IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i8p820-d533021.html
   My bibliography  Save this article

Functional Symmetry and Statistical Depth for the Analysis of Movement Patterns in Alzheimer’s Patients

Author

Listed:
  • Alicia Nieto-Reyes

    (Department of Mathematics, Statistics and Computer Science, University of Cantabria, 39005 Santander, Spain
    Current address: Faculty of Science, Avd. Los Castros s/n, 39005 Santander, Spain.)

  • Heather Battey

    (Department of Mathematics, Imperial College London, London SW7 2BX, UK)

  • Giacomo Francisci

    (Department of Mathematics, Statistics and Computer Science, University of Cantabria, 39005 Santander, Spain
    Department of Mathematics, University of Trento, 38122 Trento, Italy)

Abstract

Black-box techniques have been applied with outstanding results to classify, in a supervised manner, the movement patterns of Alzheimer’s patients according to their stage of the disease. However, these techniques do not provide information on the difference of the patterns among the stages. We make use of functional data analysis to provide insight on the nature of these differences. In particular, we calculate the center of symmetry of the underlying distribution at each stage and use it to compute the functional depth of the movements of each patient. This results in an ordering of the data to which we apply nonparametric permutation tests to check on the differences in the distribution, median and deviance from the median. We consistently obtain that the movement pattern at each stage is significantly different to that of the prior and posterior stage in terms of the deviance from the median applied to the depth. The approach is validated by simulation.

Suggested Citation

  • Alicia Nieto-Reyes & Heather Battey & Giacomo Francisci, 2021. "Functional Symmetry and Statistical Depth for the Analysis of Movement Patterns in Alzheimer’s Patients," Mathematics, MDPI, vol. 9(8), pages 1-17, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:8:p:820-:d:533021
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/8/820/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/8/820/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. López-Pintado, Sara & Romo, Juan, 2009. "On the Concept of Depth for Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 718-734.
    2. Nieto-Reyes, Alicia & Battey, Heather, 2021. "A topologically valid construction of depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    3. López-Pintado, Sara & Romo, Juan, 2011. "A half-region depth for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1679-1695, April.
    4. Antonio Cuevas & Manuel Febrero & Ricardo Fraiman, 2007. "Robust estimation and classification for functional data via projection-based depth notions," Computational Statistics, Springer, vol. 22(3), pages 481-496, September.
    5. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
    6. Anirvan Chakraborty & Probal Chaudhuri, 2014. "On data depth in infinite dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(2), pages 303-324, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Carmen Lacave & Ana Isabel Molina, 2023. "Advances in Artificial Intelligence and Statistical Techniques with Applications to Health and Education," Mathematics, MDPI, vol. 11(6), pages 1-4, March.
    2. Luis González-De La Fuente & Alicia Nieto-Reyes & Pedro Terán, 2022. "Properties of Statistical Depth with Respect to Compact Convex Random Sets: The Tukey Depth," Mathematics, MDPI, vol. 10(15), pages 1-23, August.

    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. Nieto-Reyes, Alicia & Battey, Heather, 2021. "A topologically valid construction of depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    2. Carlo Sguera & Sara López-Pintado, 2021. "A notion of depth for sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 630-649, September.
    3. Hernández Banadik, Nicolás Jorge & Muñoz García, Alberto, 2017. "Kernel depth funcions for functional data," DES - Working Papers. Statistics and Econometrics. WS 24615, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.
    5. Jiménez Recaredo, Raúl José & Elías Fernández, Antonio, 2017. "Prediction Bands for Functional Data Based on Depth Measures," DES - Working Papers. Statistics and Econometrics. WS 24606, Universidad Carlos III de Madrid. Departamento de Estadística.
    6. Carlo Sguera & Pedro Galeano & Rosa Lillo, 2014. "Spatial depth-based classification for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(4), pages 725-750, December.
    7. Alba M. Franco-Pereira & Rosa E. Lillo, 2020. "Rank tests for functional data based on the epigraph, the hypograph and associated graphical representations," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(3), pages 651-676, September.
    8. Lucas Fernandez-Piana & Marcela Svarc, 2022. "An integrated local depth measure," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(2), pages 175-197, June.
    9. Cleveland, Jason & Zhao, Weilong & Wu, Wei, 2018. "Robust template estimation for functional data with phase variability using band depth," Computational Statistics & Data Analysis, Elsevier, vol. 125(C), pages 10-26.
    10. Anirvan Chakraborty & Probal Chaudhuri, 2014. "On data depth in infinite dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(2), pages 303-324, April.
    11. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    12. Miguel Flores & Salvador Naya & Rubén Fernández-Casal & Sonia Zaragoza & Paula Raña & Javier Tarrío-Saavedra, 2020. "Constructing a Control Chart Using Functional Data," Mathematics, MDPI, vol. 8(1), pages 1-26, January.
    13. Nagy, Stanislav & Ferraty, Frédéric, 2019. "Data depth for measurable noisy random functions," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 95-114.
    14. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    15. López-Pintado, Sara & Romo, Juan, 2011. "A half-region depth for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1679-1695, April.
    16. Serfling, Robert & Wijesuriya, Uditha, 2017. "Depth-based nonparametric description of functional data, with emphasis on use of spatial depth," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 24-45.
    17. Valencia García, Dalia Jazmin & Lillo Rodríguez, Rosa Elvira & Romo, Juan, 2013. "Spearman coefficient for functions," DES - Working Papers. Statistics and Econometrics. WS ws133329, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast DD-classification of functional data," Statistical Papers, Springer, vol. 58(4), pages 1055-1089, December.
    19. Kuelbs, James & Zinn, Joel, 2015. "Half-region depth for stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 86-105.
    20. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.

    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:gam:jmathe:v:9:y:2021:i:8:p:820-:d:533021. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.