Dataset Information

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework.

ABSTRACT: Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the Representer Theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study.

SUBMITTER: Sun X

PROVIDER: S-EPMC6380514 | biostudies-literature | 2018

REPOSITORIES: biostudies-literature

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Publications

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework.

Sun Xiaoxiao X Du Pang P Wang Xiao X Ma Ping P

Journal of the American Statistical Association 20180619 524

Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squ ...[more]

PMID: 30799886

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Dataset Information

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework.

Publications

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework.

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