Skip to main content
SpringerLink
Log in

A longitudinal model for shapes through triangulation

  • Original Paper
  • Published:
AStA Advances in Statistical Analysis Aims and scope Submit manuscript

Abstract

It is known that the shapes of planar triangles can be represented by a set of points on the surface of the unit sphere. On the other hand, most of the objects can easily be triangulated and so each triangle can accordingly be treated in the context of shape analysis. There is a growing interest to fit a smooth path going through the cloud of shape data available in some time instances. To tackle this problem, we propose a longitudinal model through a triangulation procedure for the shape data. In fact, our strategy initially relies on a spherical regression model for triangles, but is extended to shape data via triangulation. Regarding modeling of directional data, we use the bivariate von Mises–Fisher distribution for density of the errors. Various forms of the composite likelihood functions, constructed by altering the assumptions considered for the angles defined for each triangle, are invoked. The proposed regression model is applied to rat skull data. Also, some simulations results are presented along with the real data results.

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

Similar content being viewed by others

References

  • Barry, S.J., Bowman, A.W.: Linear mixed models for longitudinal shape data with applications to facial modeling. Biostatistics 9(3), 555–565 (2008)

    Article  MATH  Google Scholar 

  • Besag, J.: Spatial interaction and the statistical analysis of lattice systems (with discussion). J. R. Stat. Soc. Ser. B. 36(2), 192–236 (1974)

    MathSciNet  MATH  Google Scholar 

  • Bookstein, F.L.: A statistical method for biological shape comparisons. J. Theor. Biol. 107(3), 475–520 (1984)

    Article  Google Scholar 

  • Bookstein, F.L.: Size and shape spaces for landmark data in two dimensions (with discussion). Stat. Sci. 1(2), 181–242 (1986)

    Article  MATH  Google Scholar 

  • Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press, New York (1991)

    MATH  Google Scholar 

  • Chandler, R.E., Bate, S.: Inference for clustered data using the independence loglikelihood. Biometrika 94(1), 167–183 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Cox, D.R.: Partial likelihood. Biometrika 62(2), 269–276 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. In: IEEE 11th International Conference on Computer Vision, pp. 1–7 (2007)

  • Di Marzio, M., Panzera, A., Taylor, C.C.: Non-parametric regression for circular responses. Scand. J. Stat. 40(2), 238–255 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)

    MATH  Google Scholar 

  • Fisher, R.A.: Dispersion on a sphere. Proc. R. Soc. Lond. A 217, 295–305 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher, N.I., Lee, A.J.: Regression models for an angular response. Biometrics 48(3), 665–677 (1992)

    Article  MathSciNet  Google Scholar 

  • Fletcher, P.T.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vis. 105(2), 171–185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Gould, A.L.: A regression technique for angular variates. Biometrics 25(4), 683–700 (1969)

    Article  Google Scholar 

  • Hinkle, J., Fletcher, P.T., Joshi, S.: Intrinsic polynomials for regression on Riemannian manifolds. J. Math. Imaging Vis. 50(1–2), 32–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Huckemann, S., Ziezold, H.: Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Adv. Appl. Probab. 38(2), 299–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Joe, H., Lee, Y.: On weighted of bivariate margins in pairwise likelihood. J. Multivar. Anal. 100(4), 670–685 (2009)

    Article  MATH  Google Scholar 

  • Johnson, R.A., Wehrly, T.E.: Some angular-linear distributions and related regression models. J. Am. Stat. Assoc. 73(363), 602–606 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Jupp, P.E., Kent, J.T.: Fitting smooth paths to speherical data. J. R. Stat. Soc. Ser. C. 36(1), 34–46 (1987)

    MATH  Google Scholar 

  • Jupp, P.E., Mardia, K.V.: A general correlation coefficient for directional data and related regression problems. Biometrika 67(1), 163–173 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Kendall, D.G.: The diffusion of shape. Adv. Appl. Probab. 9(3), 428–430 (1977)

    Article  Google Scholar 

  • Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Kent, J., Mardia, K., Morris, R., Aykroyd, R.: Functional models of growth for landmark data. In: Mardia, K.V., Aykroyd, R.G. (eds.) Proceedings in Functional and Spatial Data Analysis, pp. 109–115. Leeds University Press, Leeds (2001)

    Google Scholar 

  • Kume, A., Dryden, I.L., Le, H.: Shape-space smoothing splines for planar landmark data. Biometrika 94(3), 513–528 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Mardia, K.V.: Statistics of Directional Data. Academic Press, London (1972)

    MATH  Google Scholar 

  • Mardia, K.V.: Statistics of directional data (with discussion). J. R. Stat. Soc. B. 37(3), 349–393 (1975)

    MATH  Google Scholar 

  • Mardia, K.V.: Shape analysis of triangles through directional techniques. J. R. Stat. Soc. B 51(3), 449–458 (1989)

    MathSciNet  Google Scholar 

  • Mardia, K.V., Hughes, G., Taylor, C.C.: Efficiency of the pseudolikelihood for multivariate normal and von Mises distributions. Research Report 07-02, Department of Statistics, University of Leeds (2007)

  • Mardia, K.V., Hughes, G., Taylor, C.C., Singh, H.: A multivariate von Mises distribution with applications to bioinformatics. Can. J. Stat. 36(1), 99–109 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, London (2000)

    MATH  Google Scholar 

  • Mardia, K.V., Kent, J.T., Hughes, G., Taylor, C.C.: Maximum likelihood estimation using composite likelihoods for closed exponential families. Biometrika 96(4), 975–982 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Mardia, K.V., Kirkbride, J., Bookstein, F.L.: Statistics of shape, direction and cylindrical variables. J. Appl. Stat. 31(4), 465–479 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Moghimbeygi, M., Golalizadeh, M.: Longitudinal shape analysis by using the spherical coordinates. J. Appl. Stat. 44(7), 1282–1295 (2017)

    Article  MathSciNet  Google Scholar 

  • Morris, R., Kent, J., Mardia, K., Aykroyd, R.: A parallel growth model for shape. In: Arridge, S., Todd-Pokropek, A. (eds.) Proceedings in Medical Imaging Understanding and Analysis, pp. 171–174. BMVA, Bristol (2000)

    Google Scholar 

  • Peng, D., Deng, M.: A method of measuring shape similarity between multi-scale objects. In: Proceedings of the 12th International Conference on GeoComputation, Wuhan, China (2013)

  • Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis: Methods and Case Studies. Springer, New York (2002)

    Book  MATH  Google Scholar 

  • Rivest, L.P.: A distribution for dependent unit vectors. Commun. Stat. Theory Methods 17(2), 461–483 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Thompson, R., Clark, R.M.: Fitting polar wander paths. Phys. Earth Planet. Inter. 27, 1–7 (1981)

    Article  Google Scholar 

  • Trouvé, A., Vialard, F.X.: A second-order model for time-dependent data interpolation: splines on shape spaces. In: MICCAI STIA workshop, MICCAI, Beijing (2010)

  • Varin, C.: On composite marginal likelihoods. AStA Adv. Stat. Anal. 92(1), 1–28 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Varin, C., Czado, C.: A mixed autoregressive probit model for ordinal longitudinal data. Biostatistics 11(1), 127–138 (2010)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Tarbiat Modares University, Tehran, Iran

    Meisam Moghimbeygi & Mousa Golalizadeh

Authors
  1. Meisam Moghimbeygi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mousa Golalizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mousa Golalizadeh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moghimbeygi, M., Golalizadeh, M. A longitudinal model for shapes through triangulation. AStA Adv Stat Anal 103, 99–121 (2019). https://doi.org/10.1007/s10182-018-0324-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-018-0324-9

Keywords

Search

Navigation