Give the ensemble projection matrix and weights of the kernels in the library using exponential weighting.
ensemble_exp(beta_exp, error_mat, A_hat)(numeric/character) A numeric value specifying the parameter when strategy = "exp". See Details.
(matrix, n*K) A n\*K matrix indicating errors.
(list of length K) A list of projection matrices to kernel space for each kernel in the kernel library.
(matrix, n*n) The ensemble projection matrix.
(vector of length K) A vector of weights of the kernels in the library.
Exponential Weighting
Additionally, another scholar gives a new strategy to calculate weights based on the estimated errors \(\{\hat{\epsilon}_d\}_{d=1}^D\). $$u_d(\beta)=\frac{exp(-\parallel \hat{\epsilon}_d \parallel_2^2/\beta)}{\sum_{d=1}^Dexp(-\parallel \hat{\epsilon}_d \parallel_2^2/\beta)}$$
beta_exp
The value of beta_exp can be "min"=\(min\{RSS\}_{d=1}^D/10\), "med"=\(median\{RSS\}_{d=1}^D\), "max"=\(max\{RSS\}_{d=1}^D*2\) and any other positive numeric number, where \(\{RSS\} _{d=1}^D\) are the set of residual sum of squares of \(D\) base kernels.
Jeremiah Zhe Liu and Brent Coull. Robust Hypothesis Test for Nonlinear Effect with Gaussian Processes. October 2017.
Xiang Zhan, Anna Plantinga, Ni Zhao, and Michael C. Wu. A fast small-sample kernel inde- pendence test for microbiome community-level association analysis. December 2017.
Arnak S. Dalalyan and Alexandre B. Tsybakov. Aggregation by Exponential Weighting and Sharp Oracle Inequalities. In Learning Theory, Lecture Notes in Computer Science, pages 97<U+2013> 111. Springer, Berlin, Heidelberg, June 2007.
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