A Reproducing Kernel Hilbert Space Approach to Functional Linear Regression
Ming Yuan and Tony Cai
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Abstract:
We study in this paper a smoothness regularization method for
functional linear regression and provide a unified treatment for both
the prediction and estimation problems. By
developing a tool on simultaneous diagonalization of two positive
definite kernels, we obtain shaper results on the minimax rates of
convergence and show that smoothness regularized estimators
achieve the optimal rates of convergence for both prediction and
estimation under conditions weaker than those for the
functional principal components based methods developed in the literature.
Despite the generality of the method of regularization, we show that
the procedure is easily implementable. Numerical results are obtained to
illustrate the merits of the method and to demonstrate the theoretical
developments.
- Paper:
pdf file.
- Other related papers:
Cai, T. & Hall, P. (2006).
Prediction in functional linear regression.
The Annals of Statistics 34, 2159-2179.Cai, T. & Zhou, H. (2008).
Adaptive functional linear regression.
Technical report.