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.

Functional Additive Quantile Regression

“…The eigenvalues, eigenfunctions and FPC scores are estimated by replacing X i (t) with estimatedX i (t). For detailed algorithm, the readers can refer to section 3.1 in Wong et al (2018). The estimated transformed FPC scores are denoted byζ i , which serve as the predictors in the following.…”

“…In addition, it assumes the eigenvalues decay at a polynomial rate. Under this condition, we have the following lemma is Proposition 1 in Wong et al (2018).…”

“…This condition was previously used in Wong et al (2018). It imposes a weak moment condition on the functional predictor and is satisfied if X(t) is a Gaussian process.…”

“…Furthermore, Wong et al (2018) extended the model to partial linear functional additive regression with multivariate functional predictors. There are fewer works for conditonal quantile modeling with functional predictors.…”

Estimation of Sparse Functional Additive Models with Adaptive Group LASSO

“…where SSE = (y − Z S β) likely to improve prediction accuracy in practice (Sang et al, 2018;Wong et al, 2018). Therefore, when prediction of a scalar response is the main goal, it would be desirable to incorporate both scalar predictors and multiple functional predictors into the current model structure using adaptive group LASSO and smoothing spline for estimation.…”