IDEAS home Printed from https://ideas.repec.org/a/spr/sankha/v81y2019i1d10.1007_s13171-018-0137-7.html
   My bibliography  Save this article

Challenges in Topological Object Data Analysis

Author

Listed:
  • Vic Patrangenaru

    (Florida State University)

  • Peter Bubenik

    (University of Florida)

  • Robert L. Paige

    (Missouri S & T)

  • Daniel Osborne

    (Florida A&M University)

Abstract

Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using death vectors and persistence landscapes to vectorize object data and perform statistical analysis. We apply this method to some common leaf images that were previously shown to be challenging to compare using a 3D shape techniques. Surprisingly, the most persistent features are shown to be “topological noise” and the statistical analysis depends on the less persistent features which we refer to as the “geometric signal”. We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces. We introduce a new Fréchet-Morse function technique for probability distribution on a compact object space, extending the Fréchet means lo a larger number of location parameters, including Fréchet antimeans. An example of 3D data analysis to distinguish two flowers using the new location parameters associated with a Veronese-Whitney (VW) embedding of random projective shapes of 3D configurations extracted from a set of pairs of their digital camera images is also given here.

Suggested Citation

  • Vic Patrangenaru & Peter Bubenik & Robert L. Paige & Daniel Osborne, 2019. "Challenges in Topological Object Data Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 244-271, February.
    • Handle: RePEc:spr:sankha:v:81:y:2019:i:1:d:10.1007_s13171-018-0137-7
    DOI: 10.1007/s13171-018-0137-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13171-018-0137-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13171-018-0137-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ellingson, Leif & Patrangenaru, Vic & Ruymgaart, Frits, 2013. "Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 317-333.
    2. R. N. Bhattacharya & L. Ellingson & X. Liu & V. Patrangenaru & M. Crane, 2012. "Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machine vision," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 28(3), pages 222-235, May.
    3. Harman, Radoslav & Lacko, Vladimír, 2010. "On decompositional algorithms for uniform sampling from n-spheres and n-balls," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2297-2304, November.
    4. Crane, M. & Patrangenaru, V., 2011. "Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 225-237, February.
    5. Victor Patrangenaru & Robert Paige & K. David Yao & Mingfei Qiu & David Lester, 2016. "Projective shape analysis of contours and finite 3D configurations from digital camera images," Statistical Papers, Springer, vol. 57(4), pages 1017-1040, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Victor Patrangenaru & Robert Paige & K. David Yao & Mingfei Qiu & David Lester, 2016. "Projective shape analysis of contours and finite 3D configurations from digital camera images," Statistical Papers, Springer, vol. 57(4), pages 1017-1040, December.
    2. Ruite Guo & Hwiyoung Lee & Vic Patrangenaru, 2023. "Test for Homogeneity of Random Objects on Manifolds with Applications to Biological Shape Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1178-1204, August.
    3. Fabian J.E. Telschow & Michael R. Pierrynowski & Stephan F. Huckemann, 2021. "Functional inference on rotational curves under sample‐specific group actions and identification of human gait," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(4), pages 1256-1276, December.
    4. Vic Patrangenaru & Mingfei Qiu & Marius Buibas, 2014. "Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 485-506, June.
    5. Wolf-Dieter Richter, 2019. "On (p1,…,pk)-spherical distributions," Journal of Statistical Distributions and Applications, Springer, vol. 6(1), pages 1-18, December.
    6. Filippozzi, Rafaela & Gonçalves, Douglas S. & Santos, Luiz-Rafael, 2023. "First-order methods for the convex hull membership problem," European Journal of Operational Research, Elsevier, vol. 306(1), pages 17-33.
    7. Vic Patrangenaru & Yifang Deng, 2021. "Extrinsic Regression and Anti-Regression on Projective Shape Manifolds," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 629-646, June.
    8. Alexandru Agapie, 2021. "Spherical Distributions Used in Evolutionary Algorithms," Mathematics, MDPI, vol. 9(23), pages 1-15, November.
    9. Benoit Ahanda & Daniel E. Osborne & Leif Ellingson, 2022. "Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions," METRON, Springer;Sapienza Università di Roma, vol. 80(3), pages 281-303, December.
    10. Planas, Christophe & Rossi, Alessandro, 2018. "The slice sampler and centrally symmetric distributions," JRC Working Papers in Economics and Finance 2018-11, Joint Research Centre, European Commission.
    11. Yamazoe, Hiroya & Naito, Kanta, 2024. "Simultaneous confidence region of an embedded one-dimensional curve in multi-dimensional space," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    12. Luca Frigau & Claudio Conversano & Francesco Mola, 2021. "Consistent validation of gray-level thresholding image segmentation algorithms based on machine learning classifiers," Statistical Papers, Springer, vol. 62(3), pages 1363-1386, June.
    13. Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.
    14. Jean-Thomas Camino & Christian Artigues & Laurent Houssin & Stéphane Mourgues, 2019. "Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design," Computational Optimization and Applications, Springer, vol. 73(2), pages 679-705, June.
    15. Planas Christophe & Rossi Alessandro, 2024. "The slice sampler and centrally symmetric distributions," Monte Carlo Methods and Applications, De Gruyter, vol. 30(3), pages 299-313.
    16. Luis Gutiérrez & Ramsés H. Mena & Carlos Díaz-Avalos, 2020. "Linear models for statistical shape analysis based on parametrized closed curves," Statistical Papers, Springer, vol. 61(3), pages 1213-1229, June.
    17. Bhattacharya, Rabi & Oliver, Rachel, 2020. "Superiority of Bayes estimators over the MLE in high dimensional multinomial models and its implication for nonparametric Bayes theory," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    18. Abadir, Karim M. & Luati, Alessandra & Paruolo, Paolo, 2023. "GARCH density and functional forecasts," Journal of Econometrics, Elsevier, vol. 235(2), pages 470-483.
    19. Vladimír Lacko & Radoslav Harman, 2012. "A conditional distribution approach to uniform sampling on spheres and balls in L p spaces," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(7), pages 939-951, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sankha:v:81:y:2019:i:1:d:10.1007_s13171-018-0137-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.