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Abstract:
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Interaction terms are common in traditional regression models with scalar predictors and response. In functional regression, interactions have been considered in function-on-scalar or scalar-on-function regression models. But little study has been conducted for function-on-function linear regression with (two-way) interaction terms, where the coefficient functions of interactions are three dimensional. We consider function-on-function regression models with interaction effects. For an interaction model with a given form, we propose an estimation method based on the Karhunen-Loeve expansion of the signal function in response curve. In addition to good predictive property, our approach converts the estimation of three dimensional coefficient functions of interaction effects to the estimation of two and one dimensional functions, which greatly improves the computational efficiency. In practice, the forms of models are usually unspecified. We describe a stepwise selection procedure based on our estimation method and a criterion which measures the predictive performance of a model. We examine our proposed methods via simulation and two data sets in ocean environment and air pollution.
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