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.
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)
Besag, J.: Spatial interaction and the statistical analysis of lattice systems (with discussion). J. R. Stat. Soc. Ser. B. 36(2), 192–236 (1974)
Bookstein, F.L.: A statistical method for biological shape comparisons. J. Theor. Biol. 107(3), 475–520 (1984)
Bookstein, F.L.: Size and shape spaces for landmark data in two dimensions (with discussion). Stat. Sci. 1(2), 181–242 (1986)
Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press, New York (1991)
Chandler, R.E., Bate, S.: Inference for clustered data using the independence loglikelihood. Biometrika 94(1), 167–183 (2007)
Cox, D.R.: Partial likelihood. Biometrika 62(2), 269–276 (1975)
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)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)
Fisher, R.A.: Dispersion on a sphere. Proc. R. Soc. Lond. A 217, 295–305 (1953)
Fisher, N.I., Lee, A.J.: Regression models for an angular response. Biometrics 48(3), 665–677 (1992)
Fletcher, P.T.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vis. 105(2), 171–185 (2013)
Gould, A.L.: A regression technique for angular variates. Biometrics 25(4), 683–700 (1969)
Hinkle, J., Fletcher, P.T., Joshi, S.: Intrinsic polynomials for regression on Riemannian manifolds. J. Math. Imaging Vis. 50(1–2), 32–52 (2014)
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)
Joe, H., Lee, Y.: On weighted of bivariate margins in pairwise likelihood. J. Multivar. Anal. 100(4), 670–685 (2009)
Johnson, R.A., Wehrly, T.E.: Some angular-linear distributions and related regression models. J. Am. Stat. Assoc. 73(363), 602–606 (1978)
Jupp, P.E., Kent, J.T.: Fitting smooth paths to speherical data. J. R. Stat. Soc. Ser. C. 36(1), 34–46 (1987)
Jupp, P.E., Mardia, K.V.: A general correlation coefficient for directional data and related regression problems. Biometrika 67(1), 163–173 (1980)
Kendall, D.G.: The diffusion of shape. Adv. Appl. Probab. 9(3), 428–430 (1977)
Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)
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)
Kume, A., Dryden, I.L., Le, H.: Shape-space smoothing splines for planar landmark data. Biometrika 94(3), 513–528 (2007)
Mardia, K.V.: Statistics of Directional Data. Academic Press, London (1972)
Mardia, K.V.: Statistics of directional data (with discussion). J. R. Stat. Soc. B. 37(3), 349–393 (1975)
Mardia, K.V.: Shape analysis of triangles through directional techniques. J. R. Stat. Soc. B 51(3), 449–458 (1989)
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)
Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, London (2000)
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)
Mardia, K.V., Kirkbride, J., Bookstein, F.L.: Statistics of shape, direction and cylindrical variables. J. Appl. Stat. 31(4), 465–479 (2004)
Moghimbeygi, M., Golalizadeh, M.: Longitudinal shape analysis by using the spherical coordinates. J. Appl. Stat. 44(7), 1282–1295 (2017)
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)
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)
Rivest, L.P.: A distribution for dependent unit vectors. Commun. Stat. Theory Methods 17(2), 461–483 (1988)
Thompson, R., Clark, R.M.: Fitting polar wander paths. Phys. Earth Planet. Inter. 27, 1–7 (1981)
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)
Varin, C., Czado, C.: A mixed autoregressive probit model for ordinal longitudinal data. Biostatistics 11(1), 127–138 (2010)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10182-018-0324-9