Skip to main content
SpringerLink
Log in

Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions

  • Published:
METRON Aims and scope Submit manuscript

Abstract

Symmetric positive definite (SPD) matrices arise in a wide range of applications including diffusion tensor imaging (DTI), cosmic background radiation, and as covariance matrices. A complication when working with such data is that the space of SPD matrices is a manifold, so traditional statistical methods may not be directly applied. However, there are nonparametric procedures based on resampling for statistical inference for such data, but these can be slow and computationally tedious. Schwartzman (Int Stat Rev 84(3):456–486, 2016). introduced a lognormal distribution on the space of SPD matrices, providing a convenient framework for parametric inference on this space. Our goal is to check how robust confidence regions based on this distributional assumption are to a lack of lognormality. The methods are illustrated in a simulation study by examining the coverage probability of various mixtures of distributions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Bhattacharya, R.N., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-Part I. Ann. Stat. 31(1), 1–29 (2003)

    Article  Google Scholar 

  2. Bhattacharya, R.N., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-Part II. Ann. Stat. 33(3), 1225–1259 (2005)

    Article  Google Scholar 

  3. Bhattacharya, R.N., Ellingson, L., Liu, X., Patrangenaru, V., Crane, M.: Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with application to quality control by machine vision. Appl. Stoch. Models Bus. Ind. 28, 222–235 (2012)

    Article  MathSciNet  Google Scholar 

  4. Dryden, I.L., Koloydenko, A., Zhou, D.: Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. Appl. Stat. 3(3), 1102–1123 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Ellingson, L., Groisser, D., Osborne, D.E., Patrangenaru, V., Schwartzman, A.: Nonparametric bootstrap of sample means of positive definite matrices with an application to diffusion-tensor-imaging data. Commun. Stat. Simul. Comput. 46(6), 4851–4879 (2016)

    Article  MathSciNet  Google Scholar 

  6. Fisher, R. A.: Dispersion on a sphere. Proc. R. Soc. Lond. A 217295–217305 (1953). https://doi.org/10.1098/rspa.1953.0064

  7. Hendriks, H., Landsman, Z.: Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. J. Multivar. Anal. 67(2), 227–243 (1998)

    Article  MathSciNet  Google Scholar 

  8. Osborne, D.E., Patrangenaru, V., Ellingson, L., Groisser, D., Schwartzman, A.: Nonparametric two-sample tests on homogeneous riemannian manifolds, cholesky decompositions and diffusion tensor image analysis. J. Multivar. Anal. 119, 163–175 (2013)

    Article  MathSciNet  Google Scholar 

  9. Schwartzman, A.: Lognormal distributions and geometric averages of positive definite matrices. Int. Stat. Rev. 84(3), 456–486 (2016). https://doi.org/10.1111/insr.12113

    Article  MathSciNet  Google Scholar 

  10. Watson, G.S., Williams, E.J.: On the Construction of significance tests on the circle and the sphere. Biometrika 43(3–4), 344–352 (1956). https://doi.org/10.1093/biomet/43.3-4.344

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees and associate editors for their many valuable comments and suggestions that improved the manuscript.

Author information

Authors and Affiliations

  1. Bradley University, Peoria, USA

    Benoit Ahanda

  2. Florida Agricultural and Mechanical University, Tallahassee, USA

    Daniel E. Osborne

  3. Texas Tech University, Lubbock, USA

    Leif Ellingson

Authors
  1. Benoit Ahanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel E. Osborne
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leif Ellingson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leif Ellingson.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahanda, B., Osborne, D.E. & Ellingson, L. Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions. METRON 80, 281–303 (2022). https://doi.org/10.1007/s40300-022-00234-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40300-022-00234-z

Keywords

Search

Navigation