dynamics. J. Royal Stat. Soc. Ser. B (Stat. Methodol.), 2017, 79(1): 177-196

[5] Chen KH, Müller H-G. Modeling repeated functional observations. J. Am. Stat. Assoc., 2012, 107(500): 1599-1609

[6] Constantinou P, Kokoszka P, Reimherr M. Testing separability of space-time functional processes. Biometrika, 2017, 104(2): 425-437

[7] Dai X, Müller H-G, Tao W. Derivative principal component analysis for representing the time dynamics of longitudinal and functional data. Stat. Sin., 2018, 28(3): 1583-1609

[8] Ferraty F, Vieu P. Nonparametric Functional Data Analysis: Theory and Practice, 2006, New York, Springer Science & Business Media

[9] Gervini D. Robust functional estimation using the median and spherical principal components. Biometrika, 2008, 95(3): 587-600

[10] Han F, Liu H. ECA: high-dimensional elliptical component analysis in non-gaussian distributions. J. Am. Stat. Assoc., 2018, 113(521): 252-268

[11] Hörmann S, Kidziński Ł, Hallin M. Dynamic functional principal components. J. Royal Stati. Soc.: Ser. B (Stat. Methodol.), 2015, 77(2): 319-348

[12] Hsing T, Eubank R. Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators, 2015, West Sussex, Wiley

[13] Huang L, Reiss PT, Xiao L, Zipunnikov V, Lindquist MA, Crainiceanu CM. Two-way principal component analysis for matrix-variate data, with an application to functional magnetic resonance imaging data. Biostatistics, 2017, 18(2): 214-229

[14] Kokoszka P, Reimherr M. Introduction to Functional Data Analysis, 2017, Florida, CRC Press

[15] Li G, Huang JHZ, Shen HP. Exponential family functional data analysis via a low-rank model. Biometrics, 2018, 74(4): 1301-1310

[16] Li YH, Hsing T. Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Stat., 2010, 38(6): 3321-3351

[17] Lynch B, Chen K. A test of weak separability for multi-way functional data, with application to brain connectivity studies. Biometrika, 2018, 105(4): 815-831

[18] Park SY, Staicu A-M. Longitudinal functional data analysis. Stat., 2015, 4(1): 212-226

[19] Sang P, Wang L, Cao J. Parametric functional principal component analysis. Biometrics, 2017, 73(3): 802-810

[20] Scheffler A, Telesca D, Li Q, Sugar CA, Distefano C, Jeste S, Şentürk D. Hybrid principal components analysis for region-referenced longitudinal functional EEG data. Biostatistics, 2020, 21(1): 139-157

[21] Shang HL. A survey of functional principal component analysis. AStA Adv. Stat. Anal., 2014, 98(2): 121-142

[22] Shao, L., Yao, F.: Robust Functional Data Analysis for Discretely Observed Data arXiv:2305.16236 (2023)

[23] Staicu A-M, Islam MN, Dumitru R, Heugten E. Longitudinal dynamic functional regression. J. Roy. Stat. Soc.: Ser. C (Appl. Stat.), 2020, 69(1): 25-46

[24] Wang G, Liu S, Han F, Di C-Z. Robust functional principal component analysis via a functional pairwise spatial sign operator. Biometrics, 2023, 79(2): 1239-1253

[25] Wang J-L, Chiou J-M, Müller H-G. Functional data analysis. Ann. Rev. Stat. Appl., 2016, 3: 257-295

[26] Wang J, Wong RK, Zhang X. Low-rank covariance function estimation for multidimensional functional data. J. Am. Stat. Assoc., 2022, 117(538): 809-822

[27] Xu YH, Li YH, Nettleton D. Nested hierarchical functional data modeling and inference for the analysis of functional plant phenotypes. J. Am. Stat. Assoc., 2018, 113(522): 593-606

[28] Yao F, Müller H-G, Wang J-L. Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc., 2005, 100(470): 577-590

[29] Zhang J-T, Chen J, et al. . Statistical inferences for functional data. Ann. Stat., 2007, 35(3): 1052-1079

[30] Zhong R, Liu S, Li H, Zhang J. Robust functional principal component analysis for non-Gaussian longitudinal data. J. Multivar. Anal., 2022, 189 104864

Funding

National Natural Science Foundation of China(12101270)

Natural Science Foundation of the Jiangsu Higher Education Institutions of China(22KJB110001)

Ph.D. Teacher’s Research Support Project Foundation of Jiangsu Normal Universit(21XFRX019)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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