Calculate tuning parameters based on BIC.
tuning_BIC(Y, X, K_mat, lambda)(matrix, n*1) The vector of response variable.
(matrix, n*d_fix) The fixed effect matrix.
(list of matrices) A nested list of kernel term matrices, corresponding to each kernel term specified in the formula for a base kernel function in kern_func_list.
(numeric) A numeric string specifying the range of tuning parameter to be chosen. The lower limit of lambda must be above 0.
(numeric) The estimated tuning parameter.
Bayesian Information Criteria
$$\lambda_{BIC}={argmin}_{\lambda \in \Lambda}\Big\{log\; y^{\star T}(I-A_\lambda)^2y^\star+\frac{log(n)[tr(A_\lambda)+2]}{n}\Big\}$$
Philip S. Boonstra, Bhramar Mukherjee, and Jeremy M. G. Taylor. A Small-Sample Choice of the Tuning Parameter in Ridge Regression. July 2015.
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Hirotogu Akaike. Information Theory and an Extension of the Maximum Likelihood Principle. In Selected Papers of Hirotugu Akaike, Springer Series in Statistics, pages 199<U+2013>213. Springer, New York, NY, 1998.
Clifford M. Hurvich and Chih-Ling Tsai. Regression and time series model selection in small samples. June 1989.
Hurvich Clifford M., Simonoff Jeffrey S., and Tsai Chih-Ling. Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. January 2002.