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Abstract:
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Modern techniques generate highly densely observed functional data that exhibit complex local variation patterns. To build the scalar-on-function or function-on-function linear regression model for such data, we consider general coefficient functions that can be smooth, nonsmooth, or even discontinuous. The usual smoothness measures, such as the integral of squared derivatives, may not be suitable to differentiate the roughness of these functions. We propose a family of new roughness measures based on the moduli of continuity and wavelet transformation for these general functions. Using these new roughness measures, we propose new regularization and estimation methods for the scalar-on-function and function-on-function linear regression models with highly densely observed functional data. Simulation studies and real data applications illustrate that the new methods have good performance for various coefficient functions and predictor curves. Compared to the smoothness regularization and sparsity regularization, the new regularization is particularly efficient for highly densely spiky functional data.
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