Pointwise influence matrices for functional-response regression

Philip T. Reiss, Lei Huang, Pei Shien Wu, Huaihou Chen, Stan Colcombe

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1092-1101
Number of pages10
Issue number4
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2017, The International Biometric Society


  • Bivariate smoothing
  • Degrees of freedom
  • Fractional anisotropy
  • Function-on-scalar regression
  • Functional nonlinear regression
  • Neurodevelopmental trajectory
  • Tensor product spline

ASJC Scopus subject areas

  • General Agricultural and Biological Sciences
  • Applied Mathematics
  • General Biochemistry, Genetics and Molecular Biology
  • General Immunology and Microbiology
  • Statistics and Probability


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