Partial least squares for function-on-function regression via Krylov subspaces

People employ the function-on-function regression (FoFR) to model the relationship between two random curves. Fitting this model, widely used strategies include algorithms falling into the framework of functional partial least squares (FPLS, typically requiring iterative eigendecomposition). Here we introduce an FPLS route for FoFR based upon Krylov subspaces. It can be expressed in two forms equivalent to each other (in exact arithmetic): one of them is non-iterative with explicit forms of estimators and predictions, facilitating the theoretical derivation and potential extensions (to more complex modelling); the other one stabilizes numerical outputs. The consistence of estimators and predictions is established with the aid of regularity conditions. Numerical studies illustrate the competitiveness of our proposal in terms of both accuracy and running time.