On prediction rate in partial functional linear regression

We consider a prediction of a scalar variable based on both a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, it is inevitable to impose some kind of regularization. We consider two different approaches, which are shown to achieve the same convergence rate of the mean squared prediction error under respective assumptions. One is based on functional principal components regression (FPCR) and the alternative is functional ridge regression (FRR) based on Tikhonov regularization. Also, numerical studies are carried out for a simulation data and a real data.