title: Pointwise influence matrices for functional-response regression.
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.
authors.affiliation: Department of Child and Adolescent Psychiatry, New York University School of Medicine, New York, U.S.A.|Department of Population Health, New York University School of Medicine, New York, U.S.A.|Department of Statistics, University of Haifa, Haifa, Israel.
authors.forename: Philip T
authors.initials: PT
authors.lastname: Reiss
authors.affiliation: Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, U.S.A.
authors.forename: Lei
authors.initials: L
authors.lastname: Huang
authors.affiliation: Department of Child and Adolescent Psychiatry, New York University School of Medicine, New York, U.S.A.
authors.forename: Pei-Shien
authors.initials: PS
authors.lastname: Wu
authors.affiliation: Department of Biostatistics, University of Florida, Gainesville, U.S.A.
authors.forename: Huaihou
authors.initials: H
authors.lastname: Chen
authors.affiliation: Nathan S. Kline Institute for Psychiatric Research, Orangeburg, New York, U.S.A.
authors.forename: Stan
authors.initials: S
authors.lastname: Colcombe
mesh_terms: D001921:Brain / Q000648:ultrastructure*; D006801:Humans; D016014:Linear Models; D015233:Models, Statistical; D066127:White Matter / Q000254:growth & development* / Q000648:ultrastructure
keywords: Bivariate smoothing; Degrees of freedom; Fractional anisotropy; Function-on-scalar regression; Functional nonlinear regression; Neurodevelopmental trajectory; Tensor product spline