Skip to main content
SpringerLink
Log in

Supervised classification of geometrical objects by integrating currents and functional data analysis

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

This paper focuses on the application of supervised classification techniques to a set of geometrical objects (bodies) characterized by currents, in particular, discriminant analysis and some nonparametric methods. A current is a relevant mathematical object to model geometrical data, like hypersurfaces, through integration of vector fields over them. As a consequence of the choice of a vector-valued reproducing kernel Hilbert space (RKHS) as a test space to integrate over hypersurfaces, it is possible to consider that hypersurfaces are embedded in this Hilbert space. This embedding enables us to consider classification algorithms of geometrical objects. We present a method to apply supervised classification techniques in the obtained vector-valued RKHS. This method is based on the eigenfunction decomposition of the kernel. The novelty of this paper is therefore the reformulation of a size and shape supervised classification problem in functional data analysis terms using the theory of currents and vector-valued RKHSs. This approach is applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.

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

Similar content being viewed by others

References

  • Alonso AM, Casado D, Romo J (2012) Supervised classification for functional data: a weighted distance approach. Comput Stat Data Anal 56(7):2334–2346

    MathSciNet  MATH  Google Scholar 

  • Aneiros G, Bongiorno EG, Cao R, Vieu P et al (2017) Functional statistics and related fields. Springer, Berlin

    MATH  Google Scholar 

  • Araki Y, Konishi S, Kawano S, Matsui H (2009) Functional logistic discrimination via regularized basis expansions. Commun Stat Theory Methods 38(16–17):2944–2957

    MathSciNet  MATH  Google Scholar 

  • Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68(3):337–404

    MathSciNet  MATH  Google Scholar 

  • Ballester A, Valero M, Nácher B, Piérola A, Piqueras P, Sancho M, Gargallo G, González J, Alemany S (2015) 3D body databases of the spanish population and its application to the apparel industry. In: Proceedings of the 6th international conference on 3D body scanning technologies, Lugano, Switzerland

  • Ballester A, Parrilla E, Piérola A, Uriel J, Pérez C, Piqueras P, Vivas J, Alemany S (2016) Data-driven three-dimensional reconstruction of human bodies using a mobile phone app. Int J Digit Hum 1(4):361–388

    Google Scholar 

  • Barahona S, Gual-Arnau X, Ibáñez M, Simó A (2018) Unsupervised classification of children’s bodies using currents. Adv Data Anal Classif 12(2):365–397

    MathSciNet  MATH  Google Scholar 

  • Berlinet A, Thomas-Agnan C (2011) Reproducing kernel Hilbert spaces in probability and statistics. Springer, Berlin

    MATH  Google Scholar 

  • Berrendero JR, Cuevas A, Torrecilla JL (2018) On the use of reproducing kernel hilbert spaces in functional classification. J Am Stat Assoc 113(523):1210–1218

    MathSciNet  MATH  Google Scholar 

  • Bickel PJ, Li B, Tsybakov AB, van de Geer SA, Yu B, Valdés T, Rivero C, Fan J, van der Vaart A (2006) Regularization in statistics. Test 15(2):271–344

    MathSciNet  Google Scholar 

  • Boj E, Caballé A, Delicado P, Esteve A, Fortiana J (2016) Global and local distance-based generalized linear models. Test 25(1):170–195

    MathSciNet  MATH  Google Scholar 

  • Bouveyron C, Brunet-Saumard C (2014) Model-based clustering of high-dimensional data: a review. Comput Stat Data Anal 71:52–78

    MathSciNet  MATH  Google Scholar 

  • Chiou JM, Li PL (2007) Functional clustering and identifying substructures of longitudinal data. J R Stat Soc Ser B (Stat Methodol) 69(4):679–699

    MathSciNet  Google Scholar 

  • Cuadras C (1989) Distance analysis in discrimination and classification using both continuous and categorical variables. Statistical data analysis and inference. Elsevier, Amsterdam, pp 459–473

    MATH  Google Scholar 

  • Cucker F, Smale S (2001) On the mathematical foundations of learning. Am Math Soc 39(1):1–49

    MathSciNet  MATH  Google Scholar 

  • Cuesta-Albertos JA, Fraiman R (2007) Impartial trimmed k-means for functional data. Comput Stat Data Anal 51(10):4864–4877

    MathSciNet  MATH  Google Scholar 

  • Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plan Inference 147:1–23

    MathSciNet  MATH  Google Scholar 

  • Cuevas A, Febrero M, Fraiman R (2007) Robust estimation and classification for functional data via projection-based depth notions. Comput Stat 22(3):481–496

    MathSciNet  MATH  Google Scholar 

  • Dai X, Müller HG et al (2018) Principal component analysis for functional data on riemannian manifolds and spheres. Ann Stat 46(6B):3334–3361

    MathSciNet  MATH  Google Scholar 

  • Delicado P (2007) Functional k-sample problem when data are density functions. Comput Stat 22(3):391–410

    MathSciNet  MATH  Google Scholar 

  • Delicado P (2011) Dimensionality reduction when data are density functions. Comput Stat Data Anal 55(1):401–420

    MathSciNet  MATH  Google Scholar 

  • Delicado P, Vieu P (2017) Choosing the most relevant level sets for depicting a sample of densities. Comput Stat 32(3):1083–1113

    MathSciNet  MATH  Google Scholar 

  • Devarajan P, Istook CL (2004) Validation of female figure identification technique (FFIT) for apparel software. J Text Appar Technol Manag 4(1):1–23

    Google Scholar 

  • Di Marzio M, Fensore S, Panzera A, Taylor CC (2019) Kernel density classification for spherical data. Stat Probab Lett 144:23–29

    MathSciNet  MATH  Google Scholar 

  • Dryden IL, Mardia KV (2016) Statistical shape analysis: with applications. Wiley, Hoboken

    MATH  Google Scholar 

  • Durrleman S (2010) Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. Ph.D. thesis, Université Nice Sophia Antipolis

  • Durrleman S, Pennec X, Trouvé A, Ayache N (2009) Statistical models of sets of curves and surfaces based on currents. Med Image Anal 13(5):793–808

    Google Scholar 

  • Eubank R, Hsing T (2008) Canonical correlation for stochastic processes. Stoch Process Their Appl 118(9):1634–1661

    MathSciNet  MATH  Google Scholar 

  • Federer H, Fleming W (1960) Normal and integral currents. Ann Math 72:458–520

    MathSciNet  MATH  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, Berlin

    MATH  Google Scholar 

  • Fisher R (1936) The use of multiple measurements in taxonomic problems. Ann Eugen 7(2):179–188

    Google Scholar 

  • Flores M, Gual-Arnau X, Ibáñez M, Simó A (2016) Intrinsic sample mean in the space of planar shapes. Pattern Recognit 60:164–176

    MATH  Google Scholar 

  • Fraiman R, Gamboa F, Moreno L (2019) Connecting pairwise geodesic spheres by depth: DCOPS. J Multivar Anal 169:81–94

    MathSciNet  MATH  Google Scholar 

  • Garcia-Escudero LA, Gordaliza A (2005) A proposal for robust curve clustering. J Classif 22(2):185–201

    MathSciNet  MATH  Google Scholar 

  • Gerber S, Tasdizen T, Whitaker R (2009) Dimensionality reduction and principal surfaces via kernel map manifolds. In: IEEE 12th international conference on computer vision, IEEE, pp 529–536

  • Glaunès J, Joshi S (2006) Template estimation form unlabeled point set data and surfaces for computational anatomy. In: 1st MICCAI Workshop on mathematical foundations of computational anatomy: geometrical, statistical and registration methods for modeling biological shape variability

  • Goia A, Vieu P (2016) An introduction to recent advances in high/infinite dimensional statistics. J Multivar Anal 146:1–6

    MathSciNet  MATH  Google Scholar 

  • González J, Muñoz A (2010) Representing functional data in reproducing kernel hilbert spaces with applications to clustering and classification. Tech. rep., Universidad Carlos III de Madrid. Departamento de Estadística

  • Hall P, Poskitt D, Presnell B (2001) A functional data-analytic approach to signal discrimination. Technometrics 43(1):1–9

    MathSciNet  MATH  Google Scholar 

  • Horváth L, Kokoszka P (2012) Inference for functional data with applications, vol 200. Springer, Berlin

    MATH  Google Scholar 

  • Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley, Hoboken

    MATH  Google Scholar 

  • Jacques J, Preda C (2014) Functional data clustering: a survey. Adv Data Anal Classif 8(3):231–255

    MathSciNet  MATH  Google Scholar 

  • James G, Hastie T (2001) Functional linear discriminant analysis for irregularly sampled curves. J R Stat Soc Ser B (Stat Methodol) 63(3):533–550

    MathSciNet  MATH  Google Scholar 

  • Kadri H, Duflos E, Preux P, Canu S, Rakotomamonjy A, Audiffren J (2016) Operator-valued kernels for learning from functional response data. J Mach Learn Res 17(1):613–666

    MathSciNet  MATH  Google Scholar 

  • Kendall D, Barden D, Carne T, Le H (2009) Shape and shape theory, vol 500. Wiley, Hoboken

    MATH  Google Scholar 

  • Kent JT, Mardia KV, Morris RJ, Aykroyd RG (2001) Functional models of growth for landmark data. In: Proceedings in functional and spatial data analysis university of leeds, pp 109–115

  • Kneip A, Utikal KJ (2001) Inference for density families using functional principal component analysis. J Am Stat Assoc 96(454):519–542

    MathSciNet  MATH  Google Scholar 

  • Kupresanin A, Shin H, King D, Eubank R (2010) An rkhs framework for functional data analysis. J Stat Plan Inference 140(12):3627–3637

    MathSciNet  MATH  Google Scholar 

  • Lian H (2007) Nonlinear functional models for functional responses in reproducing kernel hilbert spaces. Can J Stat 35(4):597–606

    MathSciNet  MATH  Google Scholar 

  • Lin L, St Thomas B, Zhu H, Dunson DB (2017) Extrinsic local regression on manifold-valued data. J Am Stat Assoc 112(519):1261–1273

    MathSciNet  Google Scholar 

  • López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104(486):718–734

    MathSciNet  MATH  Google Scholar 

  • Loubes JM, Pelletier B (2008) A kernel-based classifier on a riemannian manifold. Stat Decis Int Math J Stoch Methods Models 26(1):35–51

    MathSciNet  MATH  Google Scholar 

  • Lukić M, Beder J (2001) Stochastic processes with sample paths in reproducing kernel hilbert spaces. Trans Am Math Soc 353(10):3945–3969

    MathSciNet  MATH  Google Scholar 

  • Marron JS, Alonso AM (2014) Overview of object oriented data analysis. Biom J 56(5):732–753

    MathSciNet  MATH  Google Scholar 

  • MATLAB (2015) version 8.6.0 (R2015b). The MathWorks Inc., Natick, MA

  • Meunier P (2000) Use of body shape information in clothing size selection. In: Proceedings of the human factors and ergonomics society annual meeting. SAGE Publications Sage CA: Los Angeles, vol 44, pp 715–718

  • Müller HG (2005) Functional modelling and classification of longitudinal data. Scand J Stat 32(2):223–240

    MathSciNet  MATH  Google Scholar 

  • Pelletier B (2006) Non-parametric regression estimation on closed Riemannian manifolds. J Nonparametric Stat 18(1):57–67

    MathSciNet  MATH  Google Scholar 

  • Peng J, Müller HG et al (2008) Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions. Ann Appl Stat 2(3):1056–1077

    MathSciNet  MATH  Google Scholar 

  • Pennec X (2006) Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis 25(1):127–154

    MathSciNet  Google Scholar 

  • Preda C (2007) Regression models for functional data by reproducing kernel hilbert spaces methods. J Stat Plan Inference 137(3):829–840

    MathSciNet  MATH  Google Scholar 

  • Preda C, Saporta G, Lévéder C (2007) PLS classification of functional data. Comput Stat 22(2):223–235

    MathSciNet  MATH  Google Scholar 

  • Quang M, Kang S, Le T (2010) Image and video colorization using vector-valued reproducing kernel Hilbert spaces. J Math Imaging Vis 37(1):49–65

    MathSciNet  MATH  Google Scholar 

  • R Core Team (2018) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  • Rham Gd (1960) Variétés différentiables. Formes, courants, formes harmoniques. Hermann et Cie, Butterworths Scientific Publications, Hermann

  • Ripley B (2007) Pattern recognition and neural networks. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Rossi F, Villa N (2008) Recent advances in the use of svm for functional data classification. Functional and operatorial statistics. Springer, Berlin, pp 273–280

    Google Scholar 

  • Saitoh S, Sawano Y (2016) Theory of reproducing kernels and applications. Springer, Berlin

    MATH  Google Scholar 

  • Schölkopf B, Smola AJ, Bach F et al (2002) Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge

    Google Scholar 

  • Senkene E, Tempel’man A (1973) Hilbert spaces of operator-valued functions. Math Trans Acad Sci Lith SSR 13(4):665–670

    MathSciNet  Google Scholar 

  • Serra J (1982) Image analysis and mathematical morphology. Academic Press, Cambridge

    MATH  Google Scholar 

  • Shin H (2008) An extension of Fisher’s discriminant analysis for stochastic processes. J Multivar Anal 99(6):1191–1216

    MathSciNet  MATH  Google Scholar 

  • Silverman B, Ramsay J (2005) Functional data analysis. Springer, Berlin

    MATH  Google Scholar 

  • Smale S, Zhou DX (2009) Geometry on probability spaces. Constr Approx 30(3):311–323

    MathSciNet  MATH  Google Scholar 

  • Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields. Methods of geometrical statistics. Wiley, Hoboken

    MATH  Google Scholar 

  • Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500):2319–2323

    Google Scholar 

  • Turaga PK, Srivastava A (2016) Riemannian computing in computer vision. Springer, Berlin

    MATH  Google Scholar 

  • Vaillant M, Glaunès J (2005) Surface matching via currents. Information processing in medical imaging. Springer, Berlin, pp 381–392

    Google Scholar 

  • Viktor HL, Paquet E, Guo H (2006) Measuring to fit: virtual tailoring through cluster analysis and classification. In: European conference on principles of data mining and knowledge discovery. Springer, Berlin, pp 395–406

  • Vinué G, Simó A, Alemany S (2016) The k-means algorithm for 3D shapes with an application to apparel design. Adv Data Anal Classif 10(1):103–132

    MathSciNet  MATH  Google Scholar 

  • Wang H, Marron J et al (2007) Object oriented data analysis: sets of trees. Ann Stat 35(5):1849–1873

    MathSciNet  MATH  Google Scholar 

  • Wang JL, Chiou JM, Müller HG (2016) Functional data analysis. Ann Rev Stat Its Appl 3:257–295

    Google Scholar 

  • Younes L (1998) Computable elastic distance between shapes. SIAM J Appl Math 58(2):565–586

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper has been partially supported by Projects DPI2017-87333-R and UJI-B2017-13. We would also like to thank the Biomechanics Institute of Valencia for providing us with the data set.

Author information

Authors and Affiliations

  1. Department of Mathematics, Universitat Jaume I, Avda. del Riu Sec s/n, 12071, Castellón, Spain

    S. Barahona & P. Centella

  2. Department of Mathematics, INIT, Universitat Jaume I, Avda. del Riu Sec s/n, 12071, Castellón, Spain

    X. Gual-Arnau

  3. Department of Mathematics, IMAC, Universitat Jaume I, Avda. del Riu Sec s/n, 12071, Castellón, Spain

    M. V. Ibáñez & A. Simó

Authors
  1. S. Barahona
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Centella
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. Gual-Arnau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. V. Ibáñez
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Simó
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Ibáñez.

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

Barahona, S., Centella, P., Gual-Arnau, X. et al. Supervised classification of geometrical objects by integrating currents and functional data analysis. TEST 29, 637–660 (2020). https://doi.org/10.1007/s11749-019-00669-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-019-00669-z

Keywords

Mathematics Subject Classification

Search

Navigation