Abstract
Motivated by mobile devices that record data at a high frequency, we propose a new methodological framework for analyzing a semi-parametric regression model that allow us to study a nonlinear relationship between a scalar response and multiple functional predictors in the presence of scalar covariates. Utilizing functional principal component analysis (FPCA) and the least-squares kernel machine method (LSKM), we are able to substantially extend the framework of semi-parametric regression models of scalar responses on scalar predictors by allowing multiple functional predictors to enter the nonlinear model. Regularization is established for feature selection in the setting of reproducing kernel Hilbert spaces. Our method performs simultaneously model fitting and variable selection on functional features. For the implementation, we propose an effective algorithm to solve related optimization problems in that iterations take place between both linear mixed-effects models and a variable selection method (e.g., sparse group lasso). We show algorithmic convergence results and theoretical guarantees for the proposed methodology. We illustrate its performance through simulation experiments and an analysis of accelerometer data.
Highlights
Data captured by mobile devices have lately received much attention in the data science community
We focus on the utility of functional principal component analysis (FPCA) to perform the decomposition of the functional Z
It is worth noting that the linear model together with the sparse group lasso (SGL) penalty selected the highest number of FPC components, yet performed the worst in terms of the model fit
Summary
Data captured by mobile devices have lately received much attention in the data science community. Oftentimes, different types of summaries of the tri-axis ACs are suggested in the literature as opposed to the utility of all three raw functionals [5,6,7,8]. These summary-databased approaches may be regarded as a quick and dirty dimension reduction strategy that comes up with summarized data with computationally manageable volumes, which would be analyzed by existing methods and software. Our contribution in this paper pertains to a new framework in that tri-axis accelerometer data are used as three-dimensional correlated functional predictors in an association analysis with a potential health outcome such as the Body Mass Index (BMI). We begin with a brief review of existing functional data models, the least-squares kernel machine model, and different variable selection techniques, which prelude the framework for this paper
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