IDEAS home Printed from https://ideas.repec.org/a/bla/biomet/v73y2017i4p1092-1101.html
   My bibliography  Save this article

Pointwise influence matrices for functional‐response regression

Author

Listed:
  • Philip T. Reiss
  • Lei Huang
  • Pei‐Shien Wu
  • Huaihou Chen
  • Stan Colcombe

Abstract

We extend the notion of an influence or hat matrix to regression with functional responses and scalar predictors. For responses depending linearly on a set of predictors, our definition is shown to reduce to the conventional influence matrix for linear models. The pointwise degrees of freedom, the trace of the pointwise influence matrix, are shown to have an adaptivity property that motivates a two‐step bivariate smoother for modeling nonlinear dependence on a single predictor. This procedure adapts to varying complexity of the nonlinear model at different locations along the function, and thereby achieves better performance than competing tensor product smoothers in an analysis of the development of white matter microstructure in the brain.

Suggested Citation

  • Philip T. Reiss & Lei Huang & Pei‐Shien Wu & Huaihou Chen & Stan Colcombe, 2017. "Pointwise influence matrices for functional‐response regression," Biometrics, The International Biometric Society, vol. 73(4), pages 1092-1101, December.
    • Handle: RePEc:bla:biomet:v:73:y:2017:i:4:p:1092-1101
    DOI: 10.1111/biom.12697
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/biom.12697
    Download Restriction: no
    File URL: https://libkey.io/10.1111/biom.12697?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
    ---><---

    References listed on IDEAS

    as
    1. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167.
    2. Simon N. Wood, 2006. "Low-Rank Scale-Invariant Tensor Product Smooths for Generalized Additive Mixed Models," Biometrics, The International Biometric Society, vol. 62(4), pages 1025-1036, December.
    3. Reiss Philip T. & Huang Lei & Mennes Maarten, 2010. "Fast Function-on-Scalar Regression with Penalized Basis Expansions," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-30, August.
    4. Simon N. Wood, 2011. "Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 3-36, January.
    5. Jeng‐Min Chiou & Hans‐Georg Müller & Jane‐Ling Wang, 2003. "Functional quasi‐likelihood regression models with smooth random effects," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 405-423, May.
    6. Luo Xiao & Yingxing Li & David Ruppert, 2013. "Fast bivariate P-splines: the sandwich smoother," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 577-599, June.
    7. repec:wyi:journl:002174 is not listed on IDEAS
    8. Hongtu Zhu & Jianqing Fan & Linglong Kong, 2014. "Spatially Varying Coefficient Model for Neuroimaging Data With Jump Discontinuities," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1084-1098, September.
    9. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506.
    10. Lucas Janson & William Fithian & Trevor J. Hastie, 2015. "Effective degrees of freedom: a flawed metaphor," Biometrika, Biometrika Trust, vol. 102(2), pages 479-485.
    11. Philip T. Reiss & R. Todd Ogden, 2009. "Smoothing parameter selection for a class of semiparametric linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 505-523, April.
    12. J. Fan & J.‐T. Zhang, 2000. "Two‐step estimation of functional linear models with applications to longitudinal data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(2), pages 303-322.
    13. S. Kaufman & S. Rosset, 2014. "When does more regularization imply fewer degrees of freedom? Sufficient conditions and counterexamples," Biometrika, Biometrika Trust, vol. 101(4), pages 771-784.
    14. Ciprian M. Crainiceanu & David Ruppert, 2004. "Likelihood ratio tests in linear mixed models with one variance component," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 165-185, February.
    15. Krivobokova, Tatyana & Kauermann, Goran, 2007. "A Note on Penalized Spline Smoothing With Correlated Errors," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1328-1337, December.
    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. María Xosé Rodríguez‐Álvarez & María Durbán & Paul H.C. Eilers & Dae‐Jin Lee & Francisco Gonzalez, 2023. "Multidimensional adaptive P‐splines with application to neurons' activity studies," Biometrics, The International Biometric Society, vol. 79(3), pages 1972-1985, September.

    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. Andrada Ivanescu & Ana-Maria Staicu & Fabian Scheipl & Sonja Greven, 2015. "Penalized function-on-function regression," Computational Statistics, Springer, vol. 30(2), pages 539-568, June.
    2. Øystein Sørensen & Anders M. Fjell & Kristine B. Walhovd, 2023. "Longitudinal Modeling of Age-Dependent Latent Traits with Generalized Additive Latent and Mixed Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 456-486, June.
    3. Reiss Philip T. & Huang Lei, 2012. "Smoothness Selection for Penalized Quantile Regression Splines," The International Journal of Biostatistics, De Gruyter, vol. 8(1), pages 1-27, May.
    4. Roel Verbelen & Katrien Antonio & Gerda Claeskens, 2018. "Unravelling the predictive power of telematics data in car insurance pricing," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 67(5), pages 1275-1304, November.
    5. Philip T. Reiss & Jeff Goldsmith & Han Lin Shang & R. Todd Ogden, 2017. "Methods for Scalar-on-Function Regression," International Statistical Review, International Statistical Institute, vol. 85(2), pages 228-249, August.
    6. Huaihou Chen & Philip T. Reiss & Thaddeus Tarpey, 2014. "Optimally weighted L-super-2 distance for functional data," Biometrics, The International Biometric Society, vol. 70(3), pages 516-525, September.
    7. Zanin, Luca & Marra, Giampiero, 2012. "Assessing the functional relationship between CO2 emissions and economic development using an additive mixed model approach," Economic Modelling, Elsevier, vol. 29(4), pages 1328-1337.
    8. Longhi, Christian & Musolesi, Antonio & Baumont, Catherine, 2014. "Modeling structural change in the European metropolitan areas during the process of economic integration," Economic Modelling, Elsevier, vol. 37(C), pages 395-407.
    9. Simon N. Wood & Natalya Pya & Benjamin Säfken, 2016. "Smoothing Parameter and Model Selection for General Smooth Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1548-1563, October.
    10. Lauren N. Berry & Nathaniel E. Helwig, 2021. "Cross-Validation, Information Theory, or Maximum Likelihood? A Comparison of Tuning Methods for Penalized Splines," Stats, MDPI, vol. 4(3), pages 1-24, September.
    11. Mestekemper, Thomas & Kauermann, Göran & Smith, Michael S., 2013. "A comparison of periodic autoregressive and dynamic factor models in intraday energy demand forecasting," International Journal of Forecasting, Elsevier, vol. 29(1), pages 1-12.
    12. Philip T. Reiss & R. Todd Ogden, 2010. "Functional Generalized Linear Models with Images as Predictors," Biometrics, The International Biometric Society, vol. 66(1), pages 61-69, March.
    13. Christian Schellhase & Göran Kauermann, 2012. "Density estimation and comparison with a penalized mixture approach," Computational Statistics, Springer, vol. 27(4), pages 757-777, December.
    14. Simon N. Wood & Zheyuan Li & Gavin Shaddick & Nicole H. Augustin, 2017. "Generalized Additive Models for Gigadata: Modeling the U.K. Black Smoke Network Daily Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1199-1210, July.
    15. Cederbaum, Jona & Scheipl, Fabian & Greven, Sonja, 2018. "Fast symmetric additive covariance smoothing," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 25-41.
    16. Arūnas P. Verbyla & Joanne Faveri & John D. Wilkie & Tom Lewis, 2018. "Tensor Cubic Smoothing Splines in Designed Experiments Requiring Residual Modelling," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 23(4), pages 478-508, December.
    17. Takuma Yoshida, 2016. "Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(4), pages 278-303, November.
    18. Lee, Wang-Sheng, 2014. "Big and Tall: Is there a Height Premium or Obesity Penalty in the Labor Market?," IZA Discussion Papers 8606, Institute of Labor Economics (IZA).
    19. Ghosal, Rahul & Maity, Arnab, 2022. "A Score Based Test for Functional Linear Concurrent Regression," Econometrics and Statistics, Elsevier, vol. 21(C), pages 114-130.
    20. 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).

    More about this item

    Statistics

    Access and download statistics

    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:bla:biomet:v:73:y:2017:i:4:p:1092-1101. 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: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0006-341X .

    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.