Partially Linear Functional Additive Models for Multivariate Functional Data

ABSTRACT

We investigate a class of partially linear functional additive models (PLFAM) that predicts a scalar response by both parametric effects of a multivariate predictor and nonparametric effects of a multivariate functional predictor. We jointly model multiple functional predictors that are cross-correlated using multivariate functional principal component analysis (mFPCA), and model the nonparametric effects of the principal component scores as additive components in the PLFAM. To address the high-dimensional nature of functional data, we let the number of mFPCA components diverge to infinity with the sample size, and adopt the component selection and smoothing operator (COSSO) penalty to select relevant components and regularize the fitting. A fundamental difference between our framework and the existing high-dimensional additive models is that the mFPCA scores are estimated with error, and the magnitude of measurement error increases with the order of mFPCA. We establish the asymptotic convergence rate for our estimator, while allowing the number of components diverge. When the number of additive components is fixed, we also establish the asymptotic distribution for the partially linear coefficients. The practical performance of the proposed methods is illustrated via simulation studies and a crop yield prediction application. Supplementary materials for this article are available online.

KEYWORDS:

• Additive model
• Functional data
• Measurement error
• Principal component analysis
• Reproducing kernel Hilbert space
• Spline

Supplemental Material

The online supplementary material contains technical proofs for the theorems, additional simulation and data analysis results, a bootstrap procedure for statistical inference, as well as R codes for the proposed methodology.

Acknowledgments

Wong’s research was partially supported by the National Science Foundation under Grants DMS-1612985 and DMS-1711952 (subcontract). Li’s research was partially supported by the National Science Foundation grant DMS-1317118. Zhu’s research was funded in part by cooperative agreement 68-7482-17-009 between the USDA Natural Resources Conservation Service and Iowa State University.

Additional information

Funding

Division of Mathematical Sciences DMS-1317118