IDEAS home Printed from https://ideas.repec.org/a/spr/testjl/v25y2016i1d10.1007_s11749-015-0470-2.html
   My bibliography  Save this article

Probability-enhanced effective dimension reduction for classifying sparse functional data

Author

Listed:
  • Fang Yao

    (University of Toronto)

  • Yichao Wu

    (North Carolina State University)

  • Jialin Zou

    (University of Toronto)

Abstract

We consider the classification of sparse functional data that are often encountered in longitudinal studies and other scientific experiments. To utilize the information from not only the functional trajectories but also the observed class labels, we propose a probability-enhanced method achieved by weighted support vector machine based on its Fisher consistency property to estimate the effective dimension reduction space. Since only a few measurements are available for some, even all, individuals, a cumulative slicing approach is suggested to borrow information across individuals. We provide justification for validity of the probability-based effective dimension reduction space, and a straightforward implementation that yields a low-dimensional projection space ready for applying standard classifiers. The empirical performance is illustrated through simulated and real examples, particularly in contrast to classification results based on the prominent functional principal component analysis.

Suggested Citation

  • Fang Yao & Yichao Wu & Jialin Zou, 2016. "Probability-enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 1-22, March.
    • Handle: RePEc:spr:testjl:v:25:y:2016:i:1:d:10.1007_s11749-015-0470-2
    DOI: 10.1007/s11749-015-0470-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11749-015-0470-2
    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/s11749-015-0470-2?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. F. Yao & E. Lei & Y. Wu, 2015. "Effective dimension reduction for sparse functional data," Biometrika, Biometrika Trust, vol. 102(2), pages 421-437.
    2. Kneip, Alois & Ramsay, James O, 2008. "Combining Registration and Fitting for Functional Models," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 1155-1165.
    3. Seung Jun Shin & Yichao Wu & Hao Helen Zhang & Yufeng Liu, 2014. "Probability-enhanced sufficient dimension reduction for binary classification," Biometrics, The International Biometric Society, vol. 70(3), pages 546-555, September.
    4. Morris, Jeffrey S. & Vannucci, Marina & Brown, Philip J. & Carroll, Raymond J., 2003. "Wavelet-Based Nonparametric Modeling of Hierarchical Functions in Colon Carcinogenesis," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 573-583, January.
    5. Junhui Wang & Xiaotong Shen & Yufeng Liu, 2008. "Probability estimation for large-margin classifiers," Biometrika, Biometrika Trust, vol. 95(1), pages 149-167.
    6. YichaoWu, & Liu, Yufeng, 2007. "Robust Truncated Hinge Loss Support Vector Machines," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 974-983, September.
    7. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    8. Zhou, Lan & Huang, Jianhua Z. & Martinez, Josue G. & Maity, Arnab & Baladandayuthapani, Veerabhadran & Carroll, Raymond J., 2010. "Reduced Rank Mixed Effects Models for Spatially Correlated Hierarchical Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 390-400.
    9. Wensheng Guo, 2002. "Functional Mixed Effects Models," Biometrics, The International Biometric Society, vol. 58(1), pages 121-128, March.
    10. 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.
    11. Francesca Chiaromonte & R. Cook, 2002. "Sufficient Dimension Reduction and Graphics in Regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 768-795, December.
    12. Fang Yao & Thomas C. M. Lee, 2006. "Penalized spline models for functional principal component analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 3-25, February.
    13. Jeffrey S. Morris & Raymond J. Carroll, 2006. "Wavelet‐based functional mixed models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 179-199, April.
    14. Gareth M. James, 2002. "Generalized linear models with functional predictors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 411-432, August.
    15. Aurore Delaigle & Peter Hall, 2012. "Achieving near perfect classification for functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(2), pages 267-286, March.
    16. John A. Rice & Colin O. Wu, 2001. "Nonparametric Mixed Effects Models for Unequally Sampled Noisy Curves," Biometrics, The International Biometric Society, vol. 57(1), pages 253-259, March.
    17. Philippe Besse & J. Ramsay, 1986. "Principal components analysis of sampled functions," Psychometrika, Springer;The Psychometric Society, vol. 51(2), pages 285-311, June.
    18. Daniel Gervini & Theo Gasser, 2004. "Self‐modelling warping functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(4), pages 959-971, November.
    19. Ferraty, F. & Vieu, P., 2003. "Curves discrimination: a nonparametric functional approach," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 161-173, October.
    20. Li, Bing & Wang, Shaoli, 2007. "On Directional Regression for Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 997-1008, September.
    21. Boente, Graciela & Fraiman, Ricardo, 2000. "Kernel-based functional principal components," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 335-345, July.
    22. He, Guozhong & Müller, Hans-Georg & Wang, Jane-Ling, 2003. "Functional canonical analysis for square integrable stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 54-77, April.
    23. Kneip A. & Utikal K. J, 2001. "Inference for Density Families Using Functional Principal Component Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 519-542, June.
    24. Peter Hall & Mohammad Hosseini‐Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126, February.
    25. Wang, Lifeng & Shen, Xiaotong, 2007. "On L1-Norm Multiclass Support Vector Machines: Methodology and Theory," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 583-594, June.
    26. J. O. Ramsay & G. Hooker & D. Campbell & J. Cao, 2007. "Parameter estimation for differential equations: a generalized smoothing approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 741-796, November.
    27. Zhu, Li-Ping & Zhu, Li-Xing & Feng, Zheng-Hui, 2010. "Dimension Reduction in Regressions Through Cumulative Slicing Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1455-1466.
    28. Gareth M. James & Trevor J. Hastie, 2001. "Functional linear discriminant analysis for irregularly sampled curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(3), pages 533-550.
    29. Daniel Gervini & Theo Gasser, 2005. "Nonparametric maximum likelihood estimation of the structural mean of a sample of curves," Biometrika, Biometrika Trust, vol. 92(4), pages 801-820, December.
    30. Lee, Yoonkyung & Lin, Yi & Wahba, Grace, 2004. "Multicategory Support Vector Machines: Theory and Application to the Classification of Microarray Data and Satellite Radiance Data," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 67-81, January.
    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. Zhang, Xin & Wang, Chong & Wu, Yichao, 2018. "Functional envelope for model-free sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 37-50.
    2. Alvarez, Agustín & Boente, Graciela & Kudraszow, Nadia, 2019. "Robust sieve estimators for functional canonical correlation analysis," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 46-62.
    3. Rodney V. Fonseca & Aluísio Pinheiro, 2020. "Wavelet estimation of the dimensionality of curve time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1175-1204, October.
    4. Chen, Di-Rong & Cheng, Kun & Liu, Chao, 2022. "Framelet block thresholding estimator for sparse functional data," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    5. Li, Pai-Ling & Chiou, Jeng-Min & Shyr, Yu, 2017. "Functional data classification using covariate-adjusted subspace projection," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 21-34.

    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. Fang Yao & Yichao Wu & Jialin Zou, 2016. "Probability-enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 1-22, March.
    2. Chen, Lu-Hung & Jiang, Ci-Ren, 2018. "Sensible functional linear discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 126(C), pages 39-52.
    3. Hans-Georg Müller & Wenjing Yang, 2010. "Dynamic relations for sparsely sampled Gaussian processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 1-29, May.
    4. Han Shang, 2014. "A survey of functional principal component analysis," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(2), pages 121-142, April.
    5. Li, Yehua & Qiu, Yumou & Xu, Yuhang, 2022. "From multivariate to functional data analysis: Fundamentals, recent developments, and emerging areas," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    6. Guochang Wang & Xinyuan Song, 2018. "Functional Sufficient Dimension Reduction for Functional Data Classification," Journal of Classification, Springer;The Classification Society, vol. 35(2), pages 250-272, July.
    7. Hongxiao Zhu & Philip J. Brown & Jeffrey S. Morris, 2012. "Robust Classification of Functional and Quantitative Image Data Using Functional Mixed Models," Biometrics, The International Biometric Society, vol. 68(4), pages 1260-1268, December.
    8. Kyunghee Han & Pantelis Z Hadjipantelis & Jane-Ling Wang & Michael S Kramer & Seungmi Yang & Richard M Martin & Hans-Georg Müller, 2018. "Functional principal component analysis for identifying multivariate patterns and archetypes of growth, and their association with long-term cognitive development," PLOS ONE, Public Library of Science, vol. 13(11), pages 1-18, November.
    9. repec:cte:wsrepe:ws131312 is not listed on IDEAS
    10. Daniel Gervini & Patrick A. Carter, 2014. "Warped functional analysis of variance," Biometrics, The International Biometric Society, vol. 70(3), pages 526-535, September.
    11. Manuel Febrero-Bande, 2016. "Comments on: Probability enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 35-40, March.
    12. Mousavi, Seyed Nourollah & Sørensen, Helle, 2017. "Multinomial functional regression with wavelets and LASSO penalization," Econometrics and Statistics, Elsevier, vol. 1(C), pages 150-166.
    13. Manuel Febrero-Bande, 2016. "Comments on: Probability enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 35-40, March.
    14. Qin Wang & Yuan Xue, 2023. "A structured covariance ensemble for sufficient dimension reduction," 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. 17(3), pages 777-800, September.
    15. Ming-Yueh Huang & Chin-Tsang Chiang, 2017. "An Effective Semiparametric Estimation Approach for the Sufficient Dimension Reduction Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1296-1310, July.
    16. Park, Yeonjoo & Simpson, Douglas G., 2019. "Robust probabilistic classification applicable to irregularly sampled functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 37-49.
    17. Lakraj, Gamage Pemantha & Ruymgaart, Frits, 2017. "Some asymptotic theory for Silverman’s smoothed functional principal components in an abstract Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 122-132.
    18. Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast DD-classification of functional data," Statistical Papers, Springer, vol. 58(4), pages 1055-1089, December.
    19. Zhang, Zhen & Müller, Hans-Georg, 2011. "Functional density synchronization," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2234-2249, July.
    20. Tengteng Xu & Riquan Zhang & Xiuzhen Zhang, 2023. "Estimation of spatial-functional based-line logit model for multivariate longitudinal data," Computational Statistics, Springer, vol. 38(1), pages 79-99, March.
    21. Peter Hall & You‐Jun Yang, 2010. "Ordering and selecting components in multivariate or functional data linear prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 93-110, January.

    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:testjl:v:25:y:2016:i:1:d:10.1007_s11749-015-0470-2. 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.