Skip to main content
SpringerLink
Log in

Probability Density Estimation on the Hyperbolic Space Applied to Radar Processing

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

Included in the following conference series:

Abstract

The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Hendriks, H.: Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. Ann. Stat. 18(2), 832–849 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Huckeman, S., Kim, P., Koo, J., Munk, A.: Mobius deconvolution on the hyperbolic plan with application to impedance density estimation. Ann. Stat. 38(4), 2465–2498 (2010)

    Article  MATH  Google Scholar 

  3. Pelletier, B.: Kernel density estimation on Riemannian manifolds. Stat. Probab. Lett. 73(3), 297–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asta, D.: Kernel Density Estimation on Symmetric Spaces. arXiv preprint arXiv:1411.4040 (2014)

  5. Said, S., Bombrun, L., Berthoumieu, Y.: New Riemannian priors on the univariate normal model. Entropy 16, 4015–4031 (2014)

    Article  MathSciNet  Google Scholar 

  6. Asta, D., Shalizi, C.: Geometric network comparison. arXiv preprint arXiv:1411.1350 (2014)

  7. Barbaresco, F.: Super resolution spectrum analysis regularization: burg, capon and ago antagonistic algorithms. In: Proceedings of EUSIPCO 1996, Trieste, pp. 2005–2008 (1996)

    Google Scholar 

  8. Barbaresco, F.: Information geometry of covariance matrix: cartan-siegel homogeneous bounded domains, mostow/berger fibration and fréchet median. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 199–255. Springer, Heidelberg (2013)

    Google Scholar 

  9. Arnaudon, M., Barbaresco, F., Yang, L.: Riemannian medians and means with applications to radar signal processing. IEEE J. Sel. Top. Sign. Proces. 7(4), 595–604 (2013)

    Article  Google Scholar 

  10. Cannon, J.W., Floyd, W.J., Kenyon, R., Parry, W.R.: Hyperbolic geometry. In: Levy, S. (ed.) Flavors of Geometry, vol. 31. MSRI Publications, Cambridge (1997)

    Google Scholar 

  11. Duin, R.P.W.: On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Trans. Comput. 25(11), 1175–1179 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Helgason, S.: Non-Euclidean Analysis, arXiv:math/0411411 [math.DG] (2004)

  13. Helgason, S.: The Abel, Fourier and Radon transform on symmetric spaces. Indagationes Math. 16(3), 531–551 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, P., Richards, D.: Deconvolution density estimation on the space of positive definite symmetric matrices. In: Nonparametric Statistics and Mixture Models: A Festschrift in Honor of Thomas P. Hettmansperger, pp. 147–168. World Scientific Publishing (2008)

    Google Scholar 

  15. Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York (1986)

    Book  MATH  Google Scholar 

  16. Anker, J.-Ph., Ostellari, P.: The heat kernel on noncompact symmetric spaces. In: Lie groups and symmetric spaces, AMS Transl. Ser. 2, vol. 210, pp. 27–46 (2003)

    Google Scholar 

  17. Grigor’yan, A.: Heat kernel and analysis on manifolds. Am. Math. Soc. 47 (2012)

    Google Scholar 

  18. Barbaresco, F.: Koszul information geometry and souriau geometric temperature/capacity of lie group thermodynamics. Entropy 16, 4521–4565 (2014)

    Article  MathSciNet  Google Scholar 

  19. Trench, W.F.: An algorithm for the inversion of finite Toeplitz matrices. J. Soc. Indust. Appl. Math. 12, 515–522 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Munkres, J.R.: Elementary Diffrentaible Topology. Annals of Mathematics Studies, vol. 54. Princeton University Press, Princeton (1966)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CMM-Centre de Morphologie Mathématique, MINES ParisTech, PSL-Research University, Fontainebleau, France

    Emmanuel Chevallier & Jesús Angulo

  2. Thales Air Systems, Surface Radar Domain, Technical Directorate, Advanced Developments Department, 91470, Limours, France

    Frédéric Barbaresco

Authors
  1. Emmanuel Chevallier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frédéric Barbaresco
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jesús Angulo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesús Angulo .

Editor information

Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Chevallier, E., Barbaresco, F., Angulo, J. (2015). Probability Density Estimation on the Hyperbolic Space Applied to Radar Processing. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_80

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25040-3_80

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25039-7

  • Online ISBN: 978-3-319-25040-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation