Conditional Local Distance Correlation for Manifold-Valued Data

  • Wenliang Pan
  • Xueqin Wang
  • Canhong Wen
  • Martin Styner
  • Hongtu ZhuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10265)


Manifold-valued data arises frequently in medical imaging, surface modeling, computational biology, and computer vision, among many others. The aim of this paper is to introduce a conditional local distance correlation measure for characterizing a nonlinear association between manifold-valued data, denoted by X, and a set of variables (e.g., diagnosis), denoted by Y, conditional on the other set of variables (e.g., gender and age), denoted by Z. Our nonlinear association measure is solely based on the distance of the space that X, Y, and Z are resided, avoiding both specifying any parametric distribution and link function and projecting data to local tangent planes. It can be easily extended to the case when both X and Y are manifold-valued data. We develop a computationally fast estimation procedure to calculate such nonlinear association measure. Moreover, we use a bootstrap method to determine its asymptotic distribution and p-value in order to test a key hypothesis of conditional independence. Simulation studies and a real data analysis are used to evaluate the finite sample properties of our methods.


Manifold-valued Local distance correlation Shape statistics 


  1. 1.
    Allen, L.S., Richey, M., Chai, Y.M., Gorski, R.A.: Sex differences in the corpus callosum of the living human being. J. Neurosci. 11(4), 933–942 (1991)Google Scholar
  2. 2.
    Banerjee, M., Chakraborty, R., Ofori, E., Okun, M.S., Viallancourt, D.E., Vemuri, B.C.: A nonlinear regression technique for manifold valued data with applications to medical image analysis. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4424–4432 (2016)Google Scholar
  3. 3.
    Bhattacharya, A., Dunson, D.B.: Nonparametric Bayesian density estimation on manifolds with applications to planar shapes. Biometrika 97(4), 851–865 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhattacharya, A., Dunson, D.B.: Nonparametric Bayes classification and hypothesis testing on manifolds. J. Multivar. Anal. 111, 1–19 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-i. Ann. Stat. 31(1), 1–29 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-ii. Ann. Stat. 33(3), 1225–1259 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cornea, E., Zhu, H., Kim, P.T., Ibrahim, J.G.: Regression models on Riemannian symmetric spaces. J. Roy. Stat. Soc. Ser. B-Stat. Methodol. 79, 463–482 (2016)CrossRefGoogle Scholar
  8. 8.
    Dale, A.M., Fischl, B., Sereno, M.I.: Cortical surface-based analysis I. Segmentation and surface reconstruction. NeuroImage 9(2), 179–194 (1999)CrossRefGoogle Scholar
  9. 9.
    Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. Int. J. Comput. Vis. 90(2), 255–266 (2010)CrossRefGoogle Scholar
  10. 10.
    Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8), 995–1005 (2004)CrossRefGoogle Scholar
  11. 11.
    Grenander, U., Miller, M.I.: Pattern Theory From Representation to Inference. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  12. 12.
    Huckemann, S., Hotz, T., Munk, A.: Intrinsic manova for Riemannian manifolds with an application to Kendall’s space of planar shapes. IEEE Trans. Pattern Anal. Mach. Intell. 32(4), 593–603 (2010)CrossRefGoogle Scholar
  13. 13.
    Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. Roy. Stat. Soc. Ser. B Methodol. 44, 71–80 (1982)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kim, H.J., Adluru, N., Bendlin, B.B., Johnson, S.C., Vemuri, B.C., Singh, V.: Canonical correlation analysis on Riemannian manifolds and its applications. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds.) ECCV 2014. LNCS, vol. 8690, pp. 251–267. Springer, Cham (2014). doi: 10.1007/978-3-319-10605-2_17 Google Scholar
  15. 15.
    Lyons, R.: Distance covariance in metric spaces. Ann. Probab. 41(5), 3284–3305 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Machado, L., Leite, F.S., Krakowski, K.: Higher-order smoothing splines versus least squares problems on Riemannian manifolds. J. Dyn. Control Syst. 16(1), 121–148 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ota, M., Obata, T., Akine, Y., Ito, H., Ikehira, H., Asada, T., Suhara, T.: Age-related degeneration of corpus callosum measured with diffusion tensor imaging. NeuroImage 31(4), 1445–1452 (2006)CrossRefGoogle Scholar
  18. 18.
    Paparoditis, E., Politis, D.: The local bootstrap for kernel estimators under general dependence conditions. Ann. Inst. Stat. Math. 52(1), 139–159 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Patrangenaru, V., Ellingson, L.: Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis. CRC Press, Boca Raton (2015)CrossRefzbMATHGoogle Scholar
  20. 20.
    Paul, L.K., Brown, W.S., Adolphs, R., Tyszka, J.M., Richards, L.J., Mukherjee, P., Sherr, E.H.: Agenesis of the corpus callosum: genetic, developmental and functional aspects of connectivity. Nat. Rev. Neurosci. 8(4), 287–299 (2007)CrossRefGoogle Scholar
  21. 21.
    Pelletier, B.: Kernel density estimation on Riemannian manifolds. Stat. Probab. Lett. 73(3), 297–304 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shuyu, L., Fang, P., Xiangqi, H., Li, D., Tianzi, J.: Shape analysis of the corpus callosum in Alzheimer’s disease, pp. 1095–1098 (2007)Google Scholar
  23. 23.
    Srivastava, A., Klassen, E.P.: Functional and Shape Data Analysis. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  24. 24.
    Su, J., Dryden, I.L., Klassen, E., Le, H., Srivastava, A.: Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds. Image Vis. Comput. 30(6), 428–442 (2012)CrossRefGoogle Scholar
  25. 25.
    Székely, G., Rizzo, M., Bakirov, N.: Measuring and testing dependence by correlation of distances. Ann. Stat. 35(6), 2769–2794 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, X., Pan, W., Hu, W., Tian, Y., Zhang, H.: Conditional distance correlation. J. Am. Stat. Assoc. 110(512), 1726 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Witelson, S.F.: Hand and sex differences in the isthmus and genu of the human corpus callosum. A postmortem morphological study. Brain 112(3), 799–835 (1989)CrossRefGoogle Scholar
  28. 28.
    Younes, L.: Shapes and Diffeomorphisms. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  29. 29.
    Yuan, Y., Zhu, H., Lin, W., Marron, J.S.: Local polynomial regression for symmetric positive definite matrices. J. Roy. Stat. Soc. Ser. B-Stat. Methodol. 74(4), 697–719 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Wenliang Pan
    • 1
    • 2
  • Xueqin Wang
    • 1
    • 2
  • Canhong Wen
    • 1
    • 2
  • Martin Styner
    • 3
  • Hongtu Zhu
    • 4
    Email author
  1. 1.Department of Statistical ScienceSun Yat-sen UniversityGuangzhouChina
  2. 2.Southern China Research Center of Statistical ScienceSun Yat-sen UniversityGuangzhouChina
  3. 3.University of North Carolina at Chapel HillChapel HillUSA
  4. 4.University of Texas MD Anderson Cancer CenterHoustonUSA

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