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The rapid advancement of functional data in various application fields has increased the demand for advanced statistical approaches that can incorporate complex structures and nonlinear associations. In this article, we propose a novel functional random forests (FunFor) approach to model the functional data response that is densely and regularly measured, as an extension of the landmark work of Breiman, who introduced traditional random forests for a univariate response. The FunFor approach is able to predict curve responses for new observations and selects important variables from a large set of scalar predictors. The FunFor approach inherits the efficiency of the traditional random forest approach in detecting complex relationships, including nonlinear and high-order interactions. Additionally, it is a non-parametric approach without the imposition of parametric and distributional assumptions. Eight simulation settings and one real-data analysis consistently demonstrate the excellent performance of the FunFor approach in various scenarios. In particular, FunFor successfully ranks the true predictors as the most important variables, while achieving the most robust variable sections and the smallest prediction errors when comparing it with three other relevant approaches. Although motivated by a biological leaf shape data analysis, the proposed FunFor approach has great potential to be widely applied in various fields due to its minimal requirement on tuning parameters and its distribution-free and model-free nature. An R package named ’FunFor’, implementing the FunFor approach, is available at GitHub.

In many practices, scientists are particularly interested in detecting which of the predictors are truly associated with a multivariate response. It is more accurate to model multiple responses as one vector rather than separating each component one by one. This is particularly true for complex traits having multiple correlated components. A Bayesian multivariate variable selection (BMVS) approach is proposed to select important predictors influencing the multivariate response from a candidate pool with ultrahigh dimension. By applying the sample‐size‐dependent spike and slab priors, the BMVS approach satisfies the strong selection consistency property under certain conditions, which represents the advantages of BMVS over other existing Bayesian multivariate regression‐based approaches. The proposed approach considers the covariance structure of multiple responses without assuming independence and integrates the estimation of covariance‐related parameters together with all regression parameters into one framework through a fast‐updating Markov chain Monte Carlo (MCMC) procedure. It is demonstrated through simulations that the BMVS approach outperforms some other relevant frequentist and Bayesian approaches. The proposed BMVS approach possesses a flexibility of wide applications, including genome‐wide association studies with multiple correlated phenotypes and a large scale of genetic variants and/or environmental variables, as demonstrated in the real data analyses section. The computer code and test data of the proposed method are available as an R package.