The effect of Hybrid Principal Components Analysis on the Signal Compression Functional Regression: With EEG-fMRI Application

Objective: In some situations that exist both scalar and functional data, called mixed and hybrid data, the hybrid PCA (HPCA) was introduced. Among the regression models for the hybrid data, we can count covariate-adjusted HPCA, the Semi-functional partial linear regression, function-on-function (FOF) regression with signal compression, and functional additive regression, models. In this article, we study the effects of HPCA decomposition of hybrid data on the prediction accuracy of the FOF regression with signal compressions. Method: We stated a two-step procedure for incorporating the HPCA in the functional regressions. The first step is reconstructing the data based on the HPCAs and the second step is merging data on the other dimensions and calculate the point-wise average of the desired functional dimension. We also choose the number of HPCA based on Mean Squared Perdition Error (MSPE). Result: In the two simulations, we show that the regression models with the first HPCA have the best accuracy prediction and model fit summaries among no HPCA and all HPCAs with a training/testing approach. Finally, we applied our methodology to the EEG-fMRI dataset. Conclusions: We conclude that our methodology improves the prediction of the experiments with the EEG datasets. And we recommend that instead of using the functional PCA on the desired dimension, reconstruct the data with HPCA and average it on the other two dimensions for functional regression models.