Advertisement

Functional test for high-dimensional covariance matrix, with application to mitochondrial calcium concentration

  • Tao Zhang
  • Zhiwen Wang
  • Yanling WanEmail author
Regular Article
  • 30 Downloads

Abstract

In this paper, we present a novel method to test equality of covariance matrices of two high-dimensional samples. The methodology applies the idea of functional data analysis into high-dimensional data study. Asymptotic results of the proposed method are established. Some simulation studies are conducted to investigate the finite sample performance of the proposed method. We illustrate our testing procedures on a mitochondrial calcium concentration data for testing equality of covariance matrices.

Keywords

High-dimensional data Functional data analysis Significance test Covariance matrix 

Notes

Acknowledgements

T. Zhang’s research was supported by National Natural Science Foundation of China (11561006, 11861014) and Natural Science Foundation of Guangxi (2018JJA110013); Z. Wang’s research was supported by National Natural Science Foundation of China (61462008), Scientific Research and Technology Development Project of Liuzhou (2016C050205) and 2015 Innovation Team Project of Guangxi University of Science and Technology (gxkjdx201504).

References

  1. Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. Wiley-Interscience, New YorkzbMATHGoogle Scholar
  2. Bai Z, Jiang D, Yao J-F, Zheng S (2009) Corrections to LRT on large-dimensional covariance matrix by RMT. Ann Stat 37:1–34MathSciNetCrossRefGoogle Scholar
  3. Benko M, Härdle W, Kneip A et al (2009) Common functional principal components. Ann Stat 37:1–34MathSciNetCrossRefGoogle Scholar
  4. Bosq D (2000) Linear processes in function spaces. Springer, BerlinCrossRefGoogle Scholar
  5. Büning H (2000) Robustness and power of parametric, nonparametric, robustified and adaptive tests—the multi-sample location problem. Stat Pap 41:381–407MathSciNetCrossRefGoogle Scholar
  6. Cai T, Liu W, Xia Y (2013) Two-sample covariance matrix testing and support recoverary in high-dimensional and sparse settings. J Am Stat Assoc 108(501):265–277CrossRefGoogle Scholar
  7. Chen K, Chen K, Müller H-G, Wang J-L (2011) Stringing high-dimensional data for functional analysis. J Am Stat Assoc 106(493):275–284MathSciNetCrossRefGoogle Scholar
  8. Chen S, Zhang L, Zhong P (2010) Tests for high-dimensional covariance matrices. J Am Stat Assoc 105:810–819MathSciNetCrossRefGoogle Scholar
  9. Febrero-Bande M, Oviedo de la Fuente M (2012) Statistical computing in functional data analysis: the R package fda. usc. J Stat Softw 51(4):1–28CrossRefGoogle Scholar
  10. Fremdt S, Steinebach JG, Horváth L, Kokoszka P (2013) Testing the equality of covariance operators in functional samples. Scand J Stat 40:138–152MathSciNetCrossRefGoogle Scholar
  11. Ferraty F (2011) Recent advances in functional data analysis and related topics. Springer, BerlinCrossRefGoogle Scholar
  12. Ferré L, Yao AF (2005) Smoothed functional inverse regression. Stat Sin 15:665–683MathSciNetzbMATHGoogle Scholar
  13. Gregory K, Carroll R, Baladandayuthapani V, Lahiri S (2015) A two-sample test for equality of means in high dimension. J Am Stat Assoc 110(510):837–849MathSciNetCrossRefGoogle Scholar
  14. Gupta AK, Tang J (1984) Distribution of likelihood ratio statistic for testing equality of covariance matrices of multivariate Gaussian models. Biometrika 71:555–559MathSciNetCrossRefGoogle Scholar
  15. Kraus D (2015) Components and completion of partially observed functional data. J R Stat Soc Ser B 77(4):777–801MathSciNetCrossRefGoogle Scholar
  16. Li J, Chen S (2012) Two sample tests for high-dimensional covariance matrices. Ann Stat 40(2):908–940MathSciNetCrossRefGoogle Scholar
  17. Li W, Qin Y (2014) Hypothesis testing for high-dimensional covariance matrices. J Multivar Anal 128:108–119MathSciNetCrossRefGoogle Scholar
  18. Panaretos VM, Kraus D, Maddocks JH (2010) Second-order comparison of Gaussian random functions and the geometry of DNA minicircles. J Am Stat Assoc 105:670–682MathSciNetCrossRefGoogle Scholar
  19. Perlman MD (1980) Unbiasedness of the likelihood ratio tests for equality of several covariance matrices and equality of several multivariate normal populations. Ann Stat 8:247–263MathSciNetCrossRefGoogle Scholar
  20. Ruiz-Meana M, Garcia-Dorado D, Pina P, Inserte J, Agullo L, Soler-soler J (2003) Cariporide preserves mitochondrial proton gradient and delays ATP depletion in cardiomyocytes during ischemic conditions. Am J Physiol Heart Circ Physiol 285(3):H999–H1006CrossRefGoogle Scholar
  21. Schott J (2007) A test for the equality of covariance matrices when the dimension is large relative to the sample sizes. Comput Stat Data Anal 51(12):6535–6542MathSciNetCrossRefGoogle Scholar
  22. Srivastava M, Yanagihara H (2010) Testing the equality of serveral covariance matrices with fewer observations than the dimension. J Multivar Anal 101(6):1319–1329CrossRefGoogle Scholar
  23. Wang G, Zhou J, Wu W, Chen M (2017) Robust functional sliced inverse regression. Stat Pap 58:227–245MathSciNetCrossRefGoogle Scholar
  24. Zhang J-T (2013) Analysis of variance for functional data. CRC Press, Boca RatonCrossRefGoogle Scholar
  25. Zhang J-T, Liang X (2014) One-way ANOVA for functional data via globalizing the pointwise F-test. Scand J Stat 41:51–71MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceGuangxi University of Science and TechnologyLiuzhouChina
  2. 2.School of Computer Science and Communication EngineeringGuangxi University of Science and TechnologyLiuzhouChina
  3. 3.School of Social SciencesGuangxi University of Science and TechnologyLiuzhouChina

Personalised recommendations