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Abstract:
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The function-on-function linear regression model is a useful tool to study the association between functional variables, and smoothness penalty is an efficient regularization method to control the smoothness of coefficient functions. But for densely observed functional data with complex local features, the coefficient function can be spiky and the usual smooth regularization will lead to over-smoothing. We will take a new perspective to explore the spiky functional data. We view the spiky function as a certain order derivative of a smooth function which is an auxiliary variable and propose to estimate the smooth function by imposing smooth regularization. The spiky coefficient is estimated by taking derivative of the smooth auxiliary function. Compared with existing methods via traditional smooth regularization or sparse regularization on wavelet domain, simulation studies and real data applications illustrate that the new method performs similarly well when all functions are smooth, and outperforms the existing methods when the intercept or the slope coefficient function is spiky.
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